Chapter 2: Formal methods of evaluating arguments

We can also connect propositions together using certain English words, such as “and” like this:

This proposition is called a complex proposition because it contains the connective “and” which connects two separate propositions. In contrast, “the floor has been mopped” and “the dishes have been washed” are what are called atomic propositions. Atomic propositions are those that do not contain any truth-functional connectives. The word “and” in this complex proposition is a truth-functional connective. A truth-functional connective is a way of connecting propositions such that the truth value of the resulting complex proposition can be determined by the truth value of the propositions that compose it. Suppose that the floor has not been mopped but the dishes have been washed. In that case, if I assert the conjunction, “the floor has been mopped and the dishes have been washed,” then I have asserted something that is false. The reason is that a conjunction, like the one above, is only true when each conjunct (i.e., each statement that is conjoined by the “and”) is true. If either one of the conjuncts is false, then the whole conjunction is false. This should be pretty obvious. If Bob and Sally split chores and Bob’s chore was to both vacuum and dust whereas Sally’s chore was to both mop and do the dishes, then if Sally said she mopped the floor and did the dishes when in reality she only did the dishes (but did not mop the floor), then Bob could rightly complain that it isn’t true that Sally both mopped the floor and did the dishes! What this shows is that conįunctions are true only if both conįuncts are true. This is true of all conjunctions. The conjunction above has a certain form—the same form as any conjunction. We can represent that form using placeholders— lowercase letters like p and q to stand for any statement whatsoever. Thus, we represent the form of a conjunction like this:

Any conjunction has this same form. For example, the complex proposition, “it is sunny and hot today,” has this same form which we can see by writing the conjunction this way:

Although we could write the conjunction that way, it is more natural in English to conjoin the adjectives “sunny” and “hot” to get “it is sunny and hot today.” Nevertheless, these two sentences mean the same thing (it’s just that one sounds more natural in English than the other). In any case, we can see that “it is sunny today” is the proposition in the “p” place of the form of the conjunction, whereas “it is hot today” is the proposition in the “q” place of the form of the conjunction. As before, this conjunction is true only if both conjuncts are true. For example, suppose that it is a sunny but bitterly cold winter’s day. In that case, while it is true that it is sunny today, it is false that it is hot today—in which case the conjunction is false. If someone were to assert that it is sunny and hot today in those circumstances, you would tell them that isn’t true. Conversely, if it were a cloudy but hot and humid summer’s day, the conjunction would still be false. The only way the statement would be true is if both conjuncts were true.

In the formal language that we are developing in this chapter, we will represent conjunctions using a symbol which looks like this: “∧”. Using this symbol, here is how we will represent a conjunction in symbolic notation:

p ∧ q

In the following sections we will introduce four basic truth-functional connectives, each of which have their own symbol and meaning. The four basic truth-functional connectives are: conjunction, disjunction, negation, and conditional. In the remainder of this section, we will discuss only conjunction.

As we’ve seen, a conjunction conjoins two separate propositions to form a complex proposition. The conjunction is true if and only if both conjuncts are true. We can represent this information using what is called a truth table. Truth tables represent how the truth value of a complex proposition depends on the truth values of the propositions that compose it. Here is the truth table for conjunction:

Here is how to understand this truth table. The header row lists the atomic propositions, p and q, that the conjunction is composed of, as well as the conjunction itself, p ∧ q. Each of the following four rows represents a possible scenario regarding the truth of each conjunct, and there are only four possible scenarios: either p and q could both be true (as in row 1), p and q could both be false (as in row 4), p could be true while q is false (row 2), or p could be false while q is true (row 3). The final column (the truth values under the conjunction, p ⋅ q) represents how the truth value of the conjunction depends on the truth value of each conjunct (p and q). As we have seen, a conjunction is true if and only if both conjuncts are true. This is what the truth table represents. Since there is only one row (one possible scenario) in which both p and q are true (i.e., row 1), that is the only circumstance in which the conjunction is true. Since in every other row at least one of the conjuncts is false, the conjunction is false in the remaining three scenarios.

At this point, some students will start to lose a handle on what we are doing with truth tables. Often, this is because one thinks the concept is much more complicated than it actually is. (For some, this may stem, in part, from a math phobia that is triggered by the use of symbolic notation.) But a truth table is actually a very simple idea: it is simply a representation of the meaning of a truth-functional operator. When I say that a conjunction is true only if both conjuncts are true, that is just what the table is representing. There is nothing more to it than that. (Later on in this chapter we will use truth tables to prove whether an argument is valid or invalid. Understanding that will require more subtlety, but what I have so far introduced is not complicated at all.)

There is more than one way to represent conjunctions in English besides the English word “and.” Below are some common English words and phrases that commonly function as truth-functional conjunctions.

It is important to point out that many times English conjunctions carry more information than simply that the two propositions are true (which is the only information carried by our symbolic connective, the dot). We can see this with English conjunctions like “but” and “however” which have a contrastive sense. If I were to say, “Bob voted, but Caroline didn’t,” then I am contrasting what Bob and Caroline did. Nevertheless, I am still asserting two independent propositions. Another kind of information that English conjunctions represent but the dot connective doesn’t is temporal information. For example, in the conjunction:

There is clearly a temporal implication that Bob brushed his teeth first and then got into bed. It might sound strange to say:

since this would seem to imply that Bob brushed his teeth while in bed. But each of these conjunctions would be represented in the same way by our dot connective, since the dot connective does not care about the temporal aspects of things. If we were to represent “Bob got into bed” with the capital letter A and “Bob brushed his teeth” with the capital letter B, then both of these propositions would be represented exactly the same, namely, like this:

A ∧ B

Sometimes a conjunction can be represented in English with just a comma or semicolon, like this:

Both of these are conjunctions that are represented in the same way.   You should see that both of them have the form, p ∧ q.

Not every conjunction is a truth-function conjunction.   We can see this by considering a proposition like the following:

If this were a truth-functional proposition, then we should be able to identify the two, independent propositions involved. But we cannot. What would those propositions be? You might think two propositions would be these:

But that cannot be right since the fact that Maya is married and that Alice is married is not the same as saying that Maya and Alice are married to each other, which is clearly the implication of the original sentence. Furthermore, if you tried to add “to each other” to each proposition, it would no longer make sense:

Perhaps we could say that the two conjuncts are “Maya is married to Alice” and “Alice is married to Maya,” but the truth values of those two conjuncts are not independent of each other since if Maya is married to Alice it must also be true that Alice is married to Maya. In contrast, the following is an example of a truth- functional conjunction:

Unlike the previous example, in this case we can clearly identify two propositions whose truth values are independent of each other:

Whether or not Maya is a woman is an issue that is totally independent of whether Alice is a woman (and vice versa). That is, the fact that Maya is a woman tells us nothing about whether Alice is a woman. In contrast, the fact that Maya is married to Alice implies that Alice is married to Maya. So the way to determine whether or not a conįunction is truth-functional is to ask whether it is formed from two propositions whose truth is independent of each other. If there are two propositions whose truth is independent of each other, then the conjunction is truth-functional; if there are not two propositions whose truth is independent of each other, the conjunction is not truth-functional.

Exercise 8: Identify which of the following sentences are truth-functional conjunctions. If the sentence is a truth-functional conjunction, identify the two conjuncts (by writing them down).

1.      Jack and Jill are coworkers.

2.      Tom is a fireman and a father.

3.      Ringo Starr and John Lennon were bandmates.

4.      Lucy loves steak and onion sandwiches.

5.      Cameron Dias has had several relationships, although she has never married.

6.      Bob and Sally kissed.

7.      A   person   who   plays   both   mandolin   and   guitar   is   a   multi- instrumentalist.

8.      No one has ever contracted rabies and lived.

9.      Jack and Jill are cowboys.

10.  Josiah is Amish; nevertheless, he is also a drug dealer.

11.  The Tigers are the best baseball team in the state, but they are not as good as the Yankees.

12.  Bob went to the beach to enjoy some rest and relaxation.

13.  Lauren isn’t the fastest runner on the team; still, she is fast enough to have made it to the national championship.

14.  The ring is beautiful, but expensive.

15.  It is sad, but true that many Americans do not know where their next meal will come from.

¹ Some philosophers would claim that a proposition is not the same as a statement, but the reasons for doing so are not relevant to what we’ll be doing in this chapter. Thus, for our purposes, we can treat a proposition as the same thing as a statement.

But another way of adding the negation is with the phrase, “it is not the case that” like this:

Either of these English sentences expresses the same proposition, which is simply the negation of the atomic proposition, “dogs are mammals.” Of course, that proposition is false since it is true that dogs are mammals. Just as we can make a true statement false by negating it, we can also make a false statement true by adding a negation. For example, the statement, “Cincinnati is the capital of Ohio” is false. But we can make that statement true by adding a negation:

There are many different ways of expressing negations in English. Here are a few ways of expressing the previous proposition in different ways in English:

Each of these English sentences express the same true proposition, which is simply the negation of the atomic proposition, “Cincinnati is the capital of Ohio.” Since that statement is false, its negation is true.

There is one respect in which negation differs from the other three truth- functional connectives that we will introduce in this chapter. Unlike the other three, negation does not connect two different propositions. Nonetheless, we call it a truth-functional connective because although it doesn’t actually connect two different propositions, it does change the truth value of propositions in a truth-functional way. That is, if we know the truth value of the proposition we are negating, then we know the truth value of the resulting negated proposition. We can represent this information in the truth table for negation. In the following table, the symbol we will use to represent negation is called the “tilde” (~). (You can find the tilde on the upper left-hand side of your keyboard.)

This truth table represents the meaning of the truth-functional connective, negation, which is represented by the tilde in our symbolic language. The header row of the table represents some proposition p (which could be any proposition) and the negation of that proposition, ~p. What the table says is simply that if a proposition is true, then the negation of that proposition is false (as in the first row of the table); and if a proposition is false, then the negation of that proposition is true (as in the second row of the table).

