Part I: Propositional Logic
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
And by letting
Lincoln will be president
be represented by
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
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
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:
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.
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
I will buy you a ticket to the concert tonight.
be represented by
Our sentence then is
And its truth table—as far as we understand right now—is:
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.
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
a is evenly divisible by 2
be represented by
Our sentence then is
And its truth table—as far as we understand right now—is:
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:
Following this pattern, we should also fill out our table about the election with:
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
is a sentence.
The semantics of the conditional are given by a truth table. For any sentences Φ and Ψ:
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.)
2.2 Alternative phrasings in English for the conditional. Only if.
English includes many alternative phrasings that appear to be equivalent to the conditional. Furthermore, in English and other natural languages, the order of the conditional will sometimes be reversed. We can capture the general sense of these cases by recognizing that each of the following phrasings would be translated as (P→Q). (In these examples, we mix English and our propositional logic, in order to illustrate the variations succinctly.)
If P, then Q.
Q, if P.
On the condition that P, Q.
Q, on the condition that P.
Given that P, Q.
Q, given that P.
Provided that P, Q.
Q, provided that P.
When P, then Q.
Q, when P.
P implies Q.
Q is implied by P.
P is sufficient for Q.
Q is necessary for P.
An oddity of English is that the word “only” changes the meaning of “if”. You can see this if you consider the following two sentences.
Fifi is a cat, if Fifi is a mammal.
Fifi is a cat only if Fifi is a mammal.
Suppose we know Fifi is an organism, but, we don’t know what kind of organism Fifi is. Fifi could be a dog, a cat, a gray whale, a ladybug, a sponge. It seems clear that the first sentence is not necessarily true. If Fifi is a gray whale, for example, then it is true that Fifi is a mammal, but false that Fifi is a cat; and so, the first sentence would be false. But the second sentence looks like it must be true (given what you and I know about cats and mammals).
We should thus be careful to recognize that “only if” does not mean the same thing as “if”. (If it did, these two sentences would have the same truth value in all situations.) In fact, it seems that “only if” can best be expressed by a conditional where the “only if” appears before the consequent (remember, the consequent is the second part of the conditional—the part that the arrows points at). Thus, sentences of this form:
P only if Q.
Only if Q, P.
are best expressed by the formula
2.3 Test your understanding of the conditional
People sometimes find conditionals confusing. In part, this seems to be because some people confuse them with another kind of truth-functional connective, which we will learn about later, called the “biconditional”. Also, sometimes “if…then…” is used in English in a different way (see section 17.7 if you are curious about alternative possible meanings). But from now on, we will understand the conditional as described above. To test whether you have properly grasped the conditional, consider the following puzzle.
We have a set of four cards in figure 2.1. Each card has the following property: it has a shape on one side, and a letter on the other side. We shuffle and mix the cards, flipping some over while we shuffle. Then, we lay out the four cards:
Given our constraint that each card has a letter on one side and a shape on the other, we know that card 1 has a shape on the unseen side; card 2 has a letter on the unseen side; and so on.
Consider now the following claim:
For each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card.
Here is our puzzle: what is the minimum number of cards that we must turn over to test whether this claim is true of all four cards; and which cards are they that we must turn over? Of course we could turn them all over, but the puzzle asks you to identify all and only the cards that will test the claim.
Stop reading now, and see if you can decide on the answer. Be warned, people generally perform poorly on this puzzle. Think about it for a while. The answer is given below in problem 1.
2.4 Alternative symbolizations for the conditional
Some logic books, and some logicians, use alternative symbolizations for the various truth-functional connectives. The meanings (that is, the truth tables) are always the same, but the symbol used may be different. For this reason, we will take the time in this text to briefly recognize alternative symbolizations.
The conditional is sometimes represented with the following symbol: “⊃”. Thus, in such a case, (P→Q) would be written
In chapter 1, we considered as an example the sentence,
The Earth is not the center of the universe.
At first glance, such a sentence might appear to be fundamentally unlike a conditional. It does not contain two sentences, but only one. There is a “not” in the sentence, but it is not connecting two sentences. However, we can still think of this sentence as being constructed with a truth functional connective, if we are willing to accept that this sentence is equivalent to the following sentence.
It is not the case that the Earth is the center of the universe.
If this sentence is equivalent to the one above, then we can treat “It is not the case” as a truth functional connective. It is traditional to replace this cumbersome English phrase with a single symbol, “¬”. Then, mixing our propositional logic with English, we would have
¬The Earth is the center of the universe.
And if we let W be a sentence in our language that has the meaning The Earth is the center of the universe, we would write
This connective is called “negation”. Its syntax is: if Φ is a sentence, then
is a sentence. We call such a sentence a “negation sentence”.
The semantics of a negation sentence is also obvious, and is given by the following truth table.
To deny a true sentence is to speak a falsehood. To deny a false sentence is to say something true.
Our syntax always is recursive. This means that syntactic rules can be applied repeatedly, to the product of the rule. In other words, our syntax tells us that if P is a sentence, then ¬P is a sentence. But now note that the same rule applies again: if ¬P is a sentence, then ¬¬P is a sentence. And so on. Similarly, if P and Q are sentences, the syntax for the conditional tells us that (P→Q) is a sentence. But then so is ¬(P→Q), and so is (¬(P→Q) → (P→Q)). And so on. If we have just a single atomic sentence, our recursive syntax will allow us to form infinitely many different sentences with negation and the conditional.
