If then else
Bare indicative conditionals are bewildering, but they become surprisingly well-behaved if we add an 'else' clause.
Intuitively, 'if A then B' doesn't make an outright claim about the world. It says that B is the case if A is the case – but what if A isn't the case?
An 'else' clause resolves this question. 'If A then B else C' makes an outright claim. It says that either B or C is the case, depending on whether A is the case. That is: the world is either an A-world, in which case it is also a B-world, or it is a ¬A-world, in which case it is a C-world. For short: (A∧B)∨(¬A∧C).
An example. Someone has drawn a card from a shuffled deck (with 26 black cards and 26 red cards). I say:
This says that the card is either a red six or a black seven.
So 'if .. then .. else ..' is truth-functional.
Probability judgements seem to confirm this. Intuitively, the probability of (1) is the probability of drawing either a red six or a black seven, which is 1/13.
But isn't 'else' just an abbreviation of 'and otherwise'? Isn't 'if A then B else C' equivalent to 'if A then B and if ¬A then C'?
It surely seems so. (1) and (2) seem interchangeable:
We get the same judgements about truth-conditions and probabilities.
This is good news for the material analysis of conditionals: the conjunction of the material conditionals A→B and ¬A→C is indeed equivalent to (A∧B)∨(¬A∧C).
But what should we say if we don't think that ordinary conditionals are material conditionals?
It's hard to see, for example, how conjoining two conditionals with a selection semantics, as in (Stalnaker 1968), could turn them into material conditionals.
Or suppose we go with an expressivist semantics in the tradition of (Adams 1975) and (Edgington 1995), on which conditionals don't express propositions at all: why do we get a proposition if we conjoin two non-propositions with 'and'?
It doesn't help much to tinker with 'and'. We can, for example, consider a discourse with two separate conditionals:
Intuitively, (3) says the same as (1) and (2). The probability that both sentences in (3) are true is 1/13.
Another puzzle:
How do we assess the probability of 'if then else' conditionals (or of conjunctions or sequences or sets of conditionals)?
There's a standard answer to how we evaluate the probability of 'if A then B' (without 'else'): we evaluate the probability of B conditional on A. We evidently can't use this "Ramsey test" method for evaluating 'if A then B else C'.
We could apply the Ramsey test in stages: first evaluate B conditional on A, then evaluate C conditional on ¬A; but then what? multiply the results?
In any case, this can't be what we're doing. Take the card example again. The probability of a six given red is 1/13 (26 red cards, two of them sixes); the probability of a seven given non-red is also 1/13; how do we get from there to an overall probability of 1/13 for the conjunction?
More dramatically, suppose we assess the probability of (4).
Using the Ramsey test, the probability that the first statement is true is 1/13. But together, the two statements say that the card is either a red six or a non-red non-six. This has probability 1/2. So the probability that the report is true increases from 1/13 to 1/2 when we add the second sentence! Adding information can never increase the probability of truth. Does the second sentence reduce the information in the report?
PS: There seem to be cases where 'if .. then .. else ..' isn't truth-functional:
Suppose you don't mow the lawn and you get nothing. Does this make (5) true? Is (5) highly probably as long as it's highly probable that you don't mow and get nothing? Maybe not. (5) seems to have a reading on which it entails that you would have gotten £1000 if you had mowed the lawn.
I suspect the difference is that this reading of (5) isn't a purely "epistemic" conditional, in contrast to (1). But I'm not sure what it is instead.