a notion of "follows from," — Leontiskos

I sympathize. I think a lot of our judgments rely on what I believe @Count Timothy von Icarus mentioned earlier under the (now somewhat unfortunate) heading "material logic", distinguished from formal logic.

A classic example is color exclusion.

When you judge that if the ball is red then it's not white ― well, to most people that feels a little more like a logical point than, say, something you learn empirically, as if you might find one day that things can be two different colors. (Insert whatever ceteris paribus you need to.)

Wittgenstein would no doubt say this comes down to understanding the grammar of color terms. (He talked about color on and off for decades, right up until the end of his life.)

Well, what do we say here ― leaving aside whether color exclusion is a tenable example? What you're after is a more robust relationship between premises and conclusions, something more like grasping why it being the case that P, in the real world, brings about Q being the case, in the real world, and then just representing that as 'P ⇒ Q' or whatever. Not just a matter of truth-values, but of an intimate connection between the conditions that 'P' and 'Q' are used to represent. Yes?

Well, what do we say here ― leaving aside whether color exclusion is a tenable example? What you're after is a more robust relationship between premises and conclusions, something more like grasping why it being the case that P, in the real world, brings about Q being the case, in the real world, and then just representing that as 'P ⇒ Q' or whatever. Not just a matter of truth-values, but of an intimate connection between the conditions that 'P' and 'Q' are used to represent. Yes? — Srap Tasmaner

These are interesting topics that Aristotle also takes up, but I don't think I'm being overly greedy in what I desire. I am not requiring a special kind of aitia/account/explanation. Here is what I said above:

Validity is a relationship between premises and conclusion. This is what I say is the common interpretation of your sources on validity:

1. Assume all the premises are true
2. See if it is inferentially possible to make the conclusion false, given the true premises
3. If it is not possible, then the argument is valid

...

...validity is an inferential relationship between premises and conclusion. — Leontiskos

As Enderton notes, validity is about deducibility. It is not merely about truth values. It is about the inferential relationship between premises and conclusion. In order to show that Q follows from P, we have to show how Q is correctly inferred from P, and we need to have evidence that ~Q cannot also be inferred from P.

A key contention of mine is that I am representing the notion of validity in formal logic better than Tones is. I don't even need to advert to real-world cases, like that of color. Even within propositional logic itself, validity has to do with "follows from" and deducibility.

and @Hanover, and @Banno, and @all participants to this thread,

I was hoping this thread would be a discussion investigating deduction, implication, and validity. I am thankful that that is what everyone is discussing and other topics. I wish I had more to add to the discussion, but I am not as well-versed in logic. I have learned what I think is a strong definition of validity, which TonesinDeepFreeze stated earlier in the thread. I encourage respectful discussion of these topics by all parties.

I encourage respectful discussion of these topics by all parties. — NotAristotle

Good lad.

I have learned — NotAristotle

Even better.

Similar to what I said earlier about the genus of discourse, some arguments are apparently neither valid nor invalid:

Now the question arises: is it invalid? I don't claim that. — Leontiskos

Probably they are not "arguments" at all.

To give another example using Srap's color idea:

• Everything which is not white contains pigment
• Numbers are not white
• Therefore, numbers contain pigment

That is the sort of thing that is occurring when one tries to claim that any argument with inconsistent premises is trivially valid. The domain of discourse when speaking about validity is arguments, and arguments do not contain premises that are known to be inconsistent. Some arguments have premises that are inconsistent but are not known to be inconsistent, and that is where reductio comes in. Are these latter kind truly arguments? Not in any perfect or ideal sense, but they are in the sense that the arguer believes the premises to be consistent.

What you're after is a more robust relationship between premises and conclusions, something more like grasping why it being the case that P, in the real world, brings about Q being the case, in the real world, and then just representing that as 'P ⇒ Q' or whatever. Not just a matter of truth-values, but of an intimate connection between the conditions that 'P' and 'Q' are used to represent. Yes? — Srap Tasmaner

...And I want to say that an argument is supposed to answer the "why" of a conclusion. Inferential argumentation is an explanation for a proposition/conclusion. Validity is one aspect of the goodness of such an explanation.

validity is about deducibility — Leontiskos

I don't even need to advert to real-world cases — Leontiskos

Well, the thing is, deducibility is for math and not much else. That's the point of my story about George, and my general view that logic is ― kinda anyway ― a special case of the probability calculus.

an argument is supposed to answer the "why" of a conclusion — Leontiskos

I agree with this in spirit, I absolutely do. I frequently use the analogy of good proofs and bad proofs in mathematics: both show that the conclusion is true, but a good proof shows why.

I'll add another point: when you say something another does not know to be false but that they are disinclined to believe, they will ask, "How do you know?" You are then supposed to provide support or evidence for what you are saying.

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.

I think it shows that 'not-A' has at least two different senses. The world is not as neat as formal logic. Formal logic may not be as neat as it might be thought to be either.