2.6 Alternative symbolizations for negation
Some texts may use “~” for negation. Thus, ¬P would be expressed with
- The answer to our card game was: you need only turn over cards 3 and 4. This might seem confusing to many people at first. But remember the meaning of the conditional: it can only be false if the first part is true and the second part is false. The sentence we want to test is “For each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card”. Let Q stand for “the card has a Q on the letter side of the card.” Let S stand for “the card has a square on the shape side of the card.” Then we could make a truth table to express the meaning of the claim being tested:
Look back at the cards. The first card has an R on the letter side. So, sentence Q is false. But then we are in a situation like the last two rows of the truth table, and the conditional cannot be false. We do not need to check that card. The second card has a square on it. That means S is true for that card. But then we are in a situation represented by either the first or third row of the truth table. Again, the claim that (Q→S) cannot be false in either case with respect to that card, so there is no point in checking that card. The third card shows a Q. It corresponds to a situation that is like either the first or second row of the truth table. We cannot tell then whether (Q→S) is true or false of that card, without turning the card over. Similarly, the last card shows a situation where S is false, so we are in a kind of situation represented by either the second or last row of the truth table. We must turn the card over to determine if (Q→S) is true or false of that card.
Try this puzzle again. Consider the following claim about those same four cards: If there is a star on the shape side of the card, then there is an R on the letter side of the card. What is the minimum number of cards that you must turn over to check this claim? What cards are they?
- Consider the following four cards in figure 2.2. Each card has a letter on one side, and a shape on the other side.
For each of the following claims, in order to determine if the claim is true of all four cards, describe (1) The minimum number of cards you must turn over to check the claim, and (2) what those cards are.
- There is not a Q on the letter side of the card.
- There is not an octagon on the shape side of the card.
- If there is a triangle on the shape side of the card, then there is a P on the letter side of the card.
- There is an R on the letter side of the card only if there is a diamond on the shape side of the card.
- There is a hexagon on the shape side of the card, on the condition that there is a P on the letter side of the card.
- There is a diamond on the shape side of the card only if there is a P on the letter side of the card.
3. Which of the following have correct syntax? Which have incorrect syntax?
4. Use the following translation key to translate the following sentences into a propositional logic.
|P||Abe is able.|
|Q||Abe is honest.|
- If Abe is honest, Abe is able.
- Abe is not able.
- Abe is not able only if Abe is not honest.
- Abe is able, provided that Abe is not honest.
- If Abe is not able then Abe is not honest.
5. Make up your own translation key to translate the following sentences into a propositional logic. Then, use your key to translate the sentences into the propositional logic. Your translation key should contain only atomic sentences. These should be all and only the atomic sentences needed to translate the following sentences of English. Don’t let it bother you that some of the sentences must be false.
- Josie is a cat.
- Josie is a mammal.
- Josie is not a mammal.
- If Josie is not a cat, then Josie is not a mammal.
- Josie is a fish.
- Provided that Josie is a mammal, Josie is not a fish.
- Josie is a cat only if Josie is a mammal.
- Josie is a fish only if Josie is not a mammal.
- It’s not the case that Josie is not a mammal.
- Josie is not a cat, if Josie is a fish.
6. This problem will make use of the principle that our syntax is recursive. Translating these sentences is more challenging. Make up your own translation key to translate the following sentences into a propositional logic. Your translation key should contain only atomic sentences; these should be all and only the atomic sentences needed to translate the following sentences of English.
- It is not the case that Tom won’t pass the exam.
- If Tom studies, Tom will pass the exam.
- It is not the case that if Tom studies, then Tom will pass the exam.
- If Tom does not study, then Tom will not pass the exam.
- If Tom studies, Tom will pass the exam—provided that he wakes in time.
- If Tom passes the exam, then if Steve studies, Steve will pass the exam.
- It is not the case that if Tom passes the exam, then if Steve studies, Steve will pass the exam.
- If Tom does not pass the exam, then if Steve studies, Steve will pass the exam.
- If Tom does not pass the exam, then it is not the case that if Steve studies, Steve will pass the exam.
- If Tom does not pass the exam, then if Steve does not study, Steve won’t pass the exam.
7. Make up your own translation key in order to translate the following sentences into English. Write out the English equivalents in English sentences that seem (as much as is possible) natural.
 One thing is a little funny about this second example with unknown number a. We will not be able to find a number that is evenly divisible by 4 and not evenly divisible by 2, so the world will never be like the second row of this truth table describes. Two things need to be said about this. First, this oddity arises because of mathematical facts, not facts of our propositional logic—that is, we need to know what “divisible” means, what “4” and “2” mean, and so on, in order to understand the sentence. So, when we see that the second row is not possible, we are basing that on our knowledge of mathematics, not on our knowledge of propositional logic. Second, some conditionals can be false. In defining the conditional, we need to consider all possible conditionals; so, we must define the conditional for any case where the antecedent is true and the consequent is false, even if that cannot happen for this specific example.