Counterexamples to Modus Ponens?August 19, 2008
Moritz has a very interesting post on McGee’s counterexample to modus ponens. One thing that came up in the comments was how things looked on the restrictor analysis of conditionals. On this view ‘if’ is not to be thought of syntactically as a connective, but rather as a device for restricting modals. The basic idea is, given a modal, O, and antecedent and consequent, p and q, ‘if’ allows us to restrict the domain of O to p-worlds. Roughly, ‘if p, Oq’ means that q is true in all the best O-accessible p-worlds. In this note I want to argue that the restrictor view will have to admit exceptions to modus ponens even for simple (non nested) conditionals.
Moritz points out that, since we are no longer dealing with a connective, it is not clear what counts as an instance of modus ponens. One reaction, then, might be to simply say that there are no instances of modus ponens for the English conditional; the rule is only applicable to connectives, so we should reserve the rule for formal connectives that behave in the right way syntactically. I think this is a little extreme – after all, modus ponens presumably originated from a rule of inference for natural language conditionals, and we seem to be able to recognise instances of it when we them in natural language reasoning.
On the syntactic understanding of modus ponens for connectives the rule is something like the following: from and infer (where is syntactically a connective.) Thus we can say, for example, that modus ponens holds for the intuitionist conditional, the Stalnaker conditional, even for classical conjunction, but not for classical disjunction, the reverse conditional () and so on. But even here we’re not home and dry. We still need to be able to distinguish the antecedent from the consequent syntactically. Does modus ponens hold for the connective defined to have the same truth conditions as ? Depending on which counts as the antecedent, we get the material conditional or the material reverse conditional – MP holds for the former but not latter.
In short, I think an explicit syntactic characterisation of MP is not the correct way to go. For now I think we can just settle for a rule of thumb: if we can recognise it as in instance of modus ponens in English, then its underlying logical form forms the basis for an admissible syntactic characterisation for modus ponens. This allows us to characterise modus ponens even for syntactic items that aren’t connectives. Indeed, if English conditionals really do have the logical form the restrictor analysis predicts, then the following schemas are plausible syntactic characterisations of MP. (Notation: if is an operator, is the new operator formed by restricting by .)
- From and , infer
- From and , infer
Now I’m not going to defend the claim that both of these form genuine rules that capture modus ponens for the English conditional. Rather, all I want is that there are no other candidate rules to fill this role which are remotely plausible. This can be argued for by brute force. We have no difficulty identifying the antecedent and the conditional, thus premisses are definitely as represented as above. Combinatorially, there are only four candidates for the conclusion that we can construct from the available items: p, q, , and . Clearly we can eliminate p and .
Let us start with (1). Since the type of ‘if’ is a relation between two propositions and an operator, (1) is only valid if no matter how we reinterpret p, q and the conclusion is true whenever the premises are. A related question, which will determine the answer above one, is to ask for what interpretations of is the above valid (i.e. fixing the interpretation of and varying only p and q.) It turns out (1) is valid relative to an interpretation of iff it has a reflexive accessibility relation. Let’s consider just the direction of the biconditional that invalidates (1): it is perfectly possible for p to be true at a world w, and for q to be true at all the accessible p-worlds, and q false at w so long as w isn’t among the accessible p-worlds. Since w is a p-world, it must be inaccessible – i.e. this happens when the accessibility relation isn’t reflexive.
To illustrate, consider an example where the restricted modal isn’t reflexive. The easiest cases to consider are doxastic and deontic modals.
- John’s a murderer
- If John’s a murderer, he ought to be in jail
- Therefore, John is in jail
This is a counterexample to (1), there is not enough evidence to put John away. You might object – surely this was not a plausible characterisation of modus ponens in the first place. What we should rather conclude from the first two premises is that John ought to be in jail. This is essentially an appeal to (2).
Unfortunately (2) admits counterexamples as well. Suppose there are accessible ~p worlds, where q is false, but q is true at all the accessible p worlds, and that p is true at the actual world. Certainly p is true, and is true because q is true in all the accessible p-worlds. However, is false, since there are accessible p-worlds where q is false.
Examples are difficult to conjure up, since modals are often sensitive to the conversational background. Whenever I assert p, I essentially restrict the -accessible worlds to p-worlds. Thus asserting and then invariably places us in a context according to which is true. Let us try anyway. Suppose that for all we know, Jones isn’t in his office. So: it is not the case that Jones must be in his office. Be that as it may, if it is 3.00pm, Jones must be in his office, because we know his lunch break ends at 2.00pm. Suppose further that unbeknownst to us, it is be 3.00pm. That sounds like a consistent story right? But now if I assert them in the following order:
- It’s 3.00pm
- If it’s 3.00pm, then Jones must be in his office
- It’s not the case that Jones must be in his office
the trio above sounds inconsistent. But strictly speaking they’re all true, provided we keep the same context throughout.
What is interesting is that the McGee counterexamples, on a plausible syntactic analysis, form more convincing counterexamples to modus ponens on characterisation (2). For on a plausible syntactic analysis of nested conditionals, ‘if p, if q, then must r‘ the inside conditional ‘if q, then must r‘ has the form s = . Since s has the form operator:proposition, ‘if p, s’ is naturally read with ‘if p’ restricting the compound operator, rather than some covert modal. Thus the embedded conditional gets the form . Essentially we are doubly restricting . The resulting McGee example is not an instance of modus ponens on characterisation (1). However it does represent a failure for the rule (2), where the modal is the compound operator .