I don't know what you mean by "minimal inconsistency guard". — TonesInDeepFreeze

Roughly that the LNC could enforce a narrow, specialized sense of consistency ― that P and ~P are inconsistent, for any P ― and this would be enough to bootstrap a more general version of inconsistency that relies on consequence, so that with a fuller system you can say A and B are inconsistent if A → C and B → ~C. It's a bootstrapping technique; start with special cases and leverage those to get the general. Special cases are easier, cheaper, in this case don't require additional resources like consequence.

It's probably all too speculative to do much with. Most of the ideas I've had in the last few minutes just recreate the fact that you can build the usual collection of logical constants with negation and one of the others (unless you want to start with the Sheffer stroke). If I were to say, maybe we need both consistency and consequence as core ideas ― that's almost all that would amount to.

I was thinking, though, that there might be a way to get negation out of a primitive sense of consequence ― not the material conditional, just an intuition of what follows from what ― something like this: any given idea (claim, thought, etc.) has a twin that is the one thing guaranteed under no circumstances to follow from it, and that would be its negation. You could define ~P roughly by partitioning the possible consequents into what can and can't follow from P, but the two buckets are different: what can follow from P might initially be empty, who knows; but what can't never starts empty.

If, like the gorillas, you didn't already have the abstract concept of negation, the bucket we're going to use to define negation would probably be full of stuff ― given any P, that bucket will have stuff that ~P follows from, in addition to ~P itself, maybe, sometimes. Example: if P is "It's sunny", our bucket of things that don't follow includes "It's cloudy", "It's nighttime", "It's raining" ― all different things that "It's not sunny" follows from.

Don't spend any time trying to make sense of all this. It's just me thinking on the forum again.

specialized sense of consistency ― that P and ~P are inconsistent, for any P — Srap Tasmaner

We have that. You want to use that to define inconsistency in general without using the notions of semantic or syntactical consequence?

you can say A and B are inconsistent if A → C and B → ~C. — Srap Tasmaner

We already have:

If G |- A -> C and G |- B -> ~C, then Gu{A B} is inconsistent.

I don't see what you're bringing.

unless you want to start with the Sheffer stroke — Srap Tasmaner

Or Nicod dagger.

we need both consistency and consequence as core ideas — Srap Tasmaner

Not getting it.

We define consistency from provability. (We could also define it from satisfiability.) Why is that lacking?

any given idea (claim, thought, etc.) has a twin that is the one thing guaranteed under no circumstances to follow from it — Srap Tasmaner

How do you know there is only one thing?

We have that. — TonesInDeepFreeze

We already have: — TonesInDeepFreeze

We define consistency from provability. — TonesInDeepFreeze

Sorry. Obviously I haven't managed to make clear what I'm trying to do here, probably because I've been writing a bunch of stuff I ended up scrapping, so I probably think I've said things I haven't.

I'm trying to figure out how we could bootstrap logic or reasoning, informal at first, of course, what we would need to do that, what the minimum is we could start with that could grow into informal reasoning. I'm not proposing an alternative to the logic we have now. So

Why is that lacking? — TonesInDeepFreeze

is not the kind of question I was addressing at all.

For example, my last post suggested a way you might leverage a primitive understanding of consequence or "follows from" to piece together negation. I don't know if that's plausible, but it hadn't occurred to me before, so that's at least a new idea.

How do you know there is only one thing? — TonesInDeepFreeze

At first probably not! But you can see how a bunch of ideas that all point to "not sunny" might eventually get you there.

And as I noted, there's some reason to think other great apes already have the ability to reason about pairs of near opposites, even without an abstract concept of negation. I was imagining a way some sense of consequence might get you from such pairs to genuine negation.

Like I said, all very speculative, and probably not worth your time.

I tried to put something together along the lines you have in mind.

The best I came up with is this:

(1) For any sentence P, the set of all sentences is partitioned into two sets: (1) the set of sentences that follow from P, call it C(P) and (2) the set of sentences that do not follow from P, call it N(P). Then instead of sentences, consider sets of sentences, let the negation of C(P) be N(P).

But that's not what you want. So, maybe we would consider the set of sentences not compatible (my word) with P such as "raining" is not compatible with "sunny" (putting aside sun showers). But that uses "not".

So I thought of this:

(2) For any sentence P, the set of the set of all sentences is partitioned into two sets: (1) the set of sentences Q such that {P Q} is satisfiable, call it C*(P) and (2) the set of sentences Q such that {P Q} is not satisfiable, call it N*(P). Then instead of sentences, consider sets of sentences, let the negation of C*(P) be N*(P). But that uses "not".

But then I thought that we should just leave it up to gorillas; and that does seem to work.

Set theory is needed for the rest of math and so is logic — Srap Tasmaner

Much of classical math existed before the introduction of set theory. So, no. Modern math is another thing.

let the negation of C(P) be N(P) — TonesInDeepFreeze

Yeah that's an interesting idea!

I guess we could assume that nothing in N(P) would follow from anything in C(P), because follow-from would already have that sort of "transitive" property that we're used to.

I've tried to work out some consequences of this, but it's still not clear to me. (I had a whole lot of ideas that just didn't work.) It's interesting though.

Much of classical math existed before the introduction of set theory. — jgill

Yeah, I get that. Looking at the reconstruction of math using set theory is one way to hunt for the difference between math and logic, that's all. Maybe not the most interesting way.

Depends on what 'needs' means.

Mathematics pretty much needs sets to work with. But if one denies that mathematics needs to be axiomatized, then mathematics does not need the set theory axioms.

If one affirms that mathematics needs to be axiomatized, then the usual axiomatization is set theory.

I would think 'follows from' is reflexive and transitive, but not symmetric.

I would need to doublecheck these (and depends on knowing more about 'follows from'):

C(P) is consistent if and only if P is not logically false.

If P is contingent, then N(P) is inconsistent.

P is logically false if and only if N(P) = 0. (explosion)

If P is logically true, then N(P) is inconsistent.

You should think of a word for 'follows from' so that it is not conflated with other common senses.

I suggest 'P raps Q' (equivalently, 'Q raps from P') instead of 'Q follows from P'.

('raps' from 'Srap')

Might be interesting to adduce a formal sentence and demonstrate somehow that it can't be said in English alone (not just that all known attempts failed). — TonesInDeepFreeze

Yep.

Am I right in understanding that the definition you gave of formal languages is strictly syntactic? It is formal iff it follows some rule for being well-formed?

If not, how does it differ?

(When I write, 'well formed formula', take that as short for 'well formed formula of the language'.)

Of course, people may have different ideas about what 'formal' means. But at least I think we would find that, for the most part at least, such things that are considered formal languages - such as languages for formal theories, computer languages, etc. - have in common that well formedness and other certain other features are algorithmically checkable.*

If a set is recursive, then there is an algorithm to determine whether something is or is not in that set. So if the set of well formed formulas is recursive, then there is an algorithm to determine whether a given sequence of symbols is or is not a well formed formula.

The desideratum is that it is algorithmically checkable whether a given string is or is not a well formed formula.

And, yes, that is all syntactical.

And the formation rules are chosen so that indeed they provide that the set of well formed formulas of is recursive. So, the rules are given as recursive definitions.

And the inference rules are recursive relations. So the set of proofs is a recursive set. So it is machine checkable whether a sequence of formulas is or is not a proof.

The point is that, with a formal language and formal proof, it is utterly objective whether a sequence of formulas is indeed a proof. A computer or a human following the instructions of an algorithm may (at least in principle) objectively check whether a given purported proof is indeed a proof.

And, yes, all of that is syntactical.

/

* We can also look at notions of 'formal' prior to the advent of recursion theory. And we can look at the general study of modern 'formal languages' that is mostly aimed at formal linguistics and computer science.

Thank you.

