[The following is summary. Boldface (except for metavariables) and bracketed commentary are my own. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]
David W. Agler
Symbolic Logic: Syntax, Semantics, and Proof
Ch.3: Truth Tables
3.5 The Material Conditional Explained (Optional)
It may not be immediately obvious why the material conditional has its particular truth evaluation. However, the reasons for it come to light when we see that the other possible evaluations would ascribe properties or behaviors to the conditional that our intuition tells us it should not have.
In section 2.4, we discussed some problems with translating the use of “if ..., then ...” formulations in English into the material conditional operator →. [The problem we noted was that in English we often use “if..., then...” formulations to state a causal relation, but the truth functionality of → does not express this causal relation in the most accurate way.] Recall the truth table we made for the material conditional:
We gave no justification for why it should have this particular evaluation (84).
Agler will now give “a more compelling case [...] for why ‘if P then Q’ corresponds to ‘P→Q’” (85). And he will address two main strategies for justifying this evaluation.
The first strategy involves us considering every possible evaluation for a binary operator. We then eliminate all the ones that do not correspond to our intuitive understanding of how conditionals should work.
So here is a table of all the possible evaluations for binary operators.
The “if ..., then ...,” formulation will have to correspond to one of these truth evaluations, since there are no other possibilities. We can eliminate many of them on the basis of certain intuitions that we are sure about. For example, our intuition tells us that the conditional will be true if both the antecedent and consequent are true. This means we can eliminate all those columns where they are both false, namely, columns 4, 8, 10, 11, 12, 13, 14, and 16. [We see that in the first row of the truth assignments that both P and Q are true. Now we want to look down the series of columns to where the evaluation is F for that first row. These we will eliminate, because our intuition tells us it should say T.]
This leaves us with the following possibilities:
We have another intuition that the conditional should be false if the antecedent is true but the consequent is false. [We see that row two of the assignments has the antecedent as true and the consequent as false.] So now we look for all those columns that evaluate this assignment as true, which are 1, 3, 6, and 7.
This leaves us with:
We will now consider column 2. We see that it has the same values as Q.
But our intuition tells us that the conditional is saying more than just repeating the value of the consequent.
Now consider column 9. We see that it has the values that we already assigned to the biconditional.
The problem is that the biconditional has properties that the conditional does not have [namely, symmetry]. So consider the sentence “If John is in Toronto, then he is in Canada” (86). It is not logically equivalent to “If John is in Canada, then he is in Toronto” (87). [However, “John is in the capital of Canada if and only if he is in Ottawa” is logically equivalent to “John is in Ottawa if and only if he is in the capital of Canada”.] So we can eliminate 9.
Now what about 15? As we can see, it has the same evaluation as conjunction.
But probably the conditional and conjunction are not logically equivalent. He explains:
If if ..., then ... statements are represented by the ‘∧’ function, then every truth-functional use of ‘if P then Q’ can be replaced by a statement of the form ‘P and Q.’ However, this does not seem to be the case. Consider the following propositions:
(3) If John is in Toronto, then he is in Canada.
(4) John is in Toronto, and he is in Canada.
Clearly, (3) and (4) do not say the same thing. (4) is true if and only if John is both in Toronto and in Canada. However, we think that (3) is true if John is in Canada but not Toronto, e.g., if he were in Vancouver.
This leaves the only possibility being 5.
[Agler then gives the second strategy for explaining why the conditional has its particular evaluation. I regret that I do not understand how this next account demonstrates why we need this particular evaluation. I will largely quote it, with my comments following each part:]
A second justification for why truth-functional uses of if if ..., then ... correspond to the ‘→’ function depends upon our understanding of logical properties. For suppose that instead of treating truth-functional uses of ‘if P then Q’ in terms of ‘→,’they were defined as the following truth function ‘→*’:
(87). [At this point I do not know why we choose this particular evaluation. This is the evaluation for conjunction, which we already said would not work for other reasons (see above). And I do not know what is meant by logical properties. I quote further:]
If this were the case, then the following two propositions would be logically inconsistent:
(5) If John is in Toronto, then he is in Canada.
(6) John is not in Toronto.
The inconsistency can be expressed in a truth table as follows:
| However, we do not regard (5) and (6) as inconsistent for suppose that John wants to convey to his friend Liz that Toronto is in Canada. But also assume that John and Liz are having this conversation, not in Toronto but in Chicago. John says to Liz, “If I’m in Toronto, then I’m in Canada.” What John says is true even though he is not in Toronto, he’s never been to Toronto, and never plans on going to Toronto. Even if the antecedent of (5) is false, the conditional should be true for, to be this different, John simply denies that he can be both in Toronto and in Canada, for Toronto is in Canada!
[So we see that these two sentences should be consistent, but they are not under this evaluation of the “if..., then...” formulation (and this evaluation is identical to that of conjunction). Therefore, it should not be this one. I am not sure then how we come to conclude that it should instead be the one we assign to the material conditional. Agler said at the beginning that it has to do with what we think about logical properties. Perhaps we are supposed to check all the other possibilities to see if they lead to inconsistencies that we think should be consistent. I am not sure.]
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.