In Section 4-2 we saw that the argument, "X. Therefore Y", is intimately related to the truth function ~(X&~Y). This truth function is so important that we are going to introduce a new connective to represent it. We will define X⊃Y to be the truth function which is logically equivalent to ~(X&~Y). You should learn its truth table definition:
Truth table Definition of ⊃
Again, the connection between X⊃Y and the argument "X Therefore Y" is that X⊃Y is a logical truth just in case the argument "X Therefore Y" is valid.
Logicians traditionally read a sentence such as 'A⊃B' with the words 'If A, then B', and the practice is to transcribe 'If. . . then . . .' sentences of English by using '⊃'. So (to use a new example) we would transcribe 'If the cat is on the mat, then the cat is asleep.' as 'A⊃B'.
In many ways, this transcription proves to be problematic. To see why, let us forget '⊃' for a moment and set out afresh to define a truth functional connective which will serve as a transcription of the English 'If. . . then . . .':
(In the next two paragraphs, think of the example, 'If the cat is on the mat, then the cat is asleep.')
|Case||A||B||If A then B|
That is, by choosing t or f for each of the boxes under 'If A then B' in the truth table, we want to write down a truth function which says as closely as possible what 'If A then B' says in English.
The only really clear-cut case is case 2, the case in which the cat is on the mat but is not asleep. In this circumstance, the sentence 'If the cat is on the mat, then the cat is asleep.' is most assuredly false. SO we have to put f for case 2 in the column under 'If A then B'. If the cat is both on the mat and is asleep, that is, if we have case 1, we may plausibly take the conditional sentence to be true. So let us put t for case 1 under 'If A then B'. But what about cases 3 and 4, the two cases in which A is false? If the cat is not on the mat, what determines whether or not the conditional, 'If the cat is on the mat, then the cat is asleep.', is true or false?
Anything we put for cases 3 and 4 is going to give us problems. Suppose we put t for case 3. This is to commit ourselves to the following: When the cat is not on the mat and the cat is asleep somewhereklse, then the conditional, 'If the cat is on the mat, then the cat is asleep.', is true. But suppose we have sprinkled the mat with catnip, which always makes the cat very lively. Then, if we are going to assign the conditional a truth value at all, it rather seems that it should count as false. On the other hand, if we put f for case 3, we will get into trouble if the mat has a cosy place by the fire which always puts the cat to sleep. For then, if we assign a truth value at all, we will want to say that the conditional is true. Similar examples show that neither t nor f will always work for case 4.
Our problem has a very simple source: 'If. . . then . . .' in English can be used to say various things, many of which are not truth functional. Whether or not an 'If. . . then . . .' sentence of English is true or false in these nontruth functional uses depends on more than just the truth values of the sentences which you put in the blanks. The truth of 'If you are five feet five inches tall, then you will not be a good basketball player.' depends on more than the truth or falsity of 'You are five feet five inches tall.' and 'You will not be a good basketball player.' It depends on the fact that there is some factual, nonlogical connection between the truth and falsity of these two component sentences.
In many cases, the truth or falsity of an English 'If. . . then . . .' sentence depends on a nonlogical connection between the truth and falsity of the sentences which one puts in the blanks. The connection is often causal, temporal, or both. Consider the claim that 'If you stub your toe, then it will hurt.' Not only does assertion of this sentence claim that there is some causal connection between stubbing your toe and its hurting, this assertion also claims that the pain will come after the stubbing. However, sentence logic is insensitive to such connections. Sentence logic is a theory only of truth functions, of connectives which are defined entirely in terms of the truth and falsity of the component sentences. So no connective defined in sentence logic can give us a good transcription of the English 'If. . . then . . .' in all its uses.
What should we do? Thus far, one choice for cases 3 and 4 seems as good (or as bad) as another. But the connection between the words '. . . therefore . . .' and 'If. . . then . . .' suggests how we should make up our minds. When we use 'If. . . then . . .' to express some causal, temporal, or other nonlogical connection between things in the world, the project of accurately transcribing into sentence logic is hopeless. But when we use 'If. . . then . . .' to express what we mean by '. . . therefore . . .' our course should be clear. To assert "X. Therefore Y", is to advance the argument with X as premise(s) and Y as conclusion. And to advance the argument, "X Therefore Y, is (in addition to asserting X) to assert that the present case is not a counterexample; that is, it is to assert that the sentence ~(X&~Y) is true. In particular, if the argument, "X. Therefore Y", is valid, there are no counterexamples, which, as we saw, comes to the same thing as ~(X&~Y) being a logical truth.