As we have seen, it is easy to form sentences in our symbolic language using the tilde. All we have to do is add a tilde to left-hand side of an existing sentence. For example, we could represent the statement “Cincinnati is the capital of Ohio” using the capital letter C, which is called a constant. In propositional logic, a constant is a capital letter that represents an atomic proposition. In that case, we could represent the statement “Cincinnati is not the capital of Ohio” like this:

Likewise, we could represent the statement “Toledo is the capital of Ohio” using the constant T. In that case, we could represent the statement “Toledo is not the capital of Ohio” like this:

We could also create a sentence that is a conjunction of these two negations, like this:

Can you figure out what this complex proposition says? (Think about it; you should be able to figure it out given your understanding of the truth-functional connectives, negation and conjunction.) The propositions says (literally): “Cincinnati is not the capital of Ohio and Toledo is not the capital of Ohio.” In later sections we will learn how to form complex propositions using various combinations of each of the four truth-functional connectives. Before we can do that, however, we need to introduce our next truth-functional connective, disjunction.

The English word that most commonly functions as disjunction is the word “or.” It is also common that the “or” is preceded by an “either” earlier in the sentence, like this:

What this sentence asserts is that one or the other (and possibly both) of these individuals tracked mud through the house. Thus, it is composed out of the following two atomic propositions:

If the fact is that Charlie tracked mud through the house, the statement is true. If the fact is that Violet tracked mud through the house, the statement is also true. This statement is only false if in fact neither Charlie nor Violet tracked mud through the house. This statement would also be true even if it was both Charlie and Violet who tracked mud through the house. Another example of a disjunction that has this same pattern can be seen in the “click it or ticket” campaign of the National Highway Traffic Safety Administration. Think about what the slogan means. What the campaign slogan is saying is:

This is a kind of warning: buckle your seatbelt or you’ll get a ticket. Think about the conditions under which this statement would be true. There are only four different scenarios:

The first and second scenarios (rows 1 and 2) are pretty straightforwardly true, according to the “click it or ticket” statement. But suppose that your seatbelt is buckled, is it still possible to get a ticket (as in the third scenario—row 3)? Of course it is! That is, the statement allows that it could both be true that your seatbelt is buckled and true that you get a ticket. How so? (Think about it for a second and you’ll probably realize the answer.) Suppose that your seatbelt is buckled but your are speeding, or your tail light is out, or you are driving under the influence of alcohol. In any of those cases, you would get a ticket even if you were wearing your seatbelt. So the disjunction, click it or ticket, clearly allows the statement to be true even when both of the disjuncts (the statements that form the disjunction) are true. The only way the disjunction would be shown to be false is if (when pulled over) you were not wearing your seatbelt and yet did not get a ticket. Thus, the only way for the disjunction to be false is when both of the disjuncts are false.

These examples reveal a pattern: a disjunction is a truth-functional statement that is true in every instance except where both of the disjuncts are false. In our symbolic language, the symbol we will use to represent a disjunction is called a “wedge” (v). (You can simply use a lowercase “v” to write the wedge.) Here is the truth table for disjunction:

As before, the header of this truth table represents two propositions (first two columns) and their disjunction (last column). The following four rows represent the conditions under which the disjunction is true. As we have seen, the disjunction is true when at least one of its disjuncts is true, including when they are both true (the first three rows). A disjunction is false only if both disjuncts are false (last row).

As we have defined it, the wedge (v) is what is called an “inclusive or.” An inclusive or is a disjunction that is true even when both disjuncts are true. However, sometimes a disjunction clearly implies that the statement is true only if either one or the other of the disjuncts is true, but not both. For example, suppose that you know that Bob placed either first or second in the race because you remember seeing a picture of him in the paper where he was standing on a podium (and you know that only the top two runners in the race get to stand on the podium). Although you can’t remember which place he was, you know that:

This is a disjunction that is built out of two different atomic propositions: 

Although it sounds awkward to write it this way in English, we could simply connect each atomic statement with an “or”:

Bob placed first in the race or Bob placed second in the race.

That sentence makes explicit the fact that this statement is a disjunction of two separate statements. However, it is also clear that in this case the disjunction would not be true if all the disjuncts were true, because it is not possible for all the disjuncts to be true, since Bob cannot have placed both first and second. Thus, it is clear in a case such as this, that the “or” is meant as what is called an “exclusive or.” An exclusive or is a disjunction that is true only if one or the other, but not both, of its disjuncts is true. When you believe the best interpretation of a disjunction is as an exclusive or, there are ways to represent that using a combination of the disjunction, conjunction and negation. The reason we interpret the wedge as an inclusive or rather than an exclusive or is that while we can build an exclusive or out of a combination of an inclusive or and other truth-functional connectives (as I’ve just pointed out), there is no way to build an inclusive or out of the exclusive or and other truth-functional connectives. We will see how to represent an exclusive or in section 2.5.

Exercise 9: Translate the following English sentences into our formal language using conjunction (the dot), negation (the tilde), or disjunction (the wedge). Use the suggested constants to stand for the atomic propositions.

1.      Either Bob will mop or Tom will mop. (B = Bob will mop; T = Tom will mop)

2.      It is not sunny today. (S = it is sunny today)

3.      It is not the case that Bob is a burglar. (B = Bob is a burglar)

4.      Harry is arriving either tonight or tomorrow night.    (A = Harry is arriving tonight; B = Harry is arriving tomorrow night)

5.      Gareth does not like his name. (G = Gareth likes his name)

6.      Either it will not rain on Monday or it will not rain on Tuesday. (M = It will rain on Monday; T = It will rain on Tuesday)

7.      Tom does not like cheesecake. (T = Tom likes cheesecake)

8.      Bob would like to have both a large cat and a small dog as a pet. (C = Bob would like to have a large cat as a pet; D = Bob would like to have a small dog as a pet)

9.      Bob Saget is not actually very funny. (B = Bob Saget is very funny)

10.  Albert Einstein did not believe in God. (A = Albert Einstein believed in God)

The first statement is not an atomic proposition, since it contains a negation, “not.” But the second statement is atomic since it does not contain any truth- functional connectives. So if the first statement is a negation, what is the non- negated, atomic statement? It is this:

I will use the constant C to represent this atomic proposition and G to represent the proposition, “Bob will play video games.” Now that we have identified our two atomic propositions, how can we build our complex sentence using only those atomic propositions and the truth-functional connectives? Let’s start with the statement “Bob will not go to class.” Since we have defined the constant “C” as “Bob will go to class” then we can easily represent the statement “Bob will not go to class” using a negation, like this:

The original sentence asserts that, but it is also asserts that Bob will play video games. That is, it is asserting both of these statements. That means we will be connecting “~C” with “G” with the dot operator. Since we have already assigned “G” to the statement “Bob will play video games,” the resulting translation should look like this:

Although sometimes we can translate sentences into our symbolic language without the use of parentheses (as we did in the previous example), many times a translation will require the use of parentheses. For example:

Notice that whereas the earlier sentence asserted that Bob will not go to class, this sentence does not. Rather, it asserts that Bob will not do both things (i.e., go to class and play video games), but only one or the other (and possibly neither). That is, this sentence does not tell us for sure that Bob will/won’t go to class or that he will/won’t play video games, but only that he won’t do both of these things. Using the same translations as before, how would we translate this sentence? It should be clear that we cannot use the same translation as before since these two sentences are not saying the same thing. Thus, we cannot use the translation:

since that translation says for sure that Bob will not go to class and that he will play video games. Thus, our translation must be different. Here is how to translate the sentence:

I have here introduced some new symbols, the parentheses. Parentheses are using in formal logic to show groupings. In this case, the parentheses represent that the conjunction, “C ⋅ G,” is grouped together and the negation ranges over that whole conįunction rather than just the first conjuct (as was the case with the previous translation). When using multiple operators, you must learn to distinguish which operator is the main operator. The main operator of a sentence is the one that ranges over (influences) the whole sentence. In this case, the main operator is the negation, since it influences the truth value of all the rest of the sentence. In contrast, in the previous example (~C ⋅ G), the main operator was the conjunction rather than the negation since it influences both parts of sentence (i.e., both the “~C” and the “G”). We can see the need for parentheses in distinguishing these two different translations. Without the use of parentheses, we would have no way to distinguish these two sentences, which clearly have different meanings.

Here is a different example where we must utilize parentheses:

Can you tell how many atomic propositions this sentence contains? It contains three atomic propositions which are:

What I’ve written in parentheses to the right of the statement is the constant that I’ll use to represent these atomic statements in my symbolic translation. Notice that the sentence is definitely not asserting that each of these statements is true. Rather, what we have to do is use these atomic propositions to capture the meaning of the original English sentence using only our truth-functional operators. In this sentence we will actually use all three truth-functional operators (disjunction, conjunction, negation). Let’s start with negation, as that one is relatively easy. Given how we have represented the atomic proposition, D, to say that Noelle will not do the dishes is simply the negation of D:

Now consider the first part of the sentence: Noelle will either feed the dogs or clean her room. You should see the “either…or” there and recognize it as a disjunction, which we represent with the wedge, like this:

Now, how are these two compound propositions, “~D” and “F v C” themselves connected? There is one word in the sentence that tips you off—the “but.” As we saw earlier, “but” is a common way of representing a conjunction in English. Thus, we have to conjoin the disjunction (F v C) and the negation (~D). You might think that we could simply conjoin the two propositions like this:

However, that translation would not be correct, because it is not what we call a well-formed formula. A well-formed formula is a sentence in our symbolic language that has exactly one interpretation or meaning. However, the translation we have given is ambiguous between two different meanings. It could mean that (Noelle will feed the dogs) or (Noelle will clean her room and not do the dishes). That statement would be true if Noelle fed the dogs and also did the dishes. We can represent this possibility symbolically, using parentheses like this:

The point of the parentheses is to group the main parts of the sentence together. In this case, we are grouping the “C ⋅ ~D” together and leaving the “F” by itself. The result is that those groupings are connected by a disjunction, which is the main operator of the sentence. In this case, there are only two groupings: “F” on the one hand, and “C ⋅ ~D” on the other hand.

But the original sentence could also mean that (Noelle will feed the dogs or clean her room) and (Noelle will not wash the dishes). In contrast with our earlier interpretation, this interpretation would be false if Noelle fed the dogs and did the dishes, since this interpretation asserts that Noelle will not do the dishes (as part of a conjunction). Here is how we would represent this interpretation symbolically:

Notice that this interpretation, unlike the last one, groups the “F v C” together and leaves the “~D” by itself. These two grouping are then connected by a conjunction, which is the main operator of this complex sentence.

The fact that our initial attempt at the translation (without using parentheses) yielded an ambiguous sentence shows the need for parentheses to disambiguate the different possibilities. Since our formal language aims at eliminating all ambiguity, we must choose one of the two groupings as the translation of our original English sentence. So, which grouping accurately captures the original sentence? It is the second translation that accurately captures the meaning of the original English sentence. That sentence clearly asserts that Noelle will not do the dishes and that is what our second translation says. In contrast, the first translation is a sentence that could be true even if Noelle did do the dishes. Given our understanding of the original English sentence, it should not be true under those circumstances since it clearly asserts that Noelle will not do the dishes.

Let’s move to a different example. Consider the sentence:

This sentence contains four atomic propositions:

As before, I’ve written the constants than I’ll use to stand for each atomic proposition to the right of each atomic proposition. You can use any letter you’d like when coming up with your own translations, as long as each atomic proposition uses a different capital letter. (I typically try to pick letters that are distinctive of each sentence, such as picking “B” for “Bob”.) So how can we use the truth functional operators to connect these atomic propositions together to yield a sentence that captures the meaning of the original English sentence? Clearly B and K are being grouped together with the conjunction “and” and S and T are also being grouped together with the conjunction “and” as well:

Furthermore, the main operator of the sentence is a disjunction, which you should be tipped off to by the phrase “either…or.” Thus, the correct translation of the sentence is:

The main operator of this sentence is the disjunction (the wedge). Again, it is the main operator because it groups together the two main sentence groupings.

Let’s finish this section with one final example. Consider the sentence:

This sentence contains four atomic propositions:

It is clear from the English (because of the “not”) that we need to negate both W and P. It is also clear from the English (because of the “and”) that W and P are grouped together. Thus, the first part of the translation should be:

It is also clear that the last part of the sentence (following the semicolon) is a grouping of V and C and that those two propositions are connected by a disjunction (because of the word “or”):

Finally, these two grouping are connected by a conjunction (because of the “however,” which is a word the often functions as a conjunction). Thus, the correct translation of the sentence is:

As we have seen in this section, translating sentences from English into our symbolic language is a process that can be captured as a series of steps:

• Step 1: Determine what the atomic propositions are.
  
  
• Step 2: Pick a unique constant to stand for each atomic proposition.
  
  
• Step 3: If the sentence contains more than two atomic propositions, determine which atomic propositions are grouped together and which truth-functional operator connects them.
  
  
• Step 4: Determine what the main operator of the sentence is (i.e., which truth functional operator connects the groups of atomic statements together).
  
  
• Step 5: Once your translation is complete, read it back and see if it accurately captures what the original English sentence conveys. If not, see if another way of grouping the parts together better captures what the original sentence conveys.

Try using these steps to create your own translations of the sentences in exercise 10 below.

Exercise 10: Translate the following English sentences into our symbolic language using any of the three truth functional operators (i.e., conjunction, negation, and disjunction). Use the constants at the end of each sentence to represent the atomic propositions they are obviously meant for. After you have translated the sentence, identify which truth- functional connective is the main operator of the sentence. (Note: not every sentence requires parentheses; a sentence requires parentheses only if it contains more than two atomic propositions.)

1.      Bob does not know how to fly an airplane or pilot a ship, but he does know how to ride a motorcycle. (A, S, M)

2.      Tom does not know how to swim or how to ride a horse. (S, H)

3.      Theresa writes poems, not novels. (P, N)

4.      Bob does not like Sally or Felicia, but he does like Alice. (S, F, A)

5.      Cricket is not widely played in the United States, but both football and baseball are. (C, F, B)

6.      Tom and Linda are friends, but Tom and Susan aren’t—although Linda and Susan are. (T, S, L)

7.      Lansing is east of Grand Rapids but west of Detroit. (E, W)

8.      Either Tom or Linda brought David home after his surgery; but it wasn’t Steve. (T, L, S)

9.      Next year, Steve will be living in either Boulder or Flagstaff, but not Phoenix or Denver. (B, F, P, D)

10.  Henry VII of England was married to Anne Boleyn and Jane Seymour, but he only executed Anne Boleyn. (A, J, E)

11.  Henry VII of England executed either Anne Boleyn and Jane Boleyn or Thomas Cromwell and Thomas More. (A, J, C, M)

12.  Children should be seen, but not heard. (S, H)

The main operator of this statement is the conjunction and the left conjunct of the conjunction is a negation. In contrast with the “not both” form, this statement asserts that p is not true, while q is true. For example, using our previous example of Carla and the cake, the sentence

would assert that Carla will not have cake and will have ice cream. This is a very different statement from ~(C ⋅ I) which, as we have seen, allows the possibility that Carla will have cake but not ice cream. Thus, again we see the importance of parentheses in our symbolic language.

Earlier (in section 2.3) we made the distinction between what I called an “exclusive or” and an “inclusive or” and I claimed that although we interpret the wedge (v) as an inclusive or, we can represent the exclusive or symbolically as well. Since we now know how to translate the “not both,” I can show you how to translate a statement that contains an exclusive or. Recall our example:

As we saw, this disjunction contains the two disjuncts, “Bob placed first in the race” (F) and “Bob placed second in the race” (S). Using the wedge, we get:

However, since the wedge is interpreted as an inclusive or, this statement would allow that Bob got both first and second in the race, which is not possible. So we need to be able to say that although Bob placed either first or second, he did not place both first and second. But that is just the “not both” locution. So, to be absolutely clear, we are asserting two things:

and

Now all we have to do is conjoin the two sentences using the dot: 

That is the correct translation of an exclusive or. Notice that when conjoining the “F v S” to the “~(F ⋅ S)” I needed to put parentheses around the “F v S” to show that it was grouped together. Thus, it would have been incorrect to write:

since that is not a well-formed formula. The problem, as before, is that this sentence is ambiguous between two sentences that have different meanings:

While both of these sentences are well-formed, only the latter is the correct translation of the exclusive or.

Let’s move on to the English locution “neither…nor” as in: Carla will eat neither cake nor ice cream.

This statement might be true if, for example, Carla was on a diet (and was sticking to her diet). Using the same method I introduced earlier, we can ask under what conditions the statement would be true or false. As before, there are only four possibilities, which I represent symbolically this time:

There is only one circumstance in which this statement is true and that is the one in which it is false that Carla eats cake and false that Carla eats ice cream. That should be obvious from the meaning of the “neither nor” locution. Thus, the correct translation of a “neither nor” statement is as a conjunction of two negations:

~C ⋅ ~I

The main operator of this statement is the dot, which is conjoining the ~C with the ~I. Thus, the form of any “neither nor” statement can always be translated as a conjunction of two negations:

~p ⋅ ~q

As we will see in a later section (where we will prove it), this statement is also equivalent to a negation of a disjunction:

~(p v q)

Thus, the English locution “neither nor” can also be translated using this statement form.

Exercise 11: For each of the following, write out what atomic proposition each constant stands for. Then translate the sentences using the constants you have defined. Finally, after you have translated the sentence, identify which truth-functional connective is the main operator of the sentence.

1.      Coral is not both a plant and an animal. (P, A)

2.      Although protozoa and chimpanzees are both eukaryotes, they are not both animals. (There are four atomic propositions here; just use A, B, C, and D for each different proposition.)

3.      Neither chimpanzees nor protozoa are prokaryotes. (C, P)

4.      China has not signed the Kyoto Protocol and neither has the United States. (C, U)

5.      Either Chevrolet or McDonald’s will support the Olympic team, but they won’t both support it. (C, M)

6.      Peter Jennings is either a liar or has a really bad memory. (L, M)

7.      Peter Jennings is neither a liar nor has a really bad memory. (L, M)

8.      Peter Jennings is both a liar and has a really bad memory. (L, M)

9.      Peter Jennings is not both a liar and a person with a really bad memory. (L, M)

10.  Chevrolet won’t support the Olympic team this year, and McDonald’s won’t either. (C, M)

11.  Mother Theresa may be a saint. Even so, she has not been canonized yet by the Catholic Church. (S, C)

12.  The best distance runner of the last two decades is either Paul Tergat or Haile Gebrselassie, but it certainly isn’t Jim Ryun. (T, G, R)

13.  Jim Ryun was the best high school miler of all time, but he ran a slower time than Alan Webb. (R, W)

14.  Neither Paul Tergat nor Haile Gebrselassie knows how to play hockey, but they both know how to play soccer. (A, B, C, D)

15.  Ethiopians are neither good bobsledders nor tennis players, but they are excellent distance runners. (B, T, D)

16.  Before Helen Keller met Annie Sullivan, she could neither speak, read, nor communicate. (S, R, C)

17.  Although Helen Keller learned to communicate, she never learned to play soccer or baseball. (C, S, B)

18.  Tom is allowed to play football or soccer, but not both. (F, S)

19.  Tom will major in either engineering and physics, or business and sociology. (E, P, B, S)

20.  Cartman is both xenophobic and racist, but he isn’t a murderer or a thief. (X, R, M, T)

So far, we have learned how to translate certain English sentences into our symbolic language, which consists of a set of constants (i.e., the capital letters that we use to represent different atomic propositions) and the truth-functional connectives. But what is the payoff of doing so? In this section we will learn what the payoff is. In short, the payoff will be that we will have a purely formal method of determining the validity of a certain class of arguments—namely, those arguments whose validity depends on the functioning of the truth- functional connectives. This is what logicians call “propositional logic” or “sentential logic.”

In the first chapter, we learned the informal test of validity, which required us to try to imagine a scenario in which the premises of the argument were true and yet the conclusion false. We saw that if we can imagine such a scenario, then the argument is invalid. On the other hand, if it is not possible to imagine a scenario in which the premises are true and yet the conclusion is false, then the argument is valid. Consider this argument:

1.      The convict escaped either by crawling through the sewage pipes or by hiding out in the back of the delivery truck.

2.      But the convict did not escape by crawling through the sewage pipes.

3.      Therefore, the convict escaped by hiding out in the back of the delivery truck.

Using the informal test of validity, we can see that if we imagine that the first premise and the second premise are true, then the conclusion must follow. However, we can also prove this argument is valid without having to imagine scenarios and ask whether the conclusion would be true in those scenarios. We can do this by a) translating this sentence into our symbolic language and then

b) using a truth table to determine whether the argument is valid. Let’s start with the translation. The first premise contains two atomic propositions. Here are the propositions and the constants that I’ll use to stand for them:

S = The convict escaped through the sewage pipes

D = The convict escaped by hiding out in the back of the delivery van

As we can see, the first premise is a disjunction and so, using the constants indicated above, we can translate that first premise as follows:

S v D

The second premise is simply the negation of S:

~S

Finally, the conclusion is simply the atomic sentence, D. Putting this all together in standard form, we have:

1.      S v D

2.      ~S

3.      ∴ D

We will use the symbol “∴“ to denote a conclusion and will read it “therefore.”

