(1) What is the meaning of the conditional?

(2) A set of premises can prove more than one conclusion. So what is "the" conclusion that "should be"?

(3) I wrote: "What are some examples of your rules that are communicated differently?"

I meant: What are some examples of your rules that are communicated clearly?

(4) How do you know what the speaker intended? What if there is not a particular speaker? People disagree about what speakers intend often. And people misunderstand and disagree as to what was said often. You say your answer goes a long way to making clear what the meanings are. Well, yes, often it's pretty clear what a speaker intends, but not so often that it would determine which arguments are valid, since too often it is quite unclear what was intended.

(5) Yes, if the expression is equivocal, then either we reject it of choose one of the candidates for its meaning. But rather how would we determine in an objective way whether there is or is not equivocation and, if so, which candidate to choose?

Formal logic does not presume to know how sort out many of the difficulties in everyday speech; only that if we are given sentences that have a formal relation, we can determine validity of arguments. That is the very point. Suppose someone writes words from a language I don't know:

tarabalu bock meras dan pelrere bosoundo tam.
erofereht, pelrere bosoundo garom

As long as I know that 'dan' means 'and' and 'erofereht' means 'therefore', I can say, "On the assumption that the foreign language expressions are declarative statements, then the argument is modus ponens, and, as an instance of modus ponens, it's valid".

That is, if I know what are the connectives and how we reckon the truth of compound statents, I can tell you about the validity of the argument.

But you have not given such forms, but rather, we have to know the meanings first before determining validity.

So, your offer requires sorting out all the problems about 'meaning' much more than the formal method does.

(6) What definition of 'formal logic' have I given that contradicts any other definition I've given?

(7) An argument form is truth preserving if and only if, for every instance of the form, there is no assignment of truth values such that the premises are all true and the conclusion is false. That is, mutatis mutandis, the same definiens as for 'valid argument'.

(8) Yes, you defined validity as rule following, but then you defined proper rule following as being truth preserving. That's your circularity. You say, a rule is proper only if it is truth preserving, and an argument is valid only if it uses only proper rules. But truth-preservation is validity.

relevant rule is correctly followed just in case.. if it were the case that all the premises were true and the relevant rule is followed, then the conclusion must also be true. — NotAristotle

You say a relevant rule is on such that if all the premises are true then the conclusion is true.

That is the ordinary definition of 'valid' in formal logic.

And it doesn't say anything about meaning other than truth and falsehood.

So what are you adding to the ordinary formal definition or how are you disagreeing with it?

You're circular. You say that an argument is valid only if the rules used are relevant, but also rules are relevant only if they are truth-preserving. But truth-preserving is the same as valid.

Relevant rules like conditionals "And" "Or" operators-- when those are used correctly the rules are followed and the argument may be considered valid. Any rule that is such that if it weren't followed, the conclusion would be different, is a relevant rule. The rules would ideally be universal and based on logical intuition; if people use different sets of rules, then the rules must be clearly communicated so that that "logic" can be understood or followed.

The meaning of the premise and conclusion depends on the expressions used (I guess this definition isn't unequivocal as it would only apply to ordinary natural language, not to formal logic). I don't know any theories of meaning so I can't answer that. If the meanings differ, then I'm not really sure what the result would be, seems like communication is out the door let alone logic if we can't agree on the same meaning of words and sentences. — NotAristotle

What is an example of rule that if it weren't followed then the conclusion would be different? Different from what?

The rules of formal systems in mathematical logic and computing are not just clearly communicated, but they can be checked algorithmically for correctness of application.

What are some examples of your rules that are communicated differently?

What if two people both like the same rules, but have different intuitions as to whether they're being correctly applied?

How are your rules for your propositional logic different from those in ordinary formal logic?

Of course, the meanings depend on the expression. But I'm asking, given an expression, what determines its meaning, or its meanings?

People disagree about meanings often. We can't do logic with expressions because people disagree about their meanings?

What relevant rules? What makes a rule relevant? Whose rules? What if people use different sets of rules from one another? What if the rules are unclear or ambiguous?

What is the meaning of a sentence? How do we unequivocally, let alone objectively, determine the meaning of sentences? What theory of meaning? What if people take different meanings of sentences from one another? What if someone takes 'valid' to mean causal connection but they don't take causality and meaning to be the same? What about people who consider 'valid' to require that all the premises are true?

Of senses of 'valid' different from yours, are they wrong? Or can there be different reasonable senses of 'valid'? The ordinary formal sense cannot be among those different reasonable senses?

They don't imply a definition of validity.

If you read chapter one, you'll understand that we have:

(1) a definition of 'valid argument'

(2) a definition of 'interpretation' such that an interpretation assigns truth values to sentence letters

(3) a stipulation for how the truth value of a compound sentence is reckoned per an interpretation, so that for any interpretation and any compound sentence, we can reckon the truth value of that sentence.

