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.

It's not raining and it's raining therefore it's not raining.. So yeah, it's "incoherent" in that its premises are inconsistent.
— Michael

Accepting that definition of "incoherent," — Hanover

Whatever @Michael meant, I don't take it as a definition. It only states:

If a set of statements is inconsistent, then it is incoherent.

It doesn't say:

A set of statements is inconsistent if and only if it is incoherent.

More generally, an expression may be incoherent but not inconsistent. Expressions that are not syntactical are incoherent but they're not even statements, so they are not even in the category of things that are consistent or inconsistent.

By using 'incoherent' rather than 'inconsistent', we lose the information that the premises are not merely incoherent, but they are, more to the point, inconsistent.

Also, @Michael, as I understand him, meant scare quotes. Indeed, I don't see the analysis of this particular matter in ordinary formal logic as being in regard to a wider rubric of 'incoherent' (that includes both not-syntactical gibberish and syntactical inconsistency) but rather in regard to inconsistency.

Also, personally, in this context, I like to mention satisfiability rather than consistency, since they are equivalent only in first order logic, and, even more basically, mentioning satisfiability rather than consistency underscores that we don't need to have a particular, or even any, deductive calculus in view.

/

I suggested the neologism 'revonah' for an argument that has an unsatisfiable set of premises.
but maybe a neologism that is more technical sounding would be better:

Df. An argument is sat-premised if and only if the set of premises is unsatisfiable.

Df. An argument is unsat-premised if and only if the set of premises is unsatisfiable.

we have (1) valid and coherent arguments and (2) valid and incoherent arguments [and] (3) valid and sound arguments and (4) valid and unsound arguments. — Hanover

Soundness is per each interpretation. But let's say we're confining to just one interpretation, so we don't have to say 'per the interpretation':

(1t) sat-premised and valid

Not every sat-premised argument is valid.

Not every valid argument is sat-premised.

(2t) unsat-premised and valid

Every unsat-premised argument is valid.

Not every valid argument is unsat-premised.

(3t) sound

Every sound argument is valid.

Not every valid argument is sound.

(4t) unsound and valid

Would you agree that:

A. All 3s are 1s, but not all 1s are 3s?
B. All 2s are 4s, but not all 4s are 2s.
C. No 1s or 3s are 4s or 2s.
D. No 4s or 2s are 1 or 3s. — Hanover

(C) and (D) are WRONG (see below).

These are all CORRECT except those marked WRONG:

(A1) For any argument, if it is (3t) then it is (1t).

(A2) It is not the case that, for any argument, if it is (1t) then it is (3t).

(B1) For any argument, if it is (2t) then it is (4t).

(B2) It is not the case that, for any argument, if it is (4t) then it is (2t).

(C1) For any argument, if it is (1t) then it is not (4t). WRONG.

There are arguments that have a satisfiable set of premises but there is at least one false premise. This is a key point in ordinary formal logic. Consider:

{"Macron is German"} is satisfiable but "Macron is German" is false. This is a key point in ordinary formal logic: A set of premises may satisfiable but still have falsehoods. Consider:

"Macron is German" is false per ordinary facts, but there are interpretations in which "Macron is German" is true.

(C2) For any argument, if it is (1t) then it is not (2t).

(C3) For any argument, if it is (3t) then it is not (4t).

(C3) For any argument, if it is (3t) then it is not (2t).

(D1) For any argument, if it is (4t) then it is not (1t). WRONG.

There are arguments that are unsound but have a satisfiable set of premises. This is a key point in ordinary formal logic: For example:

"Macron is German" is false per ordinary facts, but there are interpretations in which "Macron is German" is true.

(D2) For any argument, if it is (4t) then it is not (3t).

(D3) For any argument, if it is (2t) then it is not (1t).

(D3) For any argument, if it is (2t) then it is not (3t).

Yes, so this has inconsistent premises:

1. It is raining
2. It is not raining
3. Therefore, is is raining

And this has incoherent premises

1. Red fast what
2. Glooblefooble
3. Therefore it is raining

Indeed.

EDIT: But "Red fast what" and "Glooblefooble" are not even premises since they are not statements. So it's not even an argument, since {"Red fast what", "Glooblefooble"} is not a set of statements.

Arguments can be:
1. Valid, consistent, and sound
2. Valid, consistent, and unsound
3. Valid, inconsistent, and unsound
4. Invalid — Michael

Usually, we don't say that arguments are consistent/inconsistent. Sets of sentences are consistent/inconsistent.