The next thing we have to do is to construct a truth table. We have already seen some examples of truth tables when I defined the truth-functional connectives that I have introduced so far (conjunction, disjunction, and negation). A truth table (as we saw in section 2.2) is simply a device we use to represent how the truth value of a complex proposition depends on the truth of the propositions that compose it in every possible scenario. When constructing a truth table, the first thing to ask is how many atomic propositions need to be represented in the truth table. In this case, the answer is “two,” since there are only two atomic propositions contained in this argument (namely, S and D). Given that there are only two atomic propositions, our truth table will contain only four rows—one row for each possible scenario. There will be one row in which both S and D are true, one row in which both S and D are false, one row in which S is true and D is false, and one row in which S is false and D is true.

The two furthest left columns are what we call the reference columns of the truth table. Reference columns assign every possible arrangement of truth values to the atomic propositions of the argument (in this case, just D and S). The reference columns capture every logically possible scenario. By doing so, we can replace having to use your imagination to imagine different scenarios (as in the informal test of validity) with a mechanical procedure that doesn’t require us to imagine or even think very much at all. Thus, you can think of each row of the truth table as specifying one of the possible scenarios. That is, each row is one of the possible assignments of truth values to the atomic propositions. For example, row 1 of the truth table (the first row after the header row) is a scenario in which it is true that the convict escaped by hiding out in the back of the delivery van, and is also true that the convict escaped by crawling through the sewage pipes. In contrast, row 4 is a scenario in which the convict did neither of these things.

The next thing we need to do is figure out what the truth values of the premises and conclusion are for each row of the truth table. We are able to determine what those truth values are because we understand how the truth value of the compound proposition depends on the truth value of the atomic propositions. Given the meanings of the truth functional connectives (discussed in previous sections), we can fill out our truth table like this:

To determine the truth values for the first premise of the argument (“S v D”) we just have to know the truth values of S and D and the meaning of the truth functional connective, the disjunction. The truth table for the disjunction says that a disjunction is true as long as at least one of its disjuncts is true. Thus, every row under the “S v D” column should be true, except for the last row since on the last row both D and S are false (whereas in the first three rows at least one or the other is true). The truth values for the second premise (~S) are easy to determine: we simply look at what we have assigned to “S” in our reference column and then we negate those truth values—the Ts becomes Fs and the Fs becomes Ts. That is just what I’ve done in the fourth column of the truth table above. Finally, the conclusion in the last column of the truth table will simply

repeat what we have assigned to “D” in our reference column, since the last conclusion simply repeats the atomic proposition “D.”

The above truth table is complete. Now the question is: How do we use this completed truth table to determine whether or not the argument is valid? In order to do so, we must apply what I’ll call the “truth table test of validity.” According to the truth table test of validity, an argument is valid if and only if for every assignment of truth values to the atomic propositions, if the premises are true then the conclusion is true. An argument is invalid if there exists an assignment of truth values to the atomic propositions on which the premises are true and yet the conclusion is false. It is imperative that you understand (and not simply memorize) what these definitions mean. You should see that these definitions of validity and invalidity have a similar structure to the informal definitions of validity and invalidity (discussed in chapter 1). The similarity is that we are looking for the possibility that the premises are true and yet the conclusion is false. If this is possible, then the argument is invalid; if it isn’t possible, then the argument is valid. The difference, as I’ve noted above, is that with the truth table test of validity, we replace having to use your imagination with a mechanical procedure of assigning truth values to atomic propositions and then determining the truth values of the premises and conclusion for each of those assignments.

Applying these definitions to the above truth table, we can see that the argument is valid because there is no assignment of truth values to the atomic propositions (i.e., no row of our truth table) on which all the premises are true and yet the conclusion is false. Look at the first row. Is that a row in which all the premises are true and yet the conclusion false? No, it isn’t, because not all the premises are true in that row. In particular, “~S” is false in that row. Look at the second row. Is that a row in which all the premises are true and yet the conclusion false? No, it isn’t; although both premises are true in that row, the conclusion is also true in that row. Now consider the third row. Is that a row in which all the premises are true and yet the conclusion false? No, because it isn’t a row in which both the premises are true. Finally, consider the last row. Is that a row in which all the premises are true and yet the conclusion false? Again, the answer is “no” because the premises aren’t both true in that row. Thus, we can see that there is no row of the truth table in which the premises are all true and yet the conclusion is false. And that means the argument is valid.

Since the truth table test of validity is a formal method of evaluating an argument’s validity, we can determine whether an argument is valid just in virtue of its form, without even knowing what the argument is about! Here is an example:

1.      (A v B) v C

2.      ~A

3.      ∴ C

Here is an argument written in our symbolic language. I don’t know what A, B, and C mean (i.e., what atomic propositions they stand for), but it doesn’t matter because we can determine whether the argument is valid without having to know what A, B, and C mean. A, B, and C could be any atomic propositions whatsoever. If this argument form is invalid then whatever meaning we give to A, B, and C, the argument will always be invalid. On the other hand, if this argument form is valid, then whatever meaning we give to A, B, and C, the argument will always be valid.

The first thing to recognize about this argument is that there are three atomic propositions, A, B, and C. And that means our truth table will have 8 rows instead of only 4 rows like our last truth table. The reason we need 8 rows is that it takes twice as many rows to represent every logically possible scenario when we are working with three different propositions. Here is a simple formula that you can use to determine how many rows your truth table needs:

2n (where n is the number of atomic propositions)

You read this formula “two to the n-th power.” So if you have one atomic proposition (as in the truth table for negation), your truth table will have only two rows. If you have two atomic propositions, it will have four rows. If you have three atomic propositions, it will have 8 rows. The number of rows needed grows exponentially as the number of atomic propositions grows linearly. The table below represents the same relationship that the above formula does:

So, our truth table for the above argument needs to have 8 rows. Here is how that truth table looks:

Here is an important point to note about setting up a truth table. You need to make sure that your reference columns capture each distinct possible assignment of truth values. One way to make sure you do this is by following the same pattern each time you construct a truth table. There is no one right way of doing this, but here is how I do it (and recommend that you do it too). Construct the reference columns so that the atomic propositions are arranged alphabetically, from left to right. Then on the right-most reference column (the C column above), alternate true and false each row, all the way to the bottom. On the reference column to the left of that (the B column above), alternate two rows true, two rows false, all the way to the bottom. On the next column to the left (the A column above), alternate 4 true, 4 false, all the way to the bottom.

The next step is to determine the truth values of the premises and conclusion. Note that our first premise is a more complex sentence that consists of two

disjunctions. The main operator is the second disjunction since the two main grouping, denoted by the parentheses, are “A v B” and “C”. Notice, however, that we cannot figure out the truth values of the main operator of the sentence until we figure out the truth values of the left disjunct, “A v B.” So that is where we need to start. Thus, in the truth table below, I have filled out the truth values directly underneath the “A v B” part of the sentence by using the truth values I have assigned to A and B in the reference columns. As you can see in the truth table below, each line is true except for the last two lines, which are false, since a disjunction is only false when both of the disjuncts are false. (If you need to review the truth table for disjunction, please see section 2.3.)

Now, since we have figured out the truth values of the left disjunct, we can figure out the truth values under the main operator (which I have emphasized in bold in the truth table below). The two columns you are looking at to determine the truth values of the main operator are the “A v B” column that we have just figured out above and the “C” reference column to the left. It is imperative to understand that the truth values under the “A v B” are irrelevant once we have figured out the truth values under the main operator of the sentence. That column was only a means to an end (the end of determining the main operator) and so I have grayed those out to emphasize that we are no longer paying any attention to them. (When you are constructing your own truth tables, you may even want to erase these subsidiary columns once you’ve determined the truth values of the main operator of the sentence. Or you may simply want to circle the truth values under the main operator to distinguish them from the rest.)

Finally, we will fill out the remaining two columns, which is very straightforward. All we have to do for the “~A” is negate the truth values that we have assigned to our “A” reference column. And all we have to do for the final column “C” is simply repeat verbatim the truth values that we have assigned to our reference column “C.”

The above truth table is now complete. The next step is to apply the truth table test of validity in order to determine whether the argument is valid or invalid. Remember that what we’re looking for is a row in which the premises are true and the conclusion is false. If we find such a row, the argument is invalid. If we do not find such a row, then the argument is valid. Applying this definition to the above truth table, we can see that the argument is invalid because of the 6th row of the table (which I have highlighted). Thus, the explanation of why this argument is invalid is that the sixth row of the table shows a scenario in which the premises are both true and yet the conclusion is false.

Exercise 12: Use the truth table test of validity to determine whether or not the following arguments are valid or invalid.

1.

1.      A v B

2.      B

3.      ∴ ~A

4.

1. (A v B) ⋅ (A v C)

2. ~A

3. ∴ B v C

5.

7.

1.      ~(A ⋅ B)

2.      ∴ ~A v ~B

8.

3.

So far, we have learned how to translate and construct truth tables for three truth functional connectives. However, there is one more truth functional connective that we have not yet learned: the conditional.2 The English phrase that is most often used to express conditional statements is “if…then.” For example,

If it is raining then the ground it wet.

Like   conjunctions   and    disjunctions,   conditionals   connect   two   atomic propositions. There are two atomic propositions in the above conditional:

It is raining.

2 Actually, there is one more truth functional connective that we will not be learning and that is what is called the “biconditional” or “material equivalence.” However, since the biconditional is equivalent to a conjunction of two different conditionals, we don’t actually need it. Although I will discuss material equivalence in section 2.9, we will not be regularly using it.

The ground it wet.

The proposition that follows the “if” is called the antecedent of the conditional and the proposition that follows the “then” is call the consequent of the conditional. The conditional statement above is not asserting either of these atomic propositions. Rather, it is telling us about the relationship between them. Let’s symbolize “it is raining” as “R” and “the ground is wet” as “G.” Thus, our symbolization of the above conditional would be:

R ⊃ G

The “⊃” symbol is called the “horseshoe” and it represents what is called the “material conditional.” A material conditional is defined as being true in every case except when the antecedent is true and the consequent is false. Below is

the truth table for the material conditional. Notice that, as just stated, there is only one scenario in which we count the conditional false: when the antecedent is true and the consequent false.