(4) To determine whether an argument is valid or not, apply (1), (2), (3).

And, if I recall, I even showed in this thread how to set that up with truth tables.

/

In a nutshell: I defined 'valid argument'. And earlier in this thread, I defined 'true in an interpretation' for compound sentences. So apply the definition of 'valid argument' by considering the truth values of the compound sentences per each of the interpretations.

Meanwhile, do you have any thoughts about offering your own unequivocal definition of 'valid argument'?

I was re-composing my post while you were posting your reply.

Would you care to formalize the validity definition as it concerns arguments and do so using logical operators? — NotAristotle

Your question is answered by looking at the method of truth tables.

That is in chapter 1 of any book in 'Logic 1'.

I can't see how it could matter if we designated a name for that special class of modus ponens described in the OP, where it is structurally consistent with modus ponens but is logically inconsistent. This thread strikes me as more of a primer in formal logic nomenclature than in logic qua logic. — Hanover

Whatever "structurally consistent" means there, a clear and simple way to say it is: The argument is and instance of modus ponens.

And no instance of modus ponens is inconsistent. What are consistent or inconsistent are sets of sentences. What is inconsistent in the argument is the set of premises.

This thread strikes me as more of a primer in formal logic nomenclature than in logic qua logic. — Hanover

This thread strikes me as having posters in it that are commenting on formal logic while knowing virtually nothing about it.

The original thread question would naturally be taken by many people to pertain to ordinary formal logic. But it can also be taken to pertain to other informal contexts, including everyday speaking and reasoning, and also can be discussed in context of alternative formal logics.

But when answers were given in terms of ordinary formal logic, certain posters commented as to the formal logic, while knowing virtually nothingabout it. Thus, correct explanations of it are exemplary meaningful, informative and generous posting.

I've said it maybe fifty times in this forum: Ordinary formal logic with its material conditional does not pertain to all contexts. But that is not a basis that one should not say how ordinary formal logic handles a question and not a basis that one should not explain ordinary formal logic to people who are talking about it without knowing about it.

Or even if just one (but not all) of the premises is false and the conclusion is false (I am having trouble thinking of an example that meets this description). — NotAristotle

Having a false premise and a false conclusion does not in and of itself make an argument invalid. You have forgotten or did not understand the definition.

there may be two senses of the term "valid" in a logical context; one formal, the other informal and that evaluating an argument with either definition may cause different conclusions as to whether a given argument is valid. — NotAristotle

Of course.

But the ordinary formal definition is itself not equivocal. It is definite. It gives an 'if and only if' with a definiens in which all the terms are themselves defined back to primitive rubric.

Meanwhile, there are other formal definitions that differ from the ordinary formal definition. And they may also be not equivocal, though probably more complicated than the ordinary definition.

Meanwhile, there are different informal senses. If there is one in particular that you propose as being definite enough to avoid the kind of subjectivity equivocation in everyday discourse, then you're welcome to state it.

Therefore
NOT A is true, and A refers to nothing. — sime

Where can one read an account of ordinary modal logic, ordinary intuitionistic logic or basic Kripke semantics in which that is the case?

I am not clear on how A -> not-A "makes sense" if A is true. — NotAristotle

It makes sense in the sense of having a truth value.

an argument where all the premises are false and the conclusion is false would necessarily be valid; is that correct? — NotAristotle

No, quite incorrect. Egregiously incorrect. That you say that shows that you haven't paid attention to the numerous explanations given in this thread, let alone that you haven't paid attention to the most basic articles available on this subject.

I was thinking of:

P->not-Q
not-P
Therefore,
not-Q.

Assuming that all the premises are false and the conclusion is false, the argument must be valid. Is that correct? — NotAristotle

Not correct at all. It goes exactly against the definition of 'valid'.

In a consistent deductive system , If the sign "Not A" is either taken to be an axiom, or is inferred as a theorem, then it means that the sign "A" is non-referring — sime

In a consistent deductive system , If the sign "Not A" is either taken to be an axiom, or is inferred as a theorem, then it means that the sign "A" is non-referring and hence meaningless in that it fails to denote any element of any possible world among any set of possible worlds that constitutes a model of the axioms. By symmetry, the same could be said of the sign "Not A" being meaningless if A is taken as an axiom, but by model-theoretic traditional the sign A is said to not denote anything in a model if ~A is provable.

For instance, let the sign "A" denote the proposition that the weather is wet in some possible world. If "A" is deductively assumed or proved, then A is a tautology, meaning that the logical interpretation of "A" is stronger than being a mere possibility and denotes the weather being wet in all possible worlds. — sime

(1) You say "in a consistent deductive system" but your remarks wouldn't apply to ordinary sentential or predicate systems, but rather, more specifically to modal systems. So, your remarks don't obtain as to deductive systems in general.