All combinations:

(1) sound (thus satisfiable set of premises) and valid
(2) satisfiable set of premises, unsound, and valid
(3) satisfiable set of premises, sound, and invalid
(4) satisfiable set of premises, unsound, and invalid
(5) unsatisfiable set of premises (thus valid and unsound)

No 3 is a 4 because no argument can be both valid and invalid.
— Michael

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

Explosion is the property of a set of statements entailing all statements. But it's still the case that no argument can be both valid and invalid.

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

There are two definitions:

Df. An argument is valid if and only if there are no interpretations in which all the premises are true and the conclusion is false.

Df. A statement is valid if and only if it is true in all interpretations.

We sometimes say 'the statement is a validity' synonymously with 'the statement is valid'.

(C) and (D) are WRONG (see below). — TonesInDeepFreeze

I appreciate the deep analysis, but can you just draw me some Venn diagram circles with 1, 2, 3, and 4s on them and then I can see what can be what? It's easier on my visual brain.

We're actually debating what terms each of us can make up and the best terms that would describe whatever we're trying to say. I'll defer to yours with my backwards name and provide myself a translator so we can speak the same language. In truth, I think we largely follow what each other are saying at this point.

What I mean by "incoherent" is that which is "expressed in an incomprehensible or confusing way; unclear." @Michael's rendition of what "incoherent" might look like includes gibberish, which is a new additon to this conversation, so it might require an entirely different term. We could then start inserting such non-linguistic items such as the smell of lilac and that weird feeling of deja vu in as premises. Everyone loves a good emoji as well, so that could go in there too.

In any event, "Gloobelfooble" could indeed be a statement, inasmuch as A can be statement and Q can be a statement.

If Gloobelfooble, then Q
Gloobelfooble
Q

"that's a valid conclusion" — Hanover

That's ambiguous. It could mean two things:

(1) A certain argument that ends with that conclusion is valid.

(2) The conclusion is valid (i.e., it is a validity).

That's ambiguous. It could mean two things: — TonesInDeepFreeze

You make a valid point.

I get that joke. Thank you.

there's no such thing as an uncountable recursive set. — TonesInDeepFreeze

Of course. Nice.

No, because that would be defining 'valid argument', not 'argument' in general. — TonesInDeepFreeze

Cool. So we have {p, q, r} with r designated as the conclusion, and that's an argument, and then in addition if it is a valid argument, r is also the logical consequence of {p, q}. Thanks for clearing this up.

You have a preference for the model-theoretic account of logical consequence, if I've understood aright. it has an intuitive appeal for me. On that, account, an argument is valid iff there are not counter-examples. The SEP article notes "One of the main challenges set by the model-theoretic definition of logical consequence is to distinguish between the logical and the nonlogical vocabulary" and suggests that this might be overlooked if we "limiting the admissible models for a language". This was my puzzle. What follows refers to that article.

Another issue is the difficulty that "the actual world is not accounted for by any model", but this seems to me to be misleading; sure "each model domain is a set, but the actual world presumably contains all sets, and as a collection which includes all sets is too ‘‘large’’ to be a set", but it doesn't follow that there are any particular things int he world that we cannot include in our model. That is, while we may not be able to model everything, we can model anything.

I take it that the "Tonk" argument undermines proof-theoretical accounts by showing them to be arbitrary. My realist tendencies play a role here.

Frankly I haven't yet been able to follow the argument for bringing proof-theoretic and model-theoretic perspectives together, but there is some appeal in that, and I gather that the result would be a win for logical pluralism.

None of this is 'tight" enough for a firm conclusion, but do you have any thoughts?

I'm not going to draw diagrams.

We're actually debating what terms each of us can make up and the best terms that would describe whatever we're trying to say. — Hanover

I'm not debating that. We can make any defiinitions we want. And I am not claiming that the definition of 'valid' in ordinary formal logic is suited for many everyday senses of 'valid'.

What I mean by "incoherent" is that which is "expressed in an incomprehensible or confusing way; unclear." — Hanover

Under that definition, I don't take contradictions to be incoherent.

Jack Shaklemoff is in Kansas and Jack Shaklemoff is not in Kansas.

That is clear, comprehensible and not confusing.

And I understand that there are no interpretations in which it is true.

Moreover, consider some set of premises that are very complicated and so that it is not at first apparent whether the set is inconsistent. I don't have to wait until it is proven that the set is consistent to understand it as a set of premises. Consider:

The set of axioms of PA along with "Every even number greater than two is the sum of two primes."

I don't know whether or not that is a consistent set of sentences. But even if later we find a proof that it is inconsistent, then it still was and still will be a clear, comprehensible and not confusing set of sentences.

