Deduction and the Necessary A Posteriori

Consider: There is a cat here, therefore there is an animal here.

Assuming we want to say that this inference is valid in some sense, here are three things we might say about it:

1) It is an elliptical argument, involving an unarticulated premise, namely that cats are animals.

2) It is an enthymematic or gappy argument, involving unarticulated reasoning.

3) It is a complete argument in itself - neither (1) nor (2) is the case.

On the first approach, the deduction is clearly a priori. But this is not the only possible attitude. While the belief that cats are animals could conceivably be overturned by experience, it is arguably not a contingent fact that all the cats around are animals (cf. Kripke 1980). So rather than regarding this as part of the matter being reasoned from, we might regard it as part of the deductive apparatus.

Thus, on the second interpretation, we might regard the move from 'cat' to 'animal' as being licensed by an unarticulated principle of reasoning (which may be expressed in the form of an inference rule, or an axiom such as 'All cats are animals'). Or we might resist even this, and say that nothing is unarticulated - perhaps still allowing that the argument can be justified by the principles of reasoning which are held to be unarticulated on the second interpretation.

Note how natural these latter two approaches are; there does seem to be some sense in which the conclusion follows from the single articulated premise. Note also that the argument by itself satisfies the natural (admittedly problematic) modal characterization of validity, if we take the relevant modality to be subjunctive or metaphysical modality ("what could have been the case") rather than a priori possibility or epistemic modality ("what could be the case"): it could not have been the case that there was a cat here yesterday but no animal here yesterday.

This may suggest that, according to some intuitive and central concept of deduction, some facts about what can be deduced from what are empirical (i.e. not knowable a priori).

However, this flies rather completely in the face of previous philosophical thinking about deduction. It also raises the following puzzle: on one way of thinking about necessity, our holding it to be necessary that all cats are animals means that we have made a certain kind of connection between our cat-concept and our animal-concept (an empirically defeasible connection, held constant when describing counterfactual scenarios). But it is natural to think of this conceptual connection as partly constitutive of the content of thoughts involving these concepts - thoughts such as 'There is a cat here' and 'There is an animal here'. Thus, someone who doesn't have such a connection arguably isn't in a position to have those two thoughts at all: there would appear to be little room for them to have those exact thoughts and yet not be able to work out a priori that one implies the other.

My suggestion is that, when faced with this sort of puzzle, one should try to distinguish different ways of individuating content, some more fine-grained than others. When talking about thoughts in, e.g., the context of communication, a relatively course-grained individuation scheme is often most appropriate. When talking about the epistemology of deduction - and not just the epistemology - a more fine-grained approach is called for. (That is, an approach where what are for many purposes two instances of the same thought get treated as distinct structures.)

Insofar as this is right, the more general moral is perhaps something like: when doing philosophy, be willing to put multiple modes of content-individuation on the table - don't let one obsess you to the exclusion of all others.

Tristan Haze

Reference

Saul A. Kripke (1980). Naming and Necessity. Harvard University Press.