According to Approach0, this is new to MSE.
In classical logic, there is the notion of antecedent strengthening; namely:
$$(A\to B)\to((A\land C)\to B)$$
is valid. A proof via tableau is given below.
It was generated here.
However, in a document on nonclassical logic (which can be found here), the following "counterexample" is given:
If Romney wins the election, he'll be sworn in in January. Therefore, if Romney wins the election and dies of a heart attack the same night, he'll be sworn in in January.
Here $A$ is "Romney wins the election", $B$ is "Romney will be sworn in in January", and $C$ is "Romney dies of a heart attack the same night as he wins the election".
My question is:
How does classical logic handle this supposed counterexample? What, if anything, is wrong with it?
I have a longstanding interest in nonclassical logics. See here for instance.
I doubt I could answer this myself. I don't want to end up a crank or anything, so I'm looking for an answer with full proofs or at least references.
I hope I have provided enough context.
Please help :)
You are questioning the validity of $(A\to B)\to((A\land C)\to B)$ when $C$ implicitly contradicts $B$. Well, let's consider when it explicitly does so; namely when $C$ is $\lnot B$.
For any statement $\varphi$ we consider $\varphi\to(\lnot\varphi\to B)$ to be valid, by way of the Principle of Explosion (aka ex falso quodlibet, EFQ). When the premises contain a contradiction, then anything may be derived.
$$\begin{split}\varphi\,,\lnot\varphi&\vDash B\\[3ex]\therefore\qquad &\vDash \varphi\to(\lnot\varphi\to B)\end{split}$$
Well, since this holds for any statement $\varphi$, let's consider when $\varphi$ is $(A\to B)$. Then $\lnot\varphi$ is equivalent to $A\land\lnot B$. So by substitution we have : $$\vDash (A\to B)\to((A\land\lnot B)\to B)$$