2.3 Material implication

Material implication is the name we'll use for $\rightarrow$.

$$\begin{array}[t]{\vert c\vert c\vert c\vert\vert c\vert} \hl... ...\:F\:} & \mathrm{\:F\:} & \mathrm{\:T\:} \\ \hline \end{array}$$

1. If John ate the apple, he'll be sick.
2. Antecedent: John ate the apple
3. Consequent: He'll be sick.

Claim made: In those circumstances where the first sentence is true, the second sentence is true. So the first two lines of the truth table make perfect sense. The claim is safe when both sentences are true, and it is clearly false when the

But what about when the first sentence is false. Well if he didnt eat the apple the claim is safe whether he's sick or not. This claim only guarantees that IF he ate the apple sickness follows. So if he didnt the claim is still ``true'' according to our truth conditions.

Question: How well does this accord with our intutuions about conditionals? Answer: Not very.

(a)	Antecedent	Consequent	Conditional
T	T	If 1960 was divisible by 5, then 1960 was a leap year.	T
(b)	F	T	If Al Gore won the election of 2000, George Bush won the election of 2004.
(c)	T	F	If George Bush won the election of 2004, Al Gore won the election of 2000.

(a) just seems false. (b) is weird not clear what kind of communicative act is being performed. (c) can be true as an instance of the ``If X, I'll eat my hat'' construction.

If you kick me again, I'll punch you.

Here the right truth table seems to be: