[SL10] Material conditional

§1. Truth-functionality

In this chapter we shall explain why the truth-table of "→" should look the way it does. The explanation might not be easy to understand. But do try your best and read the whole thing slowly and carefully, more than once if necessary.

First recall the concept of a sentential connective. A sentential connective is simply a symbol or expression that connects to one or more sentence to form a new sentence. Expressions such as "John says that" and "It is true that" are both sentential connectives, as we can add these expressions to the front of a statement to form new statements:

• Snow is white.
• It is true that snow is white.
• John says that snow is white.

A truth-functional sentential connective is a special kind of sentential connective. To say that a connective is truth-functional is to say that the truth-value of the new sentence depends only on the truth-value of the component sentences, and nothing else.

So for example "~" is a truth-functional connective, because the truth-value of "~φ" is simply the opposite of "φ". It does not depend on the meaning of φ or any other features. The same applies to all the other connectives in SL. In fact, if you can write down a truth-table for a connective, then it has to be a truth-functional connective, because by definition the truth-table tells you how the truth-value of the whole sentence depends on the truth-value of the component sentences and nothing else.

But there can be connectives that are not truth-functional. Consider the sentential connective "Albert Einstein believed that". It is a sentential connective because you can put this in front of a sentence and end up with a new meaningful sentence. But it is not truth-functional because the truth of the new sentence is not determined by the truth-value of the embedded component sentence. Consider for example:

1. Albert Einstein believed that 1+1=2.
   
2. Albert Einstein believed that Lee Teng-hui is the first elected president of Taiwan.

Both embedded sentences in italics are true, but presumably statement (1) is true while statement (2) is false, since Albert Einstein probably has never heard of Lee. This shows that the connective "Albert Einstein believed that" cannot be truth-functional. If it were, the two sentences would have the same truth-value.

§2. "If ... then ..." and →

What about "if ... then ..." ? It is very unlikely that it is a truth-functional connective with the same truth-table as "→". Here is why:

1. The truth-table of "(P→Q)" tells us that the whole sentence is true whenever the antecedent and the consequent are true. But this is counter-intuitive, e.g. "If London is in the UK then blackholes exist."
2. "(P→Q)" is also true whenever "P" is false, e.g. "If London is not the capital of the United Kingdom then the earth is flat." is true, which again is problematic.

The problem seems to arise because in saying "if a then b" one seems to claim that there is some kind of connection between the truth of a and the truth of b. So whether the conditional is true or not depends on the connection, and not solely on the truth-values of a and b. This is why the connective is not truth-functional.

So why do we still use "→" to translate "if ... then ..."? The short answer is that this is the only connective in SL that is closest in meaning to "if ... then ...", so this is the best we can do in a simple logical system such as SL. Of course, this still does not explain why we should pick the particular truth-table that "→" has. The longer answer is that if we use "→" to translate "if ... then ...", we want to make sure that certain logical properties are preserved.

First of all, one thing that we accept is that "if φ then φ" is always true for any statement φ. So to preserve this fact, we need to ensure that the truth-value of "(φ→φ)" is always T whether φ is T or F. Since we are assuming that "→" is truth-functional, this implies that "(φ→ψ)" has the truth-value T whenever φ and ψ have the same truth-value, whatever that is. So we can now fill in half of the truth-table of "→":

So what we need to do is to fill in the remaining two rows. The second row of the truth-table is the easier one. A second fact about the conditional that we want to preserve is that when the antecedent of a conditional is true and the consequent is false, then the whole conditional should be false. Therefore :

To fill in the third row of the truth-table, a different kind of argument is needed. This time we consider the properties that we do not want "→" to possess. In particular, consider this sequent:

(P→Q) ⊧ (Q→P)

Surely we do not want this argument to be valid. That means there should be an assignment where the premise is T and the conclusion is F. But this is not possible if "(P→Q)" is F when "P" is F and "Q" is T. So "(P→Q)" should be T under such an assignment. So finally combining the constraints we have considered we can see why the truth-table of "→" is the way it is:

In this tutorial we have looked at arguments that intend to show how the meaning of "if ... then ..." is different from that of "→". But we can now also understand why we still use the latter to translate the former in SL. The truth-table of "→" is deliberately constructed so that (a) it is a truth-functional connective, and (b) it captures some of the core logical properties of the natural language connective.