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How might one go about forming a countermodel (against the material conditional) to show that a given argument is valid through Stalnaker's conditional? As I understand it (and I admit that my knowledge is likely too broad to say anything substantial about Stalnaker's theory of conditionals), the material conditional is incompatible with Stalnaker's conditional because one can't draw inferences from Stalnaker's conditional e.g. even if A>B and B>C, we can’t directly infer that A>C. So if trying to form a countermodel for something of the form:

p = The girl did it

q = The boy did it

  1. p (The girl did it)
  2. ∴¬p → q (If the girl didn't do it, the boy did it)

to prove the incompatibility of Stalnaker's conditional with the above, what would I have to do? I think the objective is for me to demonstrate in some way that there isn't a sense of transitivity with Stalnaker's conditional (i.e. the corner >) like there is with the conclusion that the material conditional allows us to draw i.e. that the boy did it because of the negation of the first proposition.

I have very little experience with forming proofs/countermodels and with philosophy in general, so sorry if this question seems obtuse!

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    Unlike hypothetical conditional in context-free classic logic, counterfactual conditional is context-dependent and extremely multi modal and defeasibly non-monotone, thus transitivity is easily broken as world B's background context is most likely different from world A's, same goes for the non-equivalence btw disjunction and above counterfactual causal condition, since if the girl didn't do it, her parent just showed up and did it. As hinted by Shurangama Sutra long ago: One leaves behind...attachment to incessant calculating and attains the patience with the non-production of dharmas... Commented Apr 4, 2023 at 20:46

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Stalnaker's conditional is defined by a set of axioms given in his paper "A Theory of Conditionals". The idea behind it is to express under what conditions the consequent would be true, were the antecedent to be true. It is intended to cover both indicative and counterfactual conditionals. It is usually described as a variably strict conditional, since its truth depends on what holds in relevantly close possible worlds, but not in all possible worlds.

Roughly speaking a conditional 'if A then B' holds when, in the closest possible world in which A holds, B also holds in that world. Or we may state it as that the closest possible world in which A and B hold together is closer to the actual world than the closest possible world in which A and not-B hold. As you correctly say, some familiar forms of implication that are valid for the material conditional are not valid in general for the Stalnaker conditional.

You mention hypothetical syllogism, i.e. that A > B, B > C, entails A > C. Hypothetical syllogism can fail because it may be that in the closest A-world, B holds, and in the closest B-world C holds, but in the closest A world C does not hold. Examples of this occur when C is typically or nearly always the case when B is true, but A preempts C. Stalnaker gives as an example: "If Hoover were a communist he would be a traitor; if Hoover were born in Russia he would be a communist; therefore, if Hoover were born in Russia he would be a traitor". Ernest Adams gives an example like this: "If Brown's party loses the election, Brown will resign as President; if Brown dies his party will lose the election; therefore, if Brown dies he will resign as President". Here is another: "If Mary spends lots of money on luxury goods she will be poor; if Mary wins the lottery she will spend lots of money on luxury goods; therefore, if Mary wins the lottery she will be poor".

Note in each case the A term preempts acceptance of C. Another way to generate counterexamples to hypothetical syllogism is to use uncertain propositions. A high value for P(B | A) together with a high value for P(C | B) does not entail a high value for P(C | A). Hence, it is possible for it to be highly probable that 'if A then B' and highly probable that 'if B then C' but highly improbable that 'if A then C'. Look for cases where it is highly probable that C given B, but highly improbable that C given A and B. Again, the A acts to preempt C.

You also mention that unlike the material conditional, A does not entail ¬A > B. This is so because for the Stalnaker conditional A being true at the actual world does not guarantee that in the closest possible world in which A is false, B is true. Counterexamples to this form of inference are two a penny.

Another common form of implication that fails is or-to-if, i.e. that 'A or B' entails ¬A > B. Dorothy Edgington offers an example like this: Let A be "I'm not shot in the next five minutes", and B "I'm not injured in the next five minutes". I consider 'A or B' to be true, or at least highly probable. But ¬A > B would be "if I am shot then I am not injured" which is false, or at least highly implausible. Or another: "It will rain or it will snow in London in July this year". Highly likely since it is rare to go a whole month without rain in London, even in July. But not, "if it doesn't rain in London in July it will snow".

Robert Stalnaker, "A Theory of Conditionals", 1968, in Studies in Logical Theory, ed. Nicholas Rescher, also reprinted in Stalnaker, "Knowledge and Conditionals, 2019, Oxford.
Ernest Adams, "The Logic of Conditionals", 1975, Reidel.
Dorothy Edgington, "On Condiitonals" 1995, Mind, Vol. 104, pp 235-329.
Dorothy Edgington, "Validity, Uncertainty and Vagueness", 1992, Analysis, Vol. 52, pp. 193-204.

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p = The girl did it

q = The boy did it

p (The girl did it) ∴¬p → q (If the girl didn't do it, the boy did it)

Logic is simple. You just need to ask yourself what could be the conditions under which ¬p → q would have to be true.

There is probably an infinity of solutions, but Occam probably would go for the simple premise that either the girl did it or the boy did it, i.e., p ⊻ q. So, we have:

p ⊻ q

∴ ¬p → q

You could write it in various ways...

Two:

p ⊻ q

¬p

∴ q

Three:

(¬p → q) ⊻ (¬q → p)

¬p

∴ ¬p → q

Four:

p ⊻ q

r ⊻ s

∴ ¬p → q

Etc.

Either way, it will be logically valid if it is obvious that if the premises are true, then the conclusion is necessarily true.

This is why complicated implications are never discussed. Most of them are cryptic so nobody knows if the are true or not, but this one is a no brainer.

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