So I find myself back at some foundational questions. Is there always one and only one answer to the question of an argument's being valid? And closely related, what is the logical structure of an argument, in contrast to its syntax, grammar, and semantics.

In effect, those who claim that the argument in the OP is invalid are inadvertently suggesting that there is more than one way for an argument to be valid - that it is formally valid, in propositional calculus, but that in some other logic it is invalid. In some cases, maintain this goes against some folk's own view as expressed elsewhere.

Or they may be saying that the logical form of the argument is other than that shown by parsing it in propositional calculus. In that case, there would be more than one logical form for even such a simple argument.

The support relation is also notoriously tricky to formalize (given a world full of non-black non-ravens), so there's a lot to say about that. For us, there is logic woven into it though:

"Billy's not at work today."
"How do you know?"
"I saw him at the pharmacy, waiting for a prescription."

It goes without saying that Billy can't be in two places at once. Is that a question of logic or physics (or even biology)? What's more, the story of why Billy isn't at work should cross paths with the story of how I know he isn't. ("What were you doing at the pharmacy?")

As attached as I've become, in a dilettante-ish way, to the centrality of probability, I'm beginning to suspect a good story (or "narrative" as Isaac would have said) is what we are really looking for. — Srap Tasmaner

Good post. I may have fallen too far behind in this thread, but I don't think we have to choose between logic and physics to explain such an argument. Physics provides us with a particular kind of logic which makes the argument sound.

I want to say that "flows from" or validity in logic is a specific kind of inferential relation and justification. Your story about Billy fulfills that inferential relation, albeit with some tacit premises.

Well, the thing is, deducibility is for math and not much else. — Srap Tasmaner

But why? Given the explanation, can we deduce that Billy is not at work?

I agree that the consequence relation ("follows from") is hard to formalize. Or rather, I think it is impossible to formalize.

(Feel free to ignore this post if the thread has moved too far away from it.)

Candidly, there can't be any sensible doubt that the argument in the OP is valid for formal propositional logic. So in order for those who claim it is invalid to be correct, there must be more than one form of validity, and hence logical pluralism follows.

Candidly, there can't be any sensible doubt that the argument in the OP is valid for formal propositional logic. So in order for those who claim it is invalid to be correct, there must be more than one form of validity, and hence logical pluralism follows. — Banno

That is true, but shouldn't there be a distinction not just between "valid but not sound" but also between "valid but incoherent"?

For example:

If P then not Q
P
not Q

This is valid. It is sound if P and ~ Q are true. Unsound if not.
If P and Q are the same thing such that:

If P then not P
P
Not P

This is valid and not sound, but also not coherent.

As in, "If I went to the store, I did not go to the store, and I went to the store, so I did not go to the store." That is valid, but meaningless. I have no idea what you did, whether you went to the store, didn't go to the store, and I can't understand how your going to the store made you not go to the store."

And that was the debate for 20 pages I suppose. The pluralism might not be over "validity" if you wish to protect that term to only reference formal structure, but perhaps over soundness if you want to speak of what synthetically is false versus what is analytically false.

This conversation is pedantic and legalistic if I'm understanding it correctly. We all can agree with what truth tables show and what logic dictates, but the battle might be over terms, but I might misunderstand because that was the extent of my disagreement.

The incoherently true statement is also distinct from the vacuously true statement. As in, "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia." There the antecedent cannot ever be satisfied, so it can never be true, but it's impediment to truth is due to a synthetic falsehood, but that's unlike the OP where the antecedent is premised to be false.

I'll let you guys better explain it to me if I've misunderstood this, but the contradiction and the incoherence that follows is what trips this issue up to me at least.

If P then not P
P
Not P

This is valid and not sound, but also not coherent. — Hanover

It would help if you provided a definition of 'coherent' such that its a matter of form alone.

We do have the definition per form alone of 'inconsistent' (in sentential logic, both equivalent with unsatisfiable, and reducible to per form alone).

The set of premises of the above argument is inconsistent.

/

(0) An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.