Let’s see how this applies to the above conditional, “if it is raining, then the ground is wet.” As before, we can think about the meaning of the truth functional connectives by asking whether the sentences containing those connectives would be true or false in the four possible scenarios. The first two are pretty easy. If I assert the above conditional “if it is raining then the ground is wet” when it is both raining and the ground is wet (i.e., the first line of the truth table below), then the conditional statement would be true in that scenario. However, if I assert it and it is raining but the ground isn’t wet (i.e., the second line of the truth table below), then my statement has been shown to be false. Why? Because I’m asserting that any time it is raining, the ground is wet. But if it is raining but the ground isn’t wet, then this scenario is a counterexample to my claim—it shows that my claim is false. Now consider the scenario in which it is not raining but the ground is wet. Would this scenario show that my conditional statement is false? No, it wouldn’t. The reason is that

the conditional statement R ⊃ G is only asserting something about what is the case when it is raining. So this conditional statement isn’t asserting anything

about those scenarios in which it isn’t raining. I’m only saying that when it is raining, the ground is wet. But that doesn’t mean that the ground couldn’t be wet for other reasons (e.g., a sprinkler watering the grass). So the meaning of the material conditional should count a statement true whenever its antecedent is false. Thus, in a scenario in which it is neither raining nor the ground is wet (i.e., the fourth line of the truth table), the conditional statement should still be true. Would the fact of a sunny day and dry ground show that the conditional R

> G is false? Of course not! Thus, as we’ve seen, the material conditional is

false only when the antecedent is true and the consequent is false.

It is sometimes helpful to think of the material conditional as a rule.     For example, suppose that I tell my class:

If you pass all the exams, you will pass the course.

Let’s symbolize “you pass all the exams” as “E” and “you pass the course” as “C.” We would then symbolize the conditional as:

E ⊃ C

Under what conditions would my statement E ⊃ C be shown to be false? There are four possible scenarios:

Suppose that you pass all the exams and pass the class (first row). That would confirm my conditional statement E ⊃ C. Suppose, on the other hand, that although you passed all the exams, you did not pass the class (second row).

This would should my statement is false (and you would have legitimate grounds for complaint!). How about if you don’t pass all the exams and yet you do pass the course (third row)? My statement allows this to be true and it is important to see why. When I assert E ⊃ C I am not asserting anything about the situation in

which E is false. I am simply saying that one way of passing the course is by

passing all of the exams; but that doesn’t mean there aren’t other ways of passing the course. Finally, consider the case in which you do not pass all the exams and you also do not pass the course (fourth row). For the same reason, this scenario is compatible with my statement being true. Thus, again, we see that a material conditional is false in only one circumstance: when the antecedent is true and the consequent is false.

There are other English phrases that are commonly used to express conditional statements. Here are some equivalent ways of expressing the conditional, “if it is raining then the ground is wet”:

It is raining only if the ground is wet The ground is wet if is raining

Only if the ground is wet is it raining

That it is raining implies that the ground is wet That it is raining entails that the ground is wet As long as it is raining, the ground will be wet So long as it is raining, the ground will be wet The ground is wet, provided that it is raining Whenever it is raining, the ground is wet

If it is raining, the ground is wet

All of these conditional statements are symbolized the same way, namely R ⊃ G. The antecedent of a conditional statement always lays down what logicians call a sufficient condition. A sufficient condition is a condition that suffices for some other condition to obtain. To say that x is a sufficient condition for y is to

say that any time x is present, y will thereby be present. For example, a sufficient condition for dying is being decapitated; a sufficient condition for being a U.S. citizen is being born in the U.S. The consequent of a conditional statement always lays down a necessary condition. A necessary condition is a condition that must be in present in order for some other condition to obtain. To say that x is a necessary condition for y is to say that if x were not present, y would not be present either. For example, a necessary condition for being President of the U.S. is being a U.S. citizen; a necessary condition for having a brother is having a sibling. Notice, however, that being a U.S. citizen is not a sufficient condition for being President, and having a sibling is not a sufficient condition for having a brother. Likewise, being born in the U.S. is not a necessary condition for being a U.S. citizen (people can become “naturalized citizens”), and being decapitated is not a necessary condition for dying (one can die without being decapitated).

Exercise 13: Translate the following English sentences into symbolic logic sentences using the constants indicated. Make sure you write out what the atomic propositions are. In some cases this will be straightforward, but not in every case. Remember: atomic propositions never contain any truth functional connectives—and that includes negation! Note: although many of these sentences can be translated using only the horseshoe, others require truth functional connectives other than the horseshoe.

1.      The Tigers will win only if the Indians lose their star pitcher. (T, I)

2.      Tom will pass the class provided that he does all the homework. (P, H)

3.      The car will run only if it has gas. (R, G)

4.      The fact that you are asking me about your grade implies that you care about your grade.                   (A, C)

5.      Although Frog will swim without a bathing suit, Toad will swim only if he is wearing a bathing suit.          (F, T, B)

6.      If Obama isn’t a U.S. citizen, then I’m a monkey’s uncle.   (O, M)

7.      If Toad wears his bathing suit, he doesn’t want Frog to see him in it. (T, F)

8.      If Tom doesn’t pass the exam, then he is either stupid or lazy. (P, S, L)

9.      Bekele will win the race as long as he stays healthy. (W, H)

10.  If Bekele is either sick or injured, he will not win the race. (S, I, W)

11.  Bob will become president only if he runs a good campaign and doesn’t say anything stupid. (P, C, S)

12.  If that plant has three leaves then it is poisonous. (T, P)

13.  The fact that the plant is poisonous implies that it has three leaves. (T, P)

14.  The plant is poisonous only if it has three leaves. (T, P)

15.  The plant has three leaves if it is poisonous. (T, P)

16.  Olga will swim in the open water as long as there is a shark net present. (O, N)

17.  Olga will swim in the open water only if there is shark net. (O, N)

18.  The fact that Olga is swimming implies that she is wearing a bathing suit. (O, B)

19.  If Olga is in Nice, she does not wear a bathing suit. (N, B)

20.  If Terrence pulls Philip’s finger, something bad will happen. (T, B)

The English term “unless” can be tricky to translate. For example, The Reds will win unless their starting pitcher is injured.

If we use the constant “R” to stand for the atomic proposition, “the Reds will win” and “S” to stand for the atomic proposition, “the Reds’ starting pitcher is injured,” how would we translate this sentence using truth functional connectives? Think about what the sentence is saying (think carefully). Is the sentence asserting that the Reds will win? No; it is only saying that

The Reds will win as long as their starting pitcher isn’t injured.

“As long as” denotes a conditional statement. In particular, what follows the “as long as” phrase is a sufficient condition, and as we have seen, a sufficient condition is always the antecedent of a conditional. But notice that the sufficient condition also contains a negation. Thus, the correct translation of this sentence is:

~S ⊃ R

One simple trick you can use to translate sentences which use the term “unless” is just substitute the phrase “if it’s not the case that” for the “unless.” But another trick is just to substitute an “or” for the “unless.” Although it may sound strange in English, a disjunction will always capture the truth functional meaning of “unless.” Thus, we could also correctly translate the sentence like this:

S v R

In the next section we will show how we can prove that these two sentences are equivalent using a truth table.

As we saw in the last section, two different symbolic sentences can translate the same English sentence. In the last section I claimed that “~S ⊃ R” and “S v R” are equivalent. More precisely, they are equivalent ways of capturing the truth- functional relationship between propositions. Two propositions are materially equivalent if and only if they have the same truth value for every assignment of truth values to the atomic propositions. That is, they have the same truth values on every row of a truth table. The truth table below demonstrates that “~S ⊃ R”

and “S v R” are materially equivalent.

If you look at the truth values under the main operators of each sentence, you can see that their truth values are identical on every row. That means the two statements are materially equivalent and can be used interchangeably, as far as propositional logic goes.

Let’s demonstrate material equivalence with another example. We have seen that we can translate “neither nor” statements as a conjunction of two negations. So, a statement of the form, “neither p nor q” can be translated:

~p ⋅ ~q

But another way of translating statements of this form is as a negation of a disjunction, like this:

~(p v q)

We can prove these two statements are materially equivalent with a truth table (below).

Again, as you can see from the truth table, the truth values under the main operators of each sentence are identical on every row (i.e., for every assignment of truth values to the atomic propositions).

In fact, there is a fifth truth functional connective called “material equivalence” or the “biconditional” that is defined as true when the atomic propositions share the same truth value, and false when the truth values different. Although we will not be relying on the biconditional, I provide the truth table for it below. The biconditional is represented using the symbol “≡” which is called a “tribar.”

Some common ways of expressing the biconditional in English are with the phrases “if and only if” and “just in case.” If you have been paying close attention (or do from now on out) you will see me use the phrase “if and only if” often. It is most commonly used when one is giving a definition, such as the definition of validity and also in defining the “material equivalence” in this very section. It makes sense that the biconditional would be used in this way since when we define something we are laying down an equivalent way of saying it.

Exercise 14: Construct a truth table to determine whether the following pairs of statements are materially equivalent.

1.      A ⊃ B and ~A v B

2.      ~(A ⋅ B) and ~A v ~B

3.      A ⊃ B and ~B ⊃ ~A

4.      A v ~B and B ⊃ A

5.      B ⊃ A and A ⊃ B

6.      ~(A ⊃ B) and A ⋅ ~B

7.      A v B and ~A ⋅ ~B

8.      A v (B ⋅ C) and (A v B) ⋅ (A v C)

9.      (A v B) ⋅ C and A v (B ⋅ C)

10.    ~(A v B) and ~A v B

Can you think of a statement that could never be false? How about a statement that could never be true? It is harder than you think, unless you know how to utilize the truth functional operators to construct a tautology or a contradiction. A tautology is a statement that is true in virtue of its form. Thus, we don’t even have to know what the statement means to know that it is true. In contrast, a contradiction is a statement that is false in virtue of its form. Finally, a contingent statement is a statement whose truth depends on the way the world actually is. Thus, it is a statement that could be either true or false—it just depends on what the facts actually are. In contrast, there is an important sense in which the truth of a tautology or the falsity of a contradiction doesn’t depend on how the world is. As philosophers would say, tautologies are true in every

possible world, whereas contradictions are false in every possible world. Consider a statement like:

Matt is either 40 years old or not 40 years old.