(2) With ordinary models for modal propositional logic, sentence letters themselves are members/not-members in worlds. But that ~A is an axiom or theorem doesn't entail that A is meaningless in any given model. Rather, as in classical semantics (but by more complicated considerations) A is false if and only if ~A is true.

(3) I don't know your definition of 'tautology' in modal logic. In propositional logic, a sentence is a tautology if and only if it is true in all models. I am not familiar with a notion in modal logic that being assumed makes the sentence a tautology.

(4) You said [paraphrase:] A stands for "wet in some world", then assuming A yields "wet in all worlds". That would be (where 'p' for possibly and 'n' for necessarily):

pA -> nA

And that is not generally (if at all) considered a validity.

[EDIT:] By the way, (A -> ~A) -> ~A is intuitionistically valid, perforce so is ((A -> ~A) & A) -> ~A.

As the argument forms are intuitionistically valid:

{A -> ~A}, therefore ~A, perforce {A -> ~A, A}, therefore ~A.

Classically by classical models; intuitionistically by intuitionistic models.

definition of negation in intuitionistic logic. — sime

Yes:

Df. ~P stands for P -> f

where 'f' is primitive.

But, just to note, that can be a definition in classical logic too.

So why do we accept as logically valid a premisse that will result in a logical contradiction under one value of the antecedent? — Benkei

You're mixed up as you don't know the basic concepts. Reading just a little in a textbook in the subject would help you.

No sentence that proves a contradiction is valid. And no set of sentences that proves a contradiction is satisfiable.

And a sentence or set of sentences proves a contradiction or does not prove a contradiction irrespective of any assignment of truth values. That is, if a sentence proves a contradiction, then there is simply no assignment in which that sentence is true. And if a set of sentences proves a contradiction, then there is no assignment in which all the members of the set are true.

What would be the implications if we would say for any given argument under all values of the antecedent the conclusion may not result in a logical contradiction or the argument will be deemed invalid? — Benkei

Again, if a set of premises proves a contradiction, then there is no assignment in which all the premises are true.

But, I guess what you mean is this:

Consider all and only those arguments in which the conclusion is not contradictory.

Okay, say an argument is an N-argument if and only if its conclusion is not contradictory. And say an argument is an S-argument if and only if its set of premises is satisfiable.

So "what happens"?

Any argument is an N-argument if and only if it's an S-argument.

I would expect any statement to be logically consistent under all values of the antecedent. — Benkei

I don't know what that is supposed to mean.

An antecedent is the part of a conditional that comes before '->'.

The argument under discussion:

A -> ~A
A
therefore, ~A

The only conditional there is:

A -> ~A

Its antecedent is:

A

There are two values we can assign to A: true, false

If A is false, then A -> ~A is true.

If A is true, then A -> ~A is false.

But A -> ~A is consistent in either case. It does not prove a contradiction, and it is satisfiable, since there is an assignment in which it is true, viz. the assignment that assigns false to A.

"The fact that logical inference ignores it because under one of the values of the antecedent it does make sense"

Ignores what?

A -> ~A doesn't make sense to you. But we didn't say A -> ~A makes sense only when A is false. A -> ~A makes sense whether A is true or A is false.

Your post is hopelessly confused because you don't know the basic concepts.

If you would just read a little bit in an introductory textbook, in print or online, you would know.

All my remarks pertain to ordinary usage such as found in basic textbooks:

There is no such thing as a "derivable argument" or "underivable argument". The expressions "derivable argument" and "underivable argument' make no sense.

An argument is a pair, with the first component called 'the set of premises', and the second component called 'the conclusion'. An argument is valid or invalid, and sound or not sound. There is no such thing as an argument being "derivable" or "underivable".

Derivability pertains to proof. An argument is not itself a proof. An argument is a pair, with the first component called 'the set of premises' and the second component called 'the conclusion'.

What is underivable or not, is a conclusion from a set of premises. And that pertains to proof.

(1) Example:

set of premises:
{P, Q, R -> S, ~(P & R)}

conclusion:
S

That is an argument (it happens to be invalid). Just a set of premises and a conclusion. It's not a proof.

(2) Example:

set of premises:
{Q -> R, S, P, P -> Q}

conclusion:
R

That is an argument (it happens to be valid). Just a set of premises and a conclusion. It's not a proof.


A proof (in Hilbert form) is a sequence of sentences such that each sentence is a premise or follows by an inference rule from previous sentences in the sequence:

(3) For example (with the applications of the rules annotated):

1. P -> Q (premise)
2. Q -> R (premise)
3. P (premise)
4. Q (from 3, 1 by modus ponens)
5. R (from 2, 4 by modus ponens

That is a proof. It's a proof of R from the set of premises {P, P -> Q, Q -> R}. From that proof, we establish that the argument (2) is valid, since the proof has premises only from those of (2) (we didn't need to use the premise S, by the way) and the last line of the proof is the conclusion of (2).