Putting these facts together, we see that when "If X then Y" conveys what the 'therefore' in "X. Therefore Y conveys, we can transcribe the "If X then Y' as ~(X&~Y), for which we have introduced the new symbol X⊃Y. In short, when 'If. . . then . . .' can be accurately transcribed into sentence logic at all, we need to choose t for both cases 3 and 4 to give us the truth table for X⊃Y defined as ~(X&~Y).
Logicians recognize that '⊃' is not a very faithful transcription of 'If . . . then . . .' when 'If . . . then . . .' expresses any sort of nonlogical connection. But since '⊃ agrees with 'If. . . then . . .' in the clear case 2 and the fairly clear case I, '⊃' is going to be at least as good a transcription as any alternative. And the connection with arguments at least makes '⊃' the right choice for cases 3 and 4 when there is a right choice, that is, when 'If. . . then . . .' means '. . . therefore . . .'.
We have labored over the introduction of the sentence logic connective '⊃'. Some logic texts just give you its truth table definition and are done with it. But logicians use the '⊃' so widely to transcribe the English 'If. . . then . . .' that you should appreciate as clearly as possible the (truth functional) ways in which '⊃' does and the (nontruth functional) ways in which '⊃' does not correspond to 'If. . . then . . .'.
In these respects, the English 'and' and 'or' seem very different. 'And' and 'or' seem only to have truth functional aspects, so that they seem to correspond very closely to the truth functionally defined '&' and 'v'. Now that you have been through some consciousness raising about how English can differ from logic in having nontruth functional aspects, it is time to set the record straight about the 'and' and 'or' of English.
Surely, when I assert, 'Adam exchanged vows with Eve, and they became man and wife.' I do more than assert the truth of the two sentences 'Adam exchanged vows with Eve.' and 'They became man and wife.' I assert that there is a connection, that they enter into the state of matrimony as a result of exchanging vows. Similarly, if I yell at you, 'Agree with me or I'll knock your block off!' I do more than assert that either 'You will agree with me' or 'I will knock your block off is true. I assert that nonagreement will produce a blow to your head. In these examples 'and' and 'or' convey some causal, intentional, or conventional association which goes above and beyond the truth functional combination of the truth values of the component sentences. 'And' can likewise clearly express a temporal relation which goes beyond the truth values of the components. When I say, 'Adam put on his seat belt and started the car.' I assert not only that 'Adam put Gn his seat belt.' and 'He started the car.' are both true. I also assert that the first happened before the second.
Although 'and', 'or', and 'If. . . then . . .' all have their nontruth functional aspects, in this respect 'If. . . then . . .' is the most striking. '⊃' is much weaker than 'If. . . then . . .', inasmuch as '⊃' leaves out all of the nontruth functional causal, temporal, and other connections often conveyed when we use 'If. . . then . . .'. Students sometimes wonder: If '⊃' (and '&' and 'v') are so much weaker than their English counterparts, why should we bother with them? The answer is that although truth functional sentence logic will only serve to say a small fraction of what we can say in English, what we can say with sentence logic we can say with profound clarity. In particular, this clarity serves as the basis for the beautifully clear exposition of the nature of deductive argument.
When the language of logic was discovered, its clarity so dazzled philosophers and logicians that many hoped it would ultimately replace English, at least as an all-encompassing exact language of science. Historically, it took decades to realize that the clarity comes at the price of important expressive power.
But back to '⊃'.
Here are some things you are going to need to know about the connective '⊃':
A sentence of the form X⊃Y is called a Conditional. X is called its Antecedent and Y is called its Consequent.
Look at the truth table definition of X⊃Y and you will see that, unlike conjunctions and disjunctions, conditions are not symmetric. That is, X⊃Y is not logically equivalent to Y⊃X. So we need names to distinguish between the components. This is why we call the first component the antecedent and the second component the consequent (not the conclusion conclusion is a sentence in an argument).
Probably you will most easily remember the truth table definition of the conditional if you focus on the one case in which it is false, the one case in which the conditional always agrees with English. Just remember that a conditional is false if the antecedent is true and the consequent is false, and true in all other cases. Another useful way for thinking about the definition is to remember that if the antecedent of a conditional is false, then the whole conditional is true whatever the truth value of the consequent. And if the consequent is true, then again the conditional is true, whatever the truth value of the antecedent.
Finally, you should keep in mind some logical equivalences:
The Law of the Conditional (C): X⊃Y is logically equivalent to ~(X&~Y) and (by De Morgan's law) to ~XvY.
The Law of Contraposition (CP): X⊃Y is logically equivalent to ~Y⊃~X.