Pages 150-151 of §18.3 of Introduction to Formal Logic by Peter Smith provide two justifications for the truth table of the material conditional.
In the first justification (paragraph (a) - (c) on pg. 150), Smith shows that the material conditional must have the truth table it has if it is to preserve the four basic properties of ordinary conditional: (1) modus ponens, (2) modus tollens, (3) the falsehood condition, (4) non-reversibility.
I'm unclear about where these properties come from:
- Are they simply being inferred from the natural language usage of conditionals (Jackson uses a similar line of reasoning in which he uses the term "logical intuitions")?
- How do we know that these are the only four properties that a conditional-like truth-functional connective must satisfy?
The second justification (paragraph (d) on pg. 151) is as follows (emphasis added):
(d) For a fast-track argument, take the wff (P ∧ Q) → P. Assuming the arrow is conditional-like, this wff should always be true (since necessarily, if P and Q are both true, then P in particular is true). But the values of the antecedent/ consequent in this wff can be any of T/T, or F/T, or F/F (depending on the values of P and Q). And we've just said that our conditional needs to evaluate as T in each of those three cases. So that forces the same completion of the table for our conditional-like truth function.
I don't know if I'm following this correctly, especially the last two lines.
Say P = "I have apples" and Q = "I have oranges", then:
- If I have apples, then I have apples, regardless of whether I have oranges.
- So (P ∧ Q) → P and (P ∧ ¬Q) → P are both true.
- Similarly, if I don't have apples, then I don't have have apples, regardless of whether I have oranges.
- So (¬P ∧ Q) → ¬P and (¬P ∧ ¬Q) → ¬P are both true.
Therefore, if → is conditional-like, then (P ∧ Q) → P must always be true (we can also verify this using logical equivalences: (P ∧ Q) → P ⇔ ¬(P ∧ Q) ∨ P ⇔ (¬P ∨ ¬Q) ∨ P ⇔ (¬P ∨ P) ∨ ¬Q ⇔ ⊤ ∨ ¬Q ⇔ ⊤)).
Given the definition of conjunction, the antecedent/consequent can only take the values F/F, F/T, or T/T, and, as I tried to show in the apples and oranges example above, (P ∧ Q) → P must be true in each of these cases. From this we can infer lines 1, 2, amd 4 of the material conditional's truth table:
|A||C||A → C|
But how does this force the same completion of line 3 as the first justification (which used the falsehood condition)? Is it F simply due to the fact that the antecedent/consequent can never be T/F (since the only possible combinations are F/F, F/T, and T/T)?