2.1: The Conditional
As we noted in chapter 1, there are sentences of a natural language, like English, that are not atomic sentences. Our examples included
If Lincoln wins the election, then Lincoln will be President.
The Earth is not the center of the universe.
We could treat these like atomic sentences, but then we would lose a great deal of important information. For example, the first sentence tells us something about the relationship between the atomic sentences “Lincoln wins the election” and “Lincoln will be President”. And the second sentence above will, one supposes, have an interesting relationship to the sentence, “The Earth is the center of the universe”. To make these relations explicit, we will have to understand what “if…then…” and “not” mean. Thus, it would be useful if our logical language was able to express these kinds of sentences in a way that made these elements explicit. Let us start with the first one.
The sentence, “If Lincoln wins the election, then Lincoln will be President” contains two atomic sentences, “Lincoln wins the election” and “Lincoln will be President”. We could thus represent this sentence by letting
Lincoln wins the election
be represented in our logical language by
P
And by letting
Lincoln will be president
be represented by
Q
Then, the whole expression could be represented by writing
If P then Q
It will be useful, however, to replace the English phrase “if…then…” by a single symbol in our language. The most commonly used such symbol is “→”. Thus, we would write
P→Q
One last thing needs to be observed, however. We might want to combine this complex sentence with other sentences. In that case, we need a way to identify that this is a single sentence when it is combined with other sentences. There are several ways to do this, but the most familiar (although not the most elegant) is to use parentheses. Thus, we will write our expression
(P→Q)
This kind of sentence is called a “conditional”. It is also sometimes called a “material conditional”. The first constituent sentence (the one before the arrow, which in this example is “P”) is called the “antecedent”. The second sentence (the one after the arrow, which in this example is “Q”) is called the “consequent”.
We know how to write the conditional, but what does it mean? As before, we will take the meaning to be given by the truth conditions—that is, a description of when the sentence is either true or false. We do this with a truth table. But now, our sentence has two parts that are atomic sentences, P and Q. Note that either atomic sentence could be true or false. That means, we have to consider four possible kinds of situations. We must consider when P is true and when it is false, but then we need to consider those two kinds of situations twice: once for when Q is true and once for when Q is false. Thus, the left hand side of our truth table will look like this:
P | Q | |
T | T | |
T | F | |
F | T | |
F | F |
There are four kinds of ways the world could be that we must consider.
Note that, since there are two possible truth values (true and false), whenever we consider another atomic sentence, there are twice as many ways the world could be that we should consider. Thus, for n atomic sentences, our truth table must have 2n rows. In the case of a conditional formed out of two atomic sentences, like our example of (P→Q), our truth table will have 22 rows, which is 4 rows. We see this is the case above.
Now, we must decide upon what the conditional means. To some degree this is up to us. What matters is that once we define the semantics of the conditional, we stick to our definition. But we want to capture as much of the meaning of the English “if…then…” as we can, while remaining absolutely precise in our language.
Let us consider each kind of way the world could be. For the first row of the truth table, we have that P is true and Q is true. Suppose the world is such that Lincoln wins the election, and also Lincoln will be President. Then, would I have spoken truly if I said, “If Lincoln wins the election, then Lincoln will be President”? Most people agree that I would have. Similarly, suppose that Lincoln wins the election, but Lincoln will not be President. Would the sentence “If Lincoln wins the election, then Lincoln will be President” still be true? Most agree that it would be false now. So the first rows of our truth table are uncontroversial.
P | Q | (P→Q) |
---|---|---|
T | T | T |
T | F | F |
F | T | |
F | F |
Some students, however, find it hard to determine what truth values should go in the next two rows. Note now that our principle of bivalence requires us to fill in these rows. We cannot leave them blank. If we did, we would be saying that sometimes a conditional can have no truth value; that is, we would be saying that sometimes, some sentences have no truth value. But our principle of bivalence requires that—in all kinds of situations—every sentence is either true or false, never both, never neither. So, if we are going to respect the principle of bivalence, then we have to put either T or F in for each of the last two rows.
It is helpful at this point to change our example. Let us consider two different examples to illustrate how best to fill out the remainder of the truth table for the conditional.
First, suppose I say the following to you: “If you give me $50, then I will buy you a ticket to the concert tonight.” Let
You give me $50
be represented in our logic by
R
and let
I will buy you a ticket to the concert tonight.
be represented by
S
Our sentence then is
(R→S)
And its truth table—as far as we understand right now—is:
R | S | (R→S) |
---|---|---|
T | T | T |
T | F | F |
F | T | |
F | F |
That is, if you give me the money and I buy you the ticket, my claim that “If you give me $50, then I will buy you a ticket to the concert tonight” is true. And, if you give me the money and I don’t buy you the ticket, I lied, and my claim is false. But now, suppose you do not give me $50, but I buy you a ticket for the concert as a gift. Was my claim false? No. I simply bought you the ticket as a gift, but, presumably would have bought it if you gave me the money, also. Similarly, if you don’t give me money, and I do not buy you a ticket, that seems perfectly consistent with my claim.
So, the best way to fill out the truth table is as follows.
R | S | (R→S) |
---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Second, consider another sentence, which has the advantage that it is very clear with respect to these last two rows. Assume that a is a particular natural number, only you and I don’t know what number it is (the natural numbers are the whole positive numbers: 1, 2, 3, 4…). Consider now the following sentence.
If a is evenly divisible by 4, then a is evenly divisible by 2.
(By “evenly divisible,” I mean divisible without remainder.) The first thing to ask yourself is: is this sentence true? I hope we can all agree that it is—even though we do not know what a is. Let
a is evenly divisible by 4
be represented in our logic by
U
and let
a is evenly divisible by 2
be represented by
V
Our sentence then is
(U→V)
And its truth table—as far as we understand right now—is:
U | V | (U→V) |
---|---|---|
T | T | T |
T | F | F |
F | T | |
F | F |
Now consider a case in which a is 6. This is like the third row of the truth table. It is not the case that 6 is evenly divisible by 4, but it is the case that 6 is evenly divisible by 2. And consider the case in which a is 7. This is like the fourth row of the truth table; 7 would be evenly divisible by neither 4 nor 2. But we agreed that the conditional is true—regardless of the value of a! So, the truth table must be:^{[3]}
U | V | (U→V) |
---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Following this pattern, we should also fill out our table about the election with:
P | Q | (P→Q) |
---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
If you are dissatisfied by this, it might be helpful to think of these last two rows as vacuous cases. A conditional tells us about what happens if the antecedent is true. But when the antecedent is false, we simply default to true.
We are now ready to offer, in a more formal way, the syntax and semantics for the conditional.
The syntax of the conditional is that, if Φ and Ψ are sentences, then
(Φ→Ψ)
is a sentence.
The semantics of the conditional are given by a truth table. For any sentences Φ and Ψ:
Φ | Ψ | (Φ→Ψ) |
---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Remember that this truth table is now a definition. It defines the meaning of “→”. We are agreeing to use the symbol “→” to mean this from here on out.
The elements of the propositional logic, like “→”, that we add to our language in order to form more complex sentences, are called “truth functional connectives”. I hope it is clear why: the meaning of this symbol is given in a truth function. (If you are unfamiliar or uncertain about the idea of a function, think of a function as like a machine that takes in one or more inputs, and always then gives exactly one output. For the conditional, the inputs are two truth values; and the output is one truth value. For example, put T F into the truth function called “→”, and you get out F.)