That statement is a tautology, and it has a particular form, which can be represented symbolically like this:

p v ~p

In contrast, consider a statement like:

Matt is both 40 years old and not 40 years old.

That statement is a contradiction, and it has a particular form, which can be represented symbolically like this:

p ⋅ ~p

Finally, consider a statement like:

Matt is either 39 years old or 40 years old

That statement is a contingent statement. It doesn’t have to be true (as tautologies do) or false (as contradictions do). Instead, its truth depends on the way the world is. Suppose that Matt is 39 years old. In that case, the statement is true. But suppose he is 37 years old. In that case, the statement is false (since he is neither 39 or 40). We can use truth tables to determine whether a statement is a tautology, contradiction or contingent statement. In a tautology, the truth table will be such that every row of the truth table under the main operator will be true. In a contradiction, the truth table will be such that every row of the truth table under the main operator will be false. And contingent statements will be such that there is mixture of true and false under the main operator of the statement.

The following two truth tables are examples of tautologies and contradictions, respectively.

Notice that in the second truth table, I had to do quite a lot of work before I could figure out what the truth values of the main operator were. I had to first determine the left conjunct (A v B) and then the right conjunct (~A ⋅ ~B), but in

order to figure out the truth values of the right conjunct (which is itself a

conjunct), I had to determine the negations of A and B. Constructing truth tables can sometimes be a chore, but once you understand what you are doing (and why), it certainly isn’t very difficult.

Exercise 15: Construct a truth table to determine whether the following statements are tautologies, contradictions or contingent statements.

1.      A ⊃ (A ⋅ B)

2.      (A ⋅ B) ⊃ (~A ⊃ ~B)

3.      (A ⋅ ~A) ⊃ B

4.      (A ⊃ A) ⊃ (B ⋅ ~B)

5.      (A ⋅ B) ⊃ (A v B)

6.      (A v B) ⊃ (A ⋅ B)

7.      (~A ⊃ ~B) ⊃ (~B ⊃ ~A)

8.      (A ⊃ B) ⊃ (~B ⊃ ~A)

9.      (B v ~B) ⊃ A

10. (A v B) v ~A

Although truth tables are our only formal method of deciding whether an argument is valid or invalid in propositional logic, there is another formal method of proving that an argument is valid: the method of proof. Although you cannot construct a proof to show that an argument is invalid, you can construct proofs to show that an argument is valid. The reason proofs are helpful, is that they allow us to show that certain arguments are valid much more efficiently than do truth tables. For example, consider the following argument:

1.      (R v S) ⊃ (T ⊃ K)

2.      ~K

3.      R v S        /∴ ~T

(Note: in this section I will be writing the conclusion of the argument to the right of the last premise—in this case premise 3. As before, the conclusion we are

trying to derive is denoted by the “therefore” sign, “∴”.) We could attempt to

prove this argument is valid with a truth table, but the truth table would be 16 rows long because there are four different atomic propositions that occur in this argument, R, S, T, and K. If there were 5 or 6 different atomic propositions, the truth table would be 32 or 64 lines long! However, as we will soon see, we could also prove this argument is valid with only two additional lines. That seems a much more efficient way of establishing that this argument is valid. We will do this a little later—after we have introduced the 8 valid forms of inference that you will need in order to do proofs. Each line of the proof will be justified by citing one of these rules, with the last line of the proof being the conclusion that we are trying to ultimately establish. I will introduce the 8 valid forms of inference in groups, starting with the rules that utilize the horseshoe and negation.

The first of the 8 forms of inference is “modus ponens” which is Latin for “way that affirms.” Modus ponens has the following form:

1.      p ⊃ q

2.      p

3.      ∴ q

What this form says, in words, is that if we have asserted a conditional statement (p ⊃ q) and we have also asserted the antecedent of that conditional statement

(p)   , then we are entitled to infer the consequent of that conditional statement

(q). For example, if I asserted the conditional, “if it is raining, then the ground is wet” and I also asserted “it is raining” (the antecedent of that conditional) then I (or anyone else, for that matter) am entitled to assert the consequent of the conditional, “the ground is wet.”

As with any valid forms of inference in this section, we can prove that modus ponens is valid by constructing a truth table. As you see from the truth table below, this argument form passes the truth table test of validity (since there is no row of the truth table on which the premises are all true and yet the conclusion is false).

Thus, any argument that has this same form is valid. For example, the following argument also has this same form (modus ponens):

1.      (A ⋅ B) ⊃ C

2.      (A ⋅ B)

3.      ∴ C

In this argument we can assert C according to the rule, modus ponens. This is so even though the antecedent of the conditional is itself complex (i.e., it is a conjunction). That doesn’t matter. The first premise is still a conditional statement (since the horseshoe is the main operator) and the second premise is the antecedent of that conditional statement. The rule modus ponens says that if we have that much, we are entitled to infer the consequent of the conditional.

We can actually use modus ponens in the first argument of this section:

1.      (R v S) ⊃ (T ⊃ K)

2.      ~K

3.      R v S        /∴ ~T

4.      T ⊃ K       Modus ponens, lines 1, 3

What I have done here is I have written the valid form of inference (or rule) that justifies the line I am deriving, as well as the lines to which that rule applies, to the right of the new line of the proof that I am deriving. Here I have derived “T

> K” from lines 1 and 3 of the argument by modus ponens. Notice that line 1 is

a conditional statement and line 3 is the antecedent of that conditional statement. This proof isn’t finished yet, since we have not yet derived the conclusion we are trying to derive, namely, “~T.” We need a different rule to derive that, which we will introduce next.

The next form of inference is called “modus tollens,” which is Latin for “the way that denies.” Modus tollens has the following form:

1.      p ⊃ q

2.      ~q

3.      ∴ ~p

What this form says, in words, is that if we have asserted a conditional statement (p ⊃ q) and we have also asserted the negated consequent of that conditional (~q), then we are entitled to infer the negated antecedent of that conditional statement (~p). For example, if I asserted the conditional, “if it is raining, then

the ground is wet” and I also asserted “the ground is not wet” (the negated consequent of that conditional) then I am entitled to assert the negated antecedent of the conditional, “it is not raining.” It is important to see that any argument that has this same form is a valid argument. For example, the following argument is also an argument with this same form:

1.      C ⊃ (E v F)

2.      ~(E v F)

3.      ∴ ~C

In this argument we can assert ~C according to the rule, modus tollens. This is so even though the consequent of the conditional is itself complex (i.e., it is a disjunction). That doesn’t matter. The first premise is still a conditional statement (since the horseshoe is the main operator) and the second premise is the negated consequent of that conditional statement. The rule modus tollens says that if we have that much, we are entitled to infer the negated antecedent of the conditional.

We can use modus tollens to complete the proof we started above:

1.      (R v S) ⊃ (T ⊃ K)

2.      ~K

3.      R v S        /∴ ~T

4.      T ⊃ K       Modus ponens, lines 1, 3

5.      ~T           Modus tollens, lines 2, 4

Notice that the last line of the proof is the conclusion that we are supposed to derive and that each statement that I have derived (i.e., lines 4 and 5) has a rule to the right. That rule cited is the rule that justifies the statement that is being derived and the lines cited are the previous lines of the proof where we can see that the rule applies. This is what is called a proof. A proof is a series of statements, starting with the premises and ending with the conclusion, where each additional statement after the premises is derived from some previous line(s) of the proof using one of the valid forms of inference. We will practice this some more in the exercise at the end of this section.

The next form of inference is called “hypothetical syllogism.” This is what ancient philosophers called “the chain argument” and it should be obvious why in a moment. Here is the form of the rule:

1.      p ⊃ q

2.      q ⊃ r

3.      ∴ p ⊃ r

As you can see, the conclusion of this argument links p and r together in a conditional statement. We could continue adding conditionals such as “r ⊃ s” and “s ⊃ t” and the inferences would be just as valid. And if we lined them all

up as I have below, you can see why ancient philosophers referred to this valid argument form as a “chain argument”:

p ⊃ q

q ⊃ r

r ⊃ s

s ⊃ t

∴ p ⊃ t

Notice how the consequent of each preceding conditional statement links up with the antecedent of the next conditional statement in such a way as to create a chain. The chain could be as long as we liked, but the rule that we will cite in our proofs only connects two different conditional statements together. As before, it is important to realize that any argument with this same form is a valid argument. For example,

1.      (A v B) ⊃ ~D

2.      ~D ⊃ C

3.      ∴ (A v B) ⊃ C

Notice that the consequent of the first premise and the antecedent of the second premise are exactly the same term, “~D”. That is what allows us to “link” the antecedent of the first premise and the consequent of the second premise together in a “chain” to infer the conclusion. Being able to recognize the forms of these inferences is an important skill that you will have to become proficient at in order to do proofs.

The next four forms of inference we will introduce utilize conjunction, disjunction and negation in different ways. We will start with the rule called “simplification,” which has the following form:

1.      p ⋅ q

2.      ∴ p

What this rule says, in words, is that if we have asserted a conjunction then we are entitled to infer either one of the conjuncts. This is the rule that I introduced in the first section of this chapter. It is a pretty “obvious” rule—so obvious, in fact, that we might even wonder why we have to state it. However, every form of inference that we will introduce in this section should be obvious—that is the point of calling them basic forms of inference. They are some of the simplest forms of inference, whose validity should be transparently obvious. The idea of a proof is that although the inference being made in the argument is not obvious, we can break that inference down in steps, each of which is obvious. Thus, the obvious inferences ultimately justify the non-obvious inference being made in the argument. Those obvious inferences thus function as rules that we use to justify each step of the proof. Simplification is a prime example of one of the more obvious rules.

As before, it is important to realize that any inference that has the same form as simplification is a valid inference. For example,

1.      (A v B) ⋅ ~(C ⋅ D)

2.      ∴ (A v B)

is a valid inference because it has the same form as simplification. That is, line 1 is a conjunction (since the dot is the main operator of the sentence) and line 2 is inferring one of the conjuncts of that conjunction in line 1. (Just think of the “A v B” as the “p” and the “~(C ⋅ D)” as the “q”.)