"Gloobelfooble" could indeed be a statement, inasmuch as A can be statement and Q can be a statement.

If Gloobelfooble, then Q
Gloobelfooble
Q — Hanover

That's not what @Michael meant. He didn't mean 'Gloobelfooble' as a name of a sentence or as a variable ranging over sentences, but rather as just a meaningless expression.

You have a preference for the model-theoretic account of logical consequence, if I've understood aright. — Banno

I reference it because it is rigorous, captures a common and basic intuition I share with logicians and mathematicians, and it seems the most prevalent account so that my remarks are understood in a context people know about.

But I don't claim it is the only credible account or even the best one. And of course, the intuitionist notion of model differs from the classical account, and the intuitionist notion fascinates me as do all the alternative logics though I wish I had more time to study them.

The SEP article notes "One of the main challenges set by the model-theoretic definition of logical consequence is to distinguish between the logical and the nonlogical vocabulary" — Banno

I haven't read that article in full, so I'm only off the cuff here:

Of course, models are relative to languages. "for all models" has as tacit that there is a particular language L that is addressed, so really it is, "for all models for language L".

"the admissible models for a language".

There is the notion of admissible models of set theory, but I am not familiar with a general notion of admissibility.

SEP says: "each model domain is a set, but the actual world presumably contains all sets, and as a collection which includes all sets is too ‘‘large’’ to be a set (it constitutes a proper class), the actual world is not accounted for by any model (see Shapiro 1987)."

Of course, every domain is a set, and there is no set of all sets, so there is no domain that has all sets as members. But I don't know what it means to say "all sets are in the real world". The matter raised is interesting, but I don't know enough about it. Anyway, I haven't premised anything I've said on the claim that there is a mathematical model of all of the "real world".

"the admissible models for a language"The "Tonk" argument undermines proof-theoretical accounts by showing them to be arbitrary. — Banno

I don't know enough about it.

I do know (let '*' stand for 'tonk'):

From P infer P*Q, and from P*Q infer Q

So, if 'infer' is transitive then from P infer Q.

So, from any statement, we may infer any statement.

What argument is being made about that?

But I don't know what it means to say "all sets are in the real world". — TonesInDeepFreeze

Yeah, I baulked at that too.

From what I understand, if we allow TONK as a rule then any statement is provable. If we define logical consequence in terms of proof, and since any statement is provable by TONK, any statement is a logical consequence of any other. TONK shows that not just any rule will do for a proof-theoretical approach.

There are subsequent developments in Proof-Theoretic Semantics...

The specific relationship between introduction and elimination rules as formulated in an inversion principle excludes alleged inferential definitions such as that of the connective tonk, — Proof-Theoretic Semantics

But these seem ad hoc to me... I may be just misunderstanding them.

I'm not (or mustn't...) drawing any conclusions here, just trying to make some sense of what turns out to be a surprisingly varied and lively debate.

Regarding 'formal language' from, 'Notes on Metamathematics' by William Goldfarb:

"A formal language is specified by giving an alphabet and formation rules. The
alphabet is the stock of primitive signs; it may be finite or infinite. The formation
rules serve to specify those strings of primitive signs that are the formulas of the
formal language. (A string is a finite sequence of signs, written as a concatenation
of the signs without separation.) In some books, formulas are called “well-formed
formulas”, or “wffs”, but this is redundant: to call a string a formula is to say it is
well-formed. A formal language must be effectively decidable; that is, there must
be a purely mechanical procedure, an algorithm, for determining whether or not
any given sign is in the alphabet, and whether or not any given string is a formula."

What I said is right along those lines. His account is in context of mathematical logic, but perhaps it generalizes with any needed tweaks.

Speaking roughly, truth-theoretical semantics has realism built in to it from the ground up, while proof-theoretical semantics has meaning as use built in to it from the ground up. So again I find this central issue, Davidson against Wittgenstein. I want a way to reconcile this odd dichotomy, to bring the two together.

Ok, that's neat - formal languages as decidable.

I think to accomplish something like that you'd have to stipulate Wittgenstein, almost. Not quite, because you can argue this or that about him, but if you bring in Davidson as a point of comparison it seems that Wittgenstein is harder to interpret -- or at least has more interpretations to decide from -- than Davidson.

It's not as if you can do just anything with words - TONK is useless.

Yeh. No matter what I say the keys are locked in the car I locked.

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. — Srap Tasmaner

If an epistemological theory leads us to think we don't know anything, isn't that just evidence that the theory has gone astray?

You know you are reading this.

Yep.