Ways (0) holds:

(1) The set of premises is not satisfiable and the conclusion is logically true.

(2) The set of premises is not satisfiable and the conclusion is contingent.

(3) The set of premises is not satisfiable and the conclusion is logically false.

(4) The set of premises is satisfiable and the conclusion is logically true.

(5) The set of premises is satisfiable and the conclusion is contingent, but there is no assignment in which all the premises are true and the conclusion is false.

(6) Every member of the set of premises is logically true and the conclusion is logically true.

Ways (0) does not hold:

(7) The set of premises is satisfiable and there is an interpretation in which all the premises are true and the conclusion is false.

(8) Every member of the set of premises is logically true and the conclusion is not logically true.

/

We could coin the word 'revonah' (suggesting the opposite of what Hanover likes), and say:

An argument is revonah if and only if its set of premises is not satisfiable.

(1), (2) and (3) are revonah.

If I went to the store, I did not go to the store, and I went to the store, so I did not go to the store." That is valid, but meaningless. I have no idea what you did, whether you went to the store, didn't go to the store, and I can't understand how your going to the store made you not go to the store." — Hanover

Again, valid/invalid in ordinary formal logic pertain to the entailment relation. Indeed, it would be foolish to look for information about the truth of the premises and conclusion merely from consider of validity, except to see that there are no interpretations in which all the premises are true and the conclusion is false.

But, of course, one may hold that the world 'valid' should not be used if it doesn't comport with certain everyday and philosophical senses, though, personally, I understand the notion in ordinary formal logic and allow that words have different special senses in various fields of study.

The incoherently true statement is also distinct from the vacuously true statement. As in, "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia." There the antecedent cannot ever be satisfied — Hanover

The conditional is vacuously true in all interpretations in which 'Tokyo is in Spain' is false. But it is not the case that 'Tokyo is in Spain' is false in all interpretations.

if I've misunderstood this — Hanover

You might understand if you read an introductory textbook in formal logic. You wouldn't have to accept the material, but at least you would see how it operates.

Better after the edit.

Tones made the response I would have - what is "coherent"? The argument is coherent, in so far as it is consistent with propositional logic.

So again, you seem to want two types of validity, one formal and the other informal.

As in, "If I went to the store, I did not go to the store, and I went to the store, so I did not go to the store." That is valid, but meaningless. — Hanover

It isn't meaningless. We have an idea of what it would be to go to the store, and not to go to the store. Yep, you can't do both.

What would it be for an argument not to be "coherent", in your terms? Do you just mean "valid yet unsound"? I'm not seeing what the introduction of "coherent" adds.

I'm not happy with my response to the question of distinguishing between formal and informal languages.

Your challenge could be taken as: Provide a definition such that any language is exactly one of: formal and informal.

(1) I chose the attribute of having a recursive set of formulas ('formula in the sense of 'well formed formula' in logic). But that I think it should be more general: the set of well formed expressions is a recursive set.

(2) What about formal/informal blends?

(3) Even with ordinary formal languages for logic, there may be other considerations that are required to hold for formality other than that the set of expressions is recursive. (Especially the notion of 'an effectivized language'.)

(4) Other complications.

there can't be any sensible doubt that the argument in the OP is valid for formal propositional logic. So in order for those who claim it is invalid to be correct, there must be more than one form of validity, and hence logical pluralism follows. — Banno

That argument doesn't seem for me to work.

A logical monist could say that certain supposed laws of entailment are not correct and thus not laws of logic. It doesn't follow that the monist would be in contradiction if she also said that there are certain laws of entailment that are the only correct laws of logic.

That is, it doesn't seem to me that in denying that certain supposed laws are correct one has to agree that that there are different competing sets of laws that are all correct.

what is the logical structure of an argument, in contrast to its syntax, grammar, and semantics. — Banno

I don't know what you mean.

The most basic "structure" is that an argument is an ordered pair, with the first coordinate being a set of sentences and the second coordinate being a sentence.

