I get that, but a 3 permits explosion, which can force anything anywhere. — Hanover

Well, you could have the valid but unsound argument:

1. It is raining
2. It is not raining
3. Therefore, arguments can be both valid and invalid

But regardless of how you get there, the conclusion "arguments can be both valid and invalid" is false.

Deduction should allow you to pass, by valid inference, from what you know to what you did not know. Yes?

In mathematics, these elements are well-defined. What do we know? What has been proven. How do we generate new knowledge? By formal proof.

Neither of these elements are so well-defined outside mathematics (and formal logic, of course). There is no criterion for what counts as knowledge, and probably cannot be. And that defect cannot be made up by cleverness in how we make inferences.

I see no reason to question the traditional view. "Our reasonings concerning matters of fact are merely probable," as the man said. There is deduction in math and logic; everyone else has to make do with induction, abduction, probability. — Srap Tasmaner

Are you claiming that knowledge does not exist outside mathematics? I don't see why "the elements being less well-defined" results in any serious problem here. This comes back to the Meno question I have posed to you elsewhere. One could answer that question by denying that knowledge exists.

Deduction should allow you to pass, by valid inference, from what you know to what you did not know. Yes? — Srap Tasmaner

Sure, and haven't we achieved that with Billy?

But regardless of how you get there, the conclusion "arguments can be both valid and invalid" is false. — Michael

Can we say the conclusion is valid or do we reserve the term "valid" only to argument forms and not to conclusions?

Can we say the conclusion is valid or do we reserve the term "valid" only to argument forms and not to conclusions? — Hanover

Premises and conclusions are either true or false.
Arguments are valid if the conclusion follows from the premises.
Arguments are sound if they are valid and the premises are true.

- :up:

You're claiming the statement "that's a valid conclusion" is a category error because conclusions can't be valid or not valid, but only true or false. It'd be like asking what kind of document my cat is, for example.

The statement "that's a valid conclusion" does make sense, so I would think a listener who hears that would realize immediately that the person speaking isn't using the term "valid" as a term of art, but must mean something else.

I would say that "this is a valid conclusion" means "this is the conclusion of a valid argument".

Are you claiming that knowledge does not exist outside mathematics? I don't see why "the elements being less well-defined" results in any serious problem here. — Leontiskos

While I think it's defensible to say that "knowledge does not exist outside mathematics," I don't think I have to, to show the difficulty.

Mathematical knowledge, to borrow Williamson's term, is "luminous": that is, when you know that P, you know that you know that P. That may put it too strongly: there are cases where you think you have a proof, but you don't; there are cases where someone has provided a proof, but it's complex enough that it takes a while for people to confirm that it is a proof. Nevertheless, there is an alignment of the process of knowledge production and knowledge justification, and a single standard governs both.

Outside of mathematics, there are no standards of either that garner universal approval, much less guarantee that production and justification are measured by the same standard. We may have knowledge, but in general we cannot know when we do and when we don't, and thus we cannot know when our valid arguments are sound and when they are not.

I'll throw in a side issue that emphasizes the difference. It is a wise saying that experiments which are not performed have no results. And yet, in mathematics your hypotheses can be so sharply defined that they do: a difficult theorem like Fermat's last theorem might be solved piecemeal ― you prove that if lemma X were the case, then you could prove theorem T, and then you look for ways to prove X. That is, in mathematics, it's not that unusual to prove a conditional, without knowing whether the antecedent is in fact true. I think the independence results in set theory are also different from the sort of thing we can ever hope to achieve in empirical investigations.

I'm not in love with this story. It would be nice to retreat instead to some sort of common sense that of course we know things and deduce more things in everyday life. Sure. But part of that common sense is also that there are exceptions, we turn out not to know what we think we do, we turn out not to be justified in making the inferences we do. So I end up back in the same place, because we already have a name for this sort of rule that generally works but has exceptions: that's probability. ― Philosophical attempts to close the gap and specify, in some vaguely scientific way, exactly the criteria for knowledge and inference, so that we can be on ground just as solid outside of mathematics, have not only universally failed, but there are reasons to think they must fail.

I do not see a way around making some kind of distinction here. Either only mathematics (and logic) gets knowledge and deduction ― and everything else gets rational belief and probability ― or there are two kinds of knowledge, and two kinds of deduction. Pick your poison.

Mathematical knowledge and empirical knowledge differ so greatly they barely deserve the same name. Obviously the history of philosophy includes almost every conceivable way of either affirming or denying that claim.