I don't claim to have academic definitions of 'univocal' and 'equivocal', but at a naive level, as I'm merely winging it here, it seems to me that:

'totally univocal' is redundant. An expression is univocal if and only if it has one meaning. That's total.

'totally equivocal' is hard to conceive. An expression is equivocal if and only if it has more than one meaning. What would it mean to say it is totally equivocal?

Yes, that's a common view today. Analogy is a difficulty for logic. The move towards the univocity of being in the late medieval nominalist period (important for theology, but also for how the rest of philosophy developed) was largely born out of a period in scholasticism that was intensely focused on logic (perhaps analogous to early analytic philosophy). Yet if one wants to develop a metaphysics that avoids atomism or nominalism, it might end up being quite important to have analogy as an option.

The most obvious example where this comes out is something like:
"The shot that Lee Harvey Oswald took that killed Kennedy was a good shot."

We can say this is true, because it was something that only a "good marksmen" could regularly accomplish. But, unless we really don't like Kennedy, we would not say this is "good" in a moral sense. And yet, if we are forced to claim that the "goodness" of things like "good food" and "being a good basketball player" or "being a good teacher," have nothing to do with moral goodness, I would argue that we effectively isolate moral goodness. I don't think "castrating the Good" would be too strong a term here. By my reckoning, this change in philosophy seems to be directly responsible for the descent into emotivism in ethics, until we reached a place where Moore is forced to argue that "goodness" is just a "non-natural" quality that "just is."

Not sure what you mean by this.

I guess my first thought was essentially agreement. A syllogism with two negative premises is not valid. Making the premises inconsistent doesn't seem like it should change this.

However, I understand how the claim that "if it is impossible for all the premises to be true while the conclusion is false, then an argument is valid," flows with the idea of validity as truth preservation. And I won't deny that you can find this definition in some logic textbooks. It seems like something akin to the paradoxes of material implication in terms of the "smell test" at first glance though.

Tones' argument:

An argument is valid when it is not possible for the conclusion to be false while the premises are true
An argument with contradictory/inconsistent premises cannot have (all) true premises
Therefore, an argument with contradictory/inconsistent premises cannot have a false conclusion while the premises are true
Therefore, an argument with contradictory/inconsistent premises is valid. — Leontiskos

That is equivalent with my argument. But my argument did not mention consistency.

There is a conceptual reason for that. Though, it is not incorrect to mix semantical and syntactical considerations, I prefer the clarity of keeping it to only one of them in this definition. (And of course, there are other times when we do want to use both semantical and syntactical considerations, especially as they relate to one another.)

Tones is talking about assignment or inconsistency, not necessary falseness. — Leontiskos

Again, I used the notions of interpretation (which involves truth and falsehood), not the notion of inconsistency.

This is what I say is the common interpretation of your sources on validity:

1. Assume all the premises are true
2. See if it is inferentially possible to make the conclusion false, given the true premises
3. If it is not possible, then the argument is valid — Leontiskos

Perhaps I've overlooked, but I don't recall any of my cites saying "assume", "see" and "make" - verbs.

You've reinterpreted the definition for your tendentious purpose.

Your interpretation — Leontiskos

My interpretation is literal in an example such as Mates, and virtually literal in certain others, and equivalent with the rest of them.

Your tendentious interpretation is quite a departure as it imposes a routine to be carried out, described with a series of verbs.

The cited definitions don't mention routines to be carried out.

Put my wording next to the cites. Put your interpretation next to the cites. See that yours is nowhere near as close as mine.

Your interpretation changes the ordering of the conjunction and condition — Leontiskos

(1) No, it doesn't. (2) Even if it did, it would be okay as long as the definition were equivalent.

You want to say that if we cannot assume that all the premises are true (on pain of contradiction), then the argument is valid by default. — Leontiskos

I didn't say anything about anybody assuming anything.

we don't need an additional implication operator ― that is, one that might appear in a premise, say, and another for when we make an inference. — Srap Tasmaner

No, I do distinguish between what is object-language and what is meta-language.

'->' is in the object language. Or if the object-language is English, then 'if then' is in the object language. And in ordinary formal logic, that is the material conditional.

In the meta-language, we also use 'if then'. I put it here in italics to distinguish from 'if then' in the object-language. But still, also the meta-language 'if then' is the material conditional.

In ordinary formal logic, writers don't stop regarding 'if then' as the material conditional (if then) just because it occurs in the meta-language.

For example:

(2) If the sentence "P -> Q" is false, then P is true and Q is false.

'->'is in the object-language, and 'if then' is in the meta-language. But both of them are the material conditional.

(2) If the sentence "If John went to the store, then John got bread" is false, then "John went to the store" is true and "John got bread" is false.

The outer 'if then' is in the meta-language' and the inner 'if then' is in the object language. But both of them are the material conditional.

