2.2: Propositional Logic and the Four Basic Truth Functional Connectives
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Propositional logic (also called “sentential logic”) is the area of formal logic that deals with the logical relationships between propositions. A proposition is simply what I called in section 1.1 a statement. 1 Some examples of propositions are:
Snow is white
Snow is cold
Tom is an astronaut
The floor has been mopped
The dishes have been washed
We can also connect propositions together using certain English words, such as “and” like this:
The floor has been mopped and the dishes have been washed.
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 conjunctions are true only if both conjuncts 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:
p and q
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:
It is sunny today and it is hot today .
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 called the “dot,” 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:
| p | q | p ⋅ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
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.
| but | yet | also | although |
| however | moreover | nevertheless | still |
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:
Bob brushed his teeth and got into bed
There is clearly a temporal implication that Bob brushed his teeth first and then got into bed. It might sound strange to say:
Bob got into bed and brushed his teeth
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:
While Bob vacuumed the floor, Sally washed the dishes.
Bob vacuumed the floor; Sally washed the dishes.
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:
Maya and Alice are married.
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:
Maya is married
Alice is married
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:
Maya is married to each other
Alice is married to each other
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:
Maya and Alice are women.
Unlike the previous example, in this case we can clearly identify two propositions whose truth values are independent of each other:
Maya is a woman
Alice is a woman
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 conjunction 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
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.