Is logic a tool for justification, or discovery? I'm talking about deductive logic, not inductive logic. Suppose you're given a set of premises and want to find out what follows from them, ie you want to derive a set of conclusions. In the logic textbook I'm working through, there is a chapter on Syllogistic logic which includes a section on finding conclusions from a number of premises. Further chapters cover propositional and quantificational logic using symbols, but there is no method explained there which allows you derive any conclusions from premises. Instead, the conclusion is already given, and you prove that the argument is valid or not. The technique used to do this is 'proof by contradiction'.

So it seems as though there are 2 ways of looking at logic, as a process of discovery or a process of proof or justification. I know that in deductive logic it's not really 'discovery' in the sense of finding something new that was not implicit in the premises, but that doesn't mean it's not also a kind of discovery. After all, mathematics is mainly about deduction but most of the theorems in mathematics aren't at all obvious, in most cases you can't just look at some axioms and immediately see that some theorem follows from them. You may be able to prove that a theorem is correct by trying to refute it, but how did you get to the theorem in the first place?

So I'm wondering whether proof by contradiction can also be used in syllogistic logic. The book doesn't go into this, so it seems that you need to learn both the syllogistic approach and the quantificational logic approach, but on the other hand if you're given 2 premises of a syllogism and have followed the steps correctly for deriving a conclusion, then you know it must be correct, so in a way you've already 'proved' the conclusion, in which case, proof by contradiction isn't needed.

Hi James. Deductive logic is a tool for both discovery and justification. Given the premises, you may use it to see what conclusions can be drawn from them (this may be to find something new, or merely to test the consistency of your various beliefs by clarifying their implications) - or you may have a desired conclusion in mind and seek out premises that would justify it.
Contrary to what you say, there is real discovery through deductive reasoning. Even if formally the conclusion seems obvious (ex post facto), we still need to go through the process of reasoning to realize the fact with certainty. Moreover, deductive logic is an essential tool of inductive logic, in that we need to actively test the implications of our hypotheses, to see whether they ultimately accord with empirical fact.
I am not sure what you mean by 'proof by contradiction'. Is this maybe a reference to reductio ad absurdum? In the latter, you check out whether by denying your putative conclusion and combining that denial with one of your premises, the result is a proposition that contradicts another of your premises.

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Thanks Avi. Yes, by 'proof by contradiction' I meant reductio ad absurdum. All the proofs in the propositional and predicate logic chapters follow the same pattern, first deny the conclusion, make some assumptions and then try to find a contradiction.

I'm glad to hear you say that deductive logic is a process of discovery as well as justification. I was discussing this with someone on another forum who says that deduction can't discover anything new because it only draws out the implications of the premises, and discovery basically depends on lucky guesses and noticing patterns, or in any case it goes beyond what's in the premises.

I was wondering whether there's any way of systematically finding all the conclusions from a set of premises? if there is, my book doesn't explain how.

"someone on another forum who says that deduction can't discover anything new because it only draws out the implications of the premises, and discovery basically depends on lucky guesses and noticing patterns, or in any case it goes beyond what's in the premises"
To say deduction can't discover is to confuse deduction in action and deduction as it appears on paper in books on deduction. In action, every thought requires an effort. To put together a minor and major term through a middle term in syllogism requires intelligence and effort - it does not happen automatically, and when it does happen it is not always correct.
As for the discovery as lucky guesses, noticing patterns, and going beyond what is in the premises, this is not discovery but the formulation of inductive hypotheses. To check the truth of such hypotheses, we must still deductively find out their implications, then look out for the empirical confirmation of these implications.
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James, I am wondering if you are aware that I have a website called www.thelogician.net that deals with all such issues in full detail. I recommend you look there - starting with the book Future Logic. If you have questions, please use the search facility there.

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Thanks Avi. Wow, there's a lot to absorb at your site. Your Future Logic book looks very interesting and will give me another perspective, and probably a deeper one than the book I'm using at the moment. One criticism (don't take offence) is that I don't see any problems to work through. I think logic is more of a skill (maybe like maths) than just "stuff to remember", so problems and exercises are necessary, for me anyway.

Yes, but Future Logic is big enough as it is. Really, the exercise is for you to find concrete examples, given the abstract form. This teaches you to think in formal terms, which is a powerful skill to have.

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