The next rule we will introduce is called “conjunction” and is like the reverse of simplification. (Don’t confuse the rule called conjunction with the type of complex proposition called a conjunction.) Conjunction has the following form:

1.      p

2.      q

3.      ∴ p ⋅ q

What this rule says, in words, is that if you have asserted two different propositions, then you are entitled to assert the conjunction of those two propositions. As before, it is important to realize that any inference that has the same form as conjunction is a valid inference. For example,

1.      A ⊃ B

2.      C v D

3.      ∴ (A ⊃ B) ⋅ (C v D)

is a valid inference because it has the same form as conjunction. We are simply conjoining two propositions together; it doesn’t matter whether those propositions are atomic or complex. In this case, of course, the propositions we are conjoining together are complex, but as long as those propositions have already been asserted as premises in the argument (or derived by some other valid form of inference), we can conjoin them together into a conjunction.

The next form of inference we will introduce is called “disjunctive syllogism” and it has the following form:

1.      p v q

2.      ~p

3.      ∴ q

In words, this rule states that if we have asserted a disjunction and we have asserted the negation of one of the disjuncts, then we are entitled to assert the other disjunct. Once you think about it, this inference should be pretty obvious. If we are taking for granted the truth of the premises—that either p or q is true; and that p is not true—then is has to follow that q is true in order for the original disjunction to be true. (Remember that we must assume the premises are true when evaluating whether an argument is valid.) If I assert that it is true that either Bob or Linda stole the diamond, and I assert that Bob did not steal the diamond, then it has to follow that Linda did. That is a disjunctive syllogism. As before, any argument that has this same form is a valid argument. For example,

1.      ~A v (B ⋅ C)

2.      ~~A

3.      ∴ B ⋅ C

is a valid inference because it has the same form as disjunctive syllogism. The first premise is a disjunction (since the wedge is the main operator), the second premise is simply the negation of the left disjunct, “~A”, and the conclusion is the right disjunct of the original disjunction. It may help you to see the form of

the argument if you treat “~A” as the p and “B ⋅ C” as the q. Also notice that

the second premise contains a double negation. Your English teacher may tell you never to use double negatives, but as far as logic is concerned, there is absolutely nothing wrong with a double negation. In this case, our left disjunct in premise 1 is itself a negation, while premise 2 is simply a negation of that left disjunct.

The next rule we’ll introduce is called “addition.” It is not quite as “obvious” a rule as the ones we’ve introduced above. However, once you understand the conditions under which a disjunction is true, then it should be obvious why this form of inference is valid. Addition has the following form:

1.      p

2.      ∴ p v q

What this rule says, in words, is that that if we have asserted some proposition, p, then we are entitled to assert the disjunction of that proposition p and any

other proposition q we wish. Here’s the simple justification of the rule. If we know that p is true, and a disjunction is true if at least one of the disjuncts is true, then we know that the disjunction p v q is true even if we don’t know whether q is true or false. Why? Because it doesn’t matter whether q is true or false, since we already know that p is true. The hardest thing to understand about this rule is why we would ever want to use it. The best answer I can give you for that right now is that it can help us out when doing proofs.3

As before, is it important to realize that any argument that has this same form, is a valid argument. For example,

1.      A v B

2.      ∴ (A v B) v (~C v D)

is a valid inference because it has the same form as addition. The first premise asserts a statement (which in this case is complex—a disjunction) and the conclusion is a disjunction of that statement and some other statement. In this case, that other statement is itself complex (a disjunction). But an argument or inference can have the same form, regardless of whether the components of those sentences are atomic or complex. That is the important lesson that I have been trying to drill in in this section.

The final of our 8 valid forms of inference is called “constructive dilemma” and is the most complicated of them all. It may be most helpful to introduce it using an example. Suppose I reasoned thus:

The killer is either in the attic or the basement. If the killer is in the attic then he is above me. If the killer is in the basement then his is below me. Therefore, the killer is either                                                             or                            .

Can you fill in the blanks with the phrases that would make this argument valid? I’m guessing that you can. It should be pretty obvious. The conclusion of the argument is the following:

3 A better answer is that we need this rule in order to make this set of rules that I am presenting a sound a complete set of rules. That is, without it there would be arguments that are valid but that we aren’t able to show are valid using this set of rules. In more advanced areas of logic, such as metalogic, logicians attempt to prove things about a particular system of logic, such as proving that the system is sound and complete.

The killer is either above me or below me.

That this argument is valid should be obvious (can you imagine a scenario where all the premises are true and yet the conclusion is false?). What might not be as obvious is the form that this argument has. However, you should be able to identify that form if you utilize the tools that you have learned so far. The first premise is a disjunction. The second premise is a conditional statement whose antecedent is the left disjunct of the disjunction in the first premise. And the third premise is a conditional statement whose antecedent is the right disjunct of the disjunction in the first premise. The conclusion is the disjunction of the consequents of the conditionals in premises 2 and 3. Here is this form of inference using symbols:

1.      p v q

2.      p ⊃ r

3.      q ⊃ s

4.      ∴ r v s

We have now introduced each of the 8 forms of inference. In the next section I will walk you through some basic proofs that utilize these 8 rules.

Exercise 16: Fill in the blanks with the valid form of inference that is being used and the lines the inference follows from. Note: the conclusion is written to the right of the last premise, following the “/∴“ symbols.

Example 1:

1.      M ⊃ ~N

2.      M

3.      H ⊃ N            /∴ ~H

4.      ~N      Modus ponens, 1, 2

5.      ~H      Modus tollens, 3, 4

Example 2:

1.      A v B

2.      C ⊃ D

3.      A ⊃ C

4.      ~D                /∴ B

5.      A ⊃ D            Hypothetical syllogism, 3, 2

6.      ~A                Modus tollens, 5, 4

7.      B                  Disjunctive syllogism, 1, 6

#4

1.      A ⊃ B

2.      A ⋅ ~D

3.      B ⊃ C       /∴ C ⋅ ~D

4.      A                                

5.      A ⊃ C                         

6.      C                                

7.      ~D                              

8.      C ⋅ ~D                        

#5

1.      C

2.      A ⊃ B

3.      C ⊃ D

#8

1.      C ⊃ B

2.      ~D ⋅ ~B

3.      (A ⊃ (B ⊃ C)) v D

4.      A v C             /∴ B ⊃ C

5.      ~D                             

6.      A ⊃ (B ⊃ C)                   

7.      ~B                              

8.      ~C                               

9.      A                                

10.   B ⊃ C                           

11.   (B ⊃ C) v B                           

You can think of constructing proofs as a game. The goal of the game is to derive the conclusion from the given premises using only the 8 valid rules of inference that we have introduced. Not every proof requires you to use every rule, of course. But you may use any of the rules—as along as your use of the rule is correct. Like most games, people can be better or worse at the “game” of constructing proofs. Better players will be able to a) make fewer mistakes, b) construct the proofs more quickly, and c) construct the proofs more efficiently. In order to construct proofs, it is imperative that you internalize the 8 valid forms of inference introduced in the previous section. You will be citing these forms of inference as rules that will justify each new line of your proof that you add. By “internalize” I mean that you have memorized them so well that you can see those forms manifest in various sentences almost without even thinking about it. If you internalize the rules in this way, constructing proofs will be a pleasant diversion, rather than a frustrating activity. In addition to internalizing the 8 valid forms of inference, there are a couple of different strategies that can help when you’re stuck and can’t figure out what to do next. The first is the strategy of working backwards. When we work backwards in a proof, we ask ourselves what rule we can use to derive the sentence(s) we need to derive. Here is an example:

1.      R ⋅ S

2.      T             /∴ (T v L) ⋅ (R ⋅ S)

The conclusion, which is to the right of the second premise and follows the “/∴” symbol, is a conjunction (since the dot is the main operator). If we are trying to

“work backwards,” the relevant question to ask is: What rule can we use to derive a conjunction? If you know the rules, you should know the answer to that question. There is only one rule that allows us to derive (infer) a sentence that is a conjunction. That rule is called “conjunction.” The form of the rule conjunction says that in order to derive a conjunction, we need to have each conjunct on a separate line. So, what are the two conjuncts that we would need in order to derive the conjunction that is the conclusion (i.e., “(T v L) ⋅ (R ⋅ S)”).

We would need both “T v L” on a line and “R ⋅ S” on a separate line. But look

at premise 1—we already have “R ⋅ S” on its own line! So the only other thing we need to derive is the sentence “T v L”. Once we have that on a separate

line, then we can use the rule conjunction to conjoin those two sentences to get the conclusion! So the next question we have to ask is: How can I derive the sentence “T v L”? Again, if we are working backwards, the relevant question to ask here is: What rule allows me to derive a disjunction? There are only two: constructive dilemma and addition. However, we know that we won’t be using constructive dilemma since none of the premises are conditional statements, and constructive dilemma requires conditional statements as premises. That leaves addition. Addition allows us to disjoin any statement we like to an existing statement. Since we have “T” as the second premise, the rule addition allows us to disjoin “L” to that statement. The first new line of the proof should thus look like this:

3.      T v L        Addition 2

What I have done is number a new line of the proof (continuing the numbering from the premises) and then have written the rule that justifies that new line as well as the line(s) from which that line was derived via that rule. In this case, since addition is a rule that allows you to derive a sentence directly from just one line, I have cited only one line. The next step of the proof should be clear since we have already talked through it above. All we have to do now is go directly to the conclusion, since the conclusion is a conjunction and we now have (on separate lines of the proof) each conjunct. Thus, the final line of this (quite simple) proof should look like this:

4.      (T v L) ⋅ (R ⋅ S)   Conjunction 1, 3

Again, all I’ve done is the write the new line of the proof (continuing the numbering from the previous line) and then have written the rule that justifies that new line as well as the line(s) from which that line was derived via that rule. In this case, the rule conjunction requires that we cite two lines (i.e., each conjunct that we are conjoining). So, I have to find the lines that contained “T v L” and “R ⋅ S” and cite those lines. It does not matter the order in which you

cite the lines as along as you have cited the correct lines (e.g., I could have

equally well have written, “Conjunction 3, 1” as the justification). Thus the complete proof should look like this:

1.      R ⋅ S

2.      T             /∴ (T v L) ⋅ (R ⋅ S)

3.      T v L                 Addition 2

4.      (T v L) ⋅ (R ⋅ S)   Conjunction 1, 3

That’s it. That is all there is to constructing a proof. The last line of the proof is the conclusion to be derived: check. Each line of the proof follows by the rule and the line(s) cited: check. Since both of those requirements check out, our proof is complete and correct.