You would do a lot better not to reference me by saying that you believe that what you posted is what I am looking for.

Elementary calculus does not require "actual" infinities. — jgill

Calculus uses infinite sets on day one. Even before a student gets to calculus, with analytical geometry we're using infinite sets. The real line and the real plane are infinite sets.

But it seems you mean that calculus doesn't usually mention transfinite ordinals (though the set of natural numbers is a transfinite ordinal), which is true.

I have used R, but not a transfinite number. Unless I occasionally use the "point at infinity" in complex analysis. — jgill

Points at infinity are not required to be transfinite numbers.

There's an important distinction between handwaving and BS. Handwaving involves vagueness or imprecision, where the core idea might be sound but lacks detail or rigor in its current form. BS, on the other hand, is fundamentally incorrect—an argument that doesn't hold up under scrutiny and lacks substance from the start. — keystone

That is BS. Handwaving to the extent you execute it is very much BS. The category of BS includes both falsehood and nonsense. EDIT: And the falsehood coming from you is the pretense that you're providing grounded definitions and that you're not resorting to infinite sets. Just saying [paraphrase:] "Oh, it's not really infinite, I'm talking about the process not the object" is just abracadabra BS. If you were sincere and credible then you'd state the algorithms as actual algorithms and not resort to infinite sets such as circles. Meanwhile, there is a ton of work that's been done in constructive mathematics and other alternatives, but you don't take lesson from it as you march head forward to glory.

I'm not working with Cauchy sequences themselves, but with the algorithm used to construct any arbitrary term. — keystone

Your k-BS is "working" with still undefined terminology with your special k-BS flavor of handwaving.

take issue with using transfinite numbers to describe actual abstract objects — keystone

Transfinite cardinals and ordinals are infinite sets.

cardinality of aleph_0 — keystone

aleph_0 is an infinite set. Of course, there is also an algorithm that, at any stage, outputs up to any natural number, so that for any natural number, if the algorithm runs long enough, that natural number will be outputted, but you mention "transfinite numbers" plural, so let's look at the next one, viz. aleph_1. There is no algorithm for aleph_1 such as for aleph_0, and same for even greater cardinals. You don't need to wave transfinite cardinals around as if they have anything to do with k-BS.

I believe this post aligns with the kind of response that TonesInDeepFreeze [... ] [was] looking for in this thread and in our previous thread — keystone

You believe incorrectly, and it would be better that you not wiggle in suggestions such as that your k-BS "aligns" with what I am "looking for".

I took you at face value and mistakenly entertained the idea that you could be in good faith with all the business about graphs. But then, to complete your definition, you invoked the existence of circles. But circles are infinite sets and you provided no justification for invoking them, completely out of the blue. It turned out, yet again, that indulging you is a dead end in a sinkhole - an insult to the time and thought I wasted in this and other threads with you.

real calculus is inseparably tied to 2^aleph_0 — keystone

Who needs for you to "argue" that? We already know it.



On and on with yet more pseudo math k-BS definitions when you never finished the very first definition you started.

Isn't anything communicated with absolute precision a bit mind-numbing? — keystone

Get a load of you, bestowing yourself with "absolute precision". Your k-BS is as much absolute precision as a mound of moldy fish gut is a sashimi delicacy.

Sometimes the significance of a discovery isn't recognized until many years later. — keystone

Yes, your Fields Medal is just a matter of time. Just have to get all those stuck-in-the-mud, imagine-less, establishment math fools to come around to open their eyes to your brilliance.

Okay, but I can actually see how the edited conditional could be true. For instance, if Michael is a really great citizen, then maybe he would end up being President were he American, if so, then in the ordinary sense, the sentence can be "true" based on what it means. — NotAristotle

What is the edited conditional?

The conditional is "If Michael is American then Michael is president".

That conditional is true since "Michael is American" is false. There's no need to mention any aspect of greatness of citizenship, nor "maybe", nor "end up".

I would prefer there not to be equivocal definitions of validity, it appears that there are, one formal, the other informal. — NotAristotle

The word 'valid' is equivocal.

There are different definitions and understandings of the word 'valid'.

One of the formal definitions of 'valid' is a common one. That definition is not equivocal.

Meanwhile, among the everyday senses of 'valid', which has a definition that is not equivocal?

I do understand the difference between taking 'true' as defined and taking 'true' as primitive.

Davidson against Wittgenstein. — Banno

How do you state that contest?

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.

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?

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.

I get that joke. Thank you.

"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).

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'.

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.

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)

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.

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).

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.

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.

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.

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.

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.