I think that the following three forms are truth-conditionally equivalent.

(1) p or q
(2) p or else q
(3) p or, if not p, q

I think that (2) and (3) are equivalent in a stronger sense; they express the very same proposition. That is, “else” just means “if not”. But it isn’t obvious that (1) and (3) are equivalent. In particular, saying that they are equivalent requires some constraints on our theory of conditionals.

_Digression_. If we have intuitionist doubts about conditional excluded middle then we shouldn’t think that (1) and (3) are equivalent. If we let q = ~p, then (3) is obviously an intuitionist theorem, while (1) is not. It’s an interesting question what the intuitionist should say about (2) in that case. I think they should say it is a theorem, and perhaps say that one reason classical theorists are so tempted by the purported theoremhood of _p or ~p_ is that they confuse it with the genuine theorem _p or else ~p_. But real intuitionists might have alternative views. I’m setting intuitionism aside for this post though. _End of Digression_.

That (3) entails (1) just requires that modus ponens is valid. (Assuming classical logic for the non-conditional connectives). Proof:

# Assume that (3) is true.

# Assume that p is true.

# Then (1) follows from line 2 by or-introduction.

# Drop the assumption of p, and assume not-p is true.

# Then if not-p, q is true. (Disjunctive syllogism from 1 and 4)

# So q is true (Modus ponens from 4 and 5)

# So (1) is true (from 6 by or-introduction)

# So (1) follows from (3), since it follows from either (3) combined with either p or ~p

I think this already raises a problem for those like McGee and Lycan who deny modus ponens. But most of us believe in modus ponens, so we can accept this entailment. The bigger issue is whether (1) entails (3).

Assume, as many philosophers do, that A and B can be true while If A, B is not true. Now consider what happens when we let p = ~A, and q = B. In that case (1) will be true, because the second disjunct is true. But neither disjunct of (3) is true, so (presumably) (3) is not true.

On the other hand, if we accept the principle of strong centring for indicative conditionals (i.e. that A & B entails If A, B), we can prove that (1) entails (3). Proof:

# Assume that (1) is true.

# Assume that p is true

# Now (3) follows from line 2 by or-introduction

# Drop the assumption of p, and assume ~p

# So q is true (by disjunctive syllogism from 1 and 4)

# So if ~p, q is true (by strong centring from 5 and 6)

# So (3) is true (by or-introduction from line 7)

# So (3) follows from (1), since it follows from (1) combined with either p or ~p

Since intuitively (1), (2) and (3) are truth conditionally equivalent, this seems to me to be a powerful argument for strong centring.