I have just walked you through a simple proof using the strategy of working backwards. This strategy works well as long as the conclusion we are trying to derive is complex—that is, if it contains truth functional connectives. However, sometimes our conclusion will simply be an atomic statement. In that case, we will not as easily be able to utilize the strategy of working backwards. But there is another strategy that we can utilize: the strategy of working forward. To utilize the strategy of working forward, we simply ask ourselves what rules we can apply to the existing premises to derive something, even if it isn’t the conclusion we are ultimately trying to derive. As a part of this strategy, we should typically break apart a conjunction whenever we have one as a premise of our argument. Doing this can help to see where to go next. (If you’ve ever played Scrabble, then you can think of this as rearranging your Scrabble tiles in order to see what words you can build.) Here is an example of a proof where we should utilize the strategy of working forward:

1.      A ⋅ B

2.      B ⊃ C       /∴ C

Notice that since the conclusion is atomic, we cannot utilize the strategy of working backwards. Instead, we should try working forward. As part of this strategy, we should break apart conjunctions by using the rule “simplification.” That will be the first step of our proof:

1.      A ⋅ B

2.      B ⊃ C       /∴ C

3.      A             Simplification 1

4.      B             Simplification 1

The first two lines of the proof is just breaking down the conjunction in line 1, where line 3 is just the left conjunct and line 4 is just the right conjunct. Both lines 3 and 4 follow by the same rule and the same line, in this case. The next question we ask when utilizing the strategy of working forward is: what lines of

the proof we can apply some rule to in order to derive something or other? Look at the conditional on line 2. We haven’t used that yet. So what rule can we apply to that line? You should be thinking of the rules that utilize conditional statements (modus ponens, modus tollens, and hypothetical syllogism). We can rule out hypothetical syllogism since here we have only one conditional and the rule hypothetical syllogism requires that we have two. If you look at line 4 (that we have just derived) you should see that it is the antecedent of the conditional statement on line 2. And you should know that that means we can apply the rule, modus ponens. So our next step is to do that:

1.      A ⋅ B

2.      B ⊃ C       /∴ C

3.      A             Simplification 1

4.      B             Simplification 1

5.      C             Modus ponens 2, 4

But now also notice that the line that we have just derived is in fact the conclusion of the argument. So our proof is finished.

Before the close of this section, let’s work through a bit longer proof. Remember: any proof, long or short, is the same process and utilizes the same strategy. It is just a matter a keeping track of where you are in the proof and what you’re ultimately trying to derive. So here is a bit more complex proof:

1.      (~A v B) ⊃ L

2.      ~B

3.      A ⊃ B

4.      L ⊃ (~R v D)

5.      ~D ⋅ (R v F)       /∴ (L v G) ⋅ ~R

The conclusion is a conjunction of “L v G” and “~R” so we know that if we can get each of those sentences on a separate line, then we can use the rule conjunction to derive the conclusion. That will be our long range goal here (and this is utilizing the strategy of working backwards). However, we cannot see how to directly get there from here at this point, so we will begin utilizing the strategy of working forward. The first thing we’ll do is simplify the conjunction on line 5:

6.      ~D           Simplification 5

7.      R v F        Simplification 5

Look at lines 2 and 6: they are both negated atomic propositions. Another part of the strategy of working forward is to utilize either atomic or negated atomic sentences. We should look for how we can utilize modus tollens or disjunctive syllogism by plugging these negated atomic sentences into other lines of the proof. Look at lines 2 and 3. You should see a modus tollens there. That will be our next step:

8.      ~A           Modus tollens 2, 3

The next step of this proof can be a bit tricky. There are a couple different ways we could go. One would be to utilize the rule “addition.” Can you see how we might helpfully utilize this rule using either line 6 or 8? If not, I’ll give you a hint: what if we were to use addition on line 8 to derive “~A v B”? Can you see how we could then plug that into line 1? In fact, “~A v B” is the antecedent of the conditional in line 1, so we could then use modus ponens to derive the consequent. Thus, let’s try starting with the addition on line 8:

9.      ~A v B     Addition 8

Next, we’ll utilize line 9 and line 1 with modus ponens to derive the next line:

10.    L            Modus ponens 1, 9

Notice at this point that what we have derived on line 10 is “L” and what we earlier said we needed as one of the conjuncts was “L v G”. You should recognize that we have a rule that will allow us to infer directly from “L” to “L v G”. That rule is addition (again). That will be the next line of the proof:

11.    L v G       Addition 10

At this point, our strategy should be to try to derive the other conjunct, “~R”. Notice that “~R” is contained within the sentence on line 4, but it is embedded. How can we “get it free”? Start by noticing that the ~R is a part of a disjunction, which is itself a consequent of a conditional statement. Also notice that we have already derived the antecedent of that conditional statement, which means that we can use modus ponens to derive the consequent:

12.    ~R v D    Modus ponens 4, 10

The penultimate step is to use a disjunctive syllogism to derive “~R”.

13.    ~R          Disjunctive syllogism 6, 12

The final step is simply to conjoin lines 11 and 13 to get the conclusion:

14.    (L v G) ⋅ ~R       Conjunction 11, 13 Thus, here is the completed proof:

1.      (~A v B) ⊃ L

2.      ~B

3.      A ⊃ B

4.      L ⊃ (~R v D)

5.      ~D ⋅ (R v F)       /∴ (L v G) ⋅ ~R

6.      ~D           Simplification 5

7.      R v F        Simplification 5

8.      ~A           Modus tollens 2, 3

9.      ~A v B     Addition 8

10.    L            Modus ponens 1, 9

11.    L v G       Addition 10

12.    ~R v D    Modus ponens 4, 10

13.    ~R          Disjunctive syllogism

14. (L v G) ⋅ ~R   Conjunction 11, 13

Constructing proofs is a skill that takes practice.     The following exercises will give you some practice with constructing proofs.

Exercise 17: Construct proofs for the following valid arguments. The first fifteen proofs can be complete in three or less additional lines. The next five proofs will be a bit longer. It is important to note that there is always more than one way to construct a proof. If your proof differs from the answer key, that doesn’t mean it is wrong.

#1

1.      A ⋅ B

2.      (A v C) ⊃ D      /∴ A ⋅ D

#2

1.      A

2.      B        /∴ (A v C) ⋅ B

#3

1.      D ⊃ E

2.      D ⋅ F            /∴ E

#4

1.      J ⊃ K

2.      J        /∴ K v L

#5

1.      A v B

2.      ~A ⋅ ~C        /∴ B

#6

1.      A ⊃ B

2.      ~B ⋅ ~C         /∴ ~A

#7

2.  ~A ⋅ C          /∴ ~B

#12

1.      (D v E) ⊃ (F ⋅ G)

2.      D                  /∴ F

#13

1.      T ⊃ U

2.      V v ~U

3.      ~V ⋅ ~W       /∴ ~T

#14

1.      (A v B) ⊃ ~C

2.      C v D

3.      A                  /∴ D

#15

1.      L v (M ⊃ N)

2.      ~L ⊃ (N ⊃ O)

3.      ~L                /∴ M ⊃ O

#20

1.      J ⊃ K

2.      K v L

3.      (L ⋅ ~J) ⊃ (M ⋅ ~J)

4.      ~K                /∴ M

So far in this chapter we have learned a formal method for determining whether a certain class of arguments (i.e., those that utilize only truth functional operators) are valid or invalid. That method is the truth table test of validity. We have also learned a formal method for proving arguments are valid or invalid (the method of proof). The other important skill we have learned in this chapter so far is translating sentences into propositional logic. Thus, there are three different skills that you should know how to do:

1.      Translate sentences from English into propositional logic

2.      Construct truth tables in order to determine whether an argument is valid or invalid

3.      Construct proofs to prove an argument is valid

It is important to reiterate that truth tables are the only formal method that allow us to determine whether an argument is valid or invalid; proofs can only show that an argument is valid, but not that it is invalid. You might think that you can use proofs to show that an argument is invalid—for example, if you are unable to construct a proof for an argument, that means that the argument is invalid. However, this doesn’t follow. There could be many reasons why you are unable to construct a proof, including that you just aren’t skilled enough to construct proofs. But the fact that you aren’t skilled enough to find a proof for an argument wouldn’t mean that the argument is invalid, it would just mean that you weren’t skilled enough to show that it is valid! So we cannot use one’s inability to construct a proof for an argument to establish that the argument is invalid. Again, only the truth table test of validity can establish that an argument is invalid.

The study of propositional logic has given us a way of understanding what “formal” means in the phrase, “formal logic.” We can see this clearly with the truth table test of validity. After we translate an argument into propositional

logic using constants and the truth functional connectives, we don’t need to know what the constants mean in order to know whether the argument is valid or invalid. We simply have to fill out the truth table in the mechanical way we have learned and then apply the truth table test of validity (which is also a mechanical procedure). Thus, once an argument has been translated into propositional logic, determining whether an argument passes the truth table test of validity is something a computer could easily do. The translation from English to symbolic format is not as easy for a computer to do because successfully doing so depends on understanding the nuances of English. Although today there are computer programs that are pretty good at doing this, it has taken many years to get there. In contrast, any simple computer program from half a century ago could easily construct and evaluate a truth table using the truth table test of validity because this doesn’t take any understanding—it is simply a mechanical procedure. There are many different programs, many of which are readily available on the web, that allow you to construct and evaluate truth tables.

In contrast, the informal test of validity (from chapter 1) requires that we understand the meaning of the statements involved in the argument in order for us to be try to imagine the premises as true and the conclusion as false. Since this test requires the use of our imagination, it clearly also requires that we understand the meanings of the statements in the argument. The truth table test of validity does not require any of this. Since the truth table method does not require understanding of the meaning of the statements involved in the argument, but only an awareness of their logical form, we refer to it as a formal logic. Formal logic is a kind of logic that looks only at the form, rather than the content (meaning) of the statements. We can easily see this by constructing an argument where the atomic propositions use silly, made-up words, such as those from Lewis Carroll’s “Jabberwocky”:

1.      If toves are slithy, then the borogoves are mimsy

2.      Borogoves are not mimsy

3.      Therefore, toves are not slithy

If we translate “toves are slithy” at “T” and “borogoves are mimsy” as “B” then the form of this argument is clearly modus tollens, which is one of the 8 valid forms of inference:

1.      T ⊃ B

2.      ~B

3.      ∴ ~T

We can thus see that this argument is valid even though we have no idea what “toves” or “borogoves” are or what “slithy” and “mimsy” mean. Thus, propositional logic, which includes the truth table test of validity, is a kind of formal logic, whereas the informal test of validity is not. There are other kinds of formal logic besides propositional logic.