Another way: An argument is a non-empty set of sentences with exactly one of the members designated as the conclusion.

Your challenge could be taken as: Provide a definition such that any language is exactly one of: formal and informal. — TonesInDeepFreeze

Well expressed; and my hunch is that we cannot provide any such clear cut distinction. So we might stipulate that formal languages are those with recursive formation rules. I can imagine a contrarian logician developing a system that undermined some aspect of that - perhaps, by some novelty, having an uncountable number of formation rules, or some such oddity.

This speculation came about after struggling with two SEP articles. The first was Logical Consequence, which I read with a view to trying to get a handle on what the recent thinking is on what it is for a conclusion to follow from a premise. This led me to the article on Logical Form, were I ran afoul of differentiating Syntax, grammar and semantics. I might have been clearer if I had, after that article, asked what logical structure is.

An argument is a non-empty set of sentences with exactly one of the members designated as the conclusion. — TonesInDeepFreeze

Perhaps one might ask, is that designation arbitrary? Why this sentence rather that that one? Is there more, such that the designated sentence is in addition a Logical Consequence (whatever that is) of the others?

An example: supose we have the sentences {p, q, r} and designate r as the conclusion. Is that an argument, or is there something more, such that in addition, r is the "logical consequence" of {p.q}? This seems to be the sort of thing that relevant logic is chasing. If we have {"Sydney is in Australia", "Some swans are black", "The cat is on the mat"} and designate "The cat is on the mat" as the conclusion, do we then have an argument, albeit a very bad one? Or is there more to an argument than just a grouping of sentences with one designated as the conclusion?

my hunch is that we cannot provide any such clear cut distinction — Banno

My hunch is that we can; but my amended attempt might not be satisfactory.

uncountable number of formation rules — Banno

Even just an uncountable set of symbols knocks it out of being formal. For example, in logic, we can have languages with uncountably many symbols (and useful to have for certain purposes, such as in model theory to derive a model upon which to base non-standard analysis), but such a language is not considered formal, since there's no such thing as an uncountable recursive set.

Perhaps one might ask, is that designation arbitrary? — Banno

Yes. It should be.

supose we have the sentences {p, q, r} and designate r as the conclusion. Is that an argument, or is there something more, such that in addition, r is the "logical consequence" of {p.q}? — Banno

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

relevant logic — Banno

Don't know how it goes specifically with relevance logic. But my guess is that even in relevance logic, 'argument' would not mean just valid argument.

I take the idea as being as general as possible: The one thing all arguments have in common is having a set of premises and a conclusion. (Sometimes a set of premises and a non-empty set of conclusions*.) Then we find definitions of various notions of validity: whether classical, intuitionistic, relevance, multi-value, etc.

* But I've heard of a notion in which the set of conclusions could be empty.

If P then not P
P
Not P

This is valid and not sound, but also not coherent. — Hanover

P → ¬P
∴ ¬P ∨ ¬P
∴ ¬P
P
∴ ¬P

Or more simply:

¬P
P
∴ ¬P

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

— Leontiskos

Given the explanation, can we deduce that Billy is not at work? — Leontiskos

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.

There is deduction in math and logic; everyone else has to make do with induction, abduction, probability. — Srap Tasmaner

That is -- making shit up and then seeing if it works(and finding that it usually does not). Though in school I call it "Guess and check"

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," we can then say we have (1) valid and coherent arguments and (2) valid and incoherent arguments.

We can also have (3) valid and sound arguments and (4) valid and unsound arguments.

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.

(Venn diagram is: 3 is a circle within the 1 circle and 2 is a circle within a 4 circle).

The OP is a 2, but not all 2s are a 4, so just calling it valid but unsound doesn't capture its special class.

Maybe we should could call 2s a "NotAristotle" after the creator of this thread. Or, is there already another name for 2s.

Disagreement with what I've said here?

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

OP's argument is (3), and is an example of the principle of explosion.

Then all 3s imply that 3 is a 4.

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

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.