the problem of deductionWatch
No doubt most of us are aware of the problem of induction, but has anyone dealt with the difficulties involved in deductive reasoning? Most of it seems tautological to me and just as irrational (seems odd to call a type of reasoning irrational) as induction. Mill seems to be the best (and only, System of Logic 1843) critic I could really find on the subject, everyone else assumes that mathematics is necessarily deductive.
Well the typical criticism from Hume is that induction cannot be proven to be true except on the basis of circular reasoning (i.e you can't just say induction is true because it works), although it is a useful habit. Does deduction not suffer from exactly the same problem as this?
When we talk about arguments in (deductive) logics, however, we don't need to use deduction to "prove" that our deductions work because they work virtually by definition. Consider the below argument:
where is the set of premises and is the conclusion.
One way of looking at entailment in "normal" logics is consider the set formed by the premises and the negation of the conclusion,
and to ask "is it possible for these both to be true?" (or, if you're into metaphysics, "is there a possible world in which these are both true?") If it is not possible (the set is unsatisfiable) - there is no possible world in which these are both true - then the premises entail the conclusion (if they are true it is not possible for the conclusion to be false). Once we've established such a definition of entailment, all that is left is to develop rules that preserve this property when moving from one line to the next. These are precisely the "valid" rules of inference of that logic.
I suppose you could then turn around and say "why did you choose that definition of entailment? What's to stop you picking a different one?" Now I'm no logician so my answer may not be satisfactory, but I would say that this definition of entailment is just the one we actually do happen to be interested in when we talking about arguments in real life. We could pick a different definition if we wanted, but what use would it be in argument when it allowed true premises with false conclusions?