I think it shows that 'not-A' has at least two different senses. — Janus

Propositional logic deals in propositions. Your piece has the form of a modus ponens, but doesn't deal in propositions. That makes it interesting in several ways. But "not-a" is pretty well defined in propositional logic, in various equivalent ways. And by that I mean that the things we can do with negation in propositional logic are set. There are not different senses of "not-A" in propositional calculus.

The argument in the OP is for all intents a propositional argument. It is an instance of the application of modus ponens. And it is valid. The thread should have finished at 's post. The subsequent discussion displays ignorance of basic logic rather than any failing of that logic.

1.Life therefore death
2.Life
Therefore
3.Death. — Janus

is not an example of 'not-A', nor of propositional logic, although it is a striking example of the creativity of language.

Formal logic can set out some of the structures we might wish to find in natural languages.

The confused ignorance on display hereabouts might turn folk off looking at logic on detail, or encourage them to think it useless. The lesson from this thread, if there is one, might be that if folk begin by misunderstanding logic, they cannot conclude that logic tells us nothing about language.

Those who pretend to be defending a supposed common sense logic that is incompatible with formal logic are doing a disservice to rationality.

@TonesInDeepFreeze and to a lesser extent @Michael have presented a patient, consistent and correct account of the validity of the argument in the OP in the face of some extraordinary rubbish from folk we might have expected to know better.

A sad thread, this one. A low point in the history of the forums.

I don't know what you mean. Example? — TonesInDeepFreeze

https://en.wikipedia.org/wiki/Proof_by_contradiction

1. A → ¬A
2. A ⊢ ¬A
3. A ⊨ ¬A
4. A ∴ ¬A — Michael

Good list.

(1) P -> Q ... is a sentence that is interpreted as true if and only if either P is interpreted as false or Q is interpreted as true.

Indeed, sometimes '->' is not primitive but is defined:

Df. P -> Q stands for ~P v Q
or
Df. (P -> Q) <-> (~P v Q)

(2) If G is a finite set of formulas and P is a formula:

G |- P means there is a proof of P from G

(3) If G is a set of formulas and P is a formula

G |= P means P is entailed by G. (where entailment is semantic)

(4) G ∴ P might be the same as (2) or the same as (3) depending on the author.

I don't need to read a whole article that I've read before. I'm just wondering whether you'd give a particular example.

I affirm that it is valid by any of these considerations:

(1) Apply the definition of 'valid argument'.
— TonesInDeepFreeze

And that is the option we are talking about, nitpicker.
— Leontiskos

Three options have been given: modus ponens, explosion, and the definition of validity. TonesInDeepFreeze's is the latter, — Leontiskos

That gives the impression that I opt for the latter more than the others. But that is not the case:

I started in the thread by pointing out that the argument is modus ponens. Then I was challenged about that and more about the question of validity came up. Then I adduced the definition of validity and showed that the argument is valid. [EDIT: That's not correct. I started in the thread by both an appeal to the definition of validity and that the argument is an instance of modus ponens, and the fact that the premises are inconsistent does not disqualify the argument from being valid. In any case, whatever approach, yes, finally to show the validity of an argument boils down to showing that the definiens of the definition holds for the argument. But my point stands that it's not like I just chose one of the three options, as in subsequent post I especially stressed modus ponens.]

It is not "nitpicking" that I now mention that, you putting words in my mouth, distorting, confused and clueless about basic formal logic, side stepping, intellectually dishonest, would be conversation controlling, tendentious distraction.

From the post you sidestepped:

Your interpretation is mistaken because validity is an inferential relationship between premises and conclusion. You would establish an inferential relationship without examining the inferential structure and relations. To say, "The premises are contradictory, therefore an inferential relationship between premises and conclusion holds," is to establish an inferential relationship without recourse to inferential relations.
— Leontiskos — Leontiskos

I addressed the matter of a 'relation'. You sidestepped that. And a bunch more that you sidestepped. Including that my definition is virtually the same as Mates and equivalent with the others.

The sources I cited include a notion of "follows from," which obviously excludes Tones' approach of relying on the degenerative case of the material conditional. When A is false (A→B) is true, but B does not follow from A. — Leontiskos

Wrong. As been explained to tedium. And you are seriously confused in distorting me:

When A is false, A -> B is true.

And B follows from {A -> B, A}.

I have not at all taken that to provide that B follows from A.

You're ridiculous.

As Enderton notes, validity is about deducibility. — Leontiskos

He said that validity and deducibility turn out to be equivalent. We easily may define (and many or most authors do) 'valid formula' and 'valid argument' without need to mention deducibility. Indeed, Enderton defines 'valid formula' (if I recall, he doesn't define 'argument' in that book) without mention of deducibility. It is only later that he remarks that validity turns out to be equivalent with deducibility.

I refer only to sentences here, not formulas in general, to keep it simple:

Df. A sentence is valid if and only if it is true per all interpretations and assignments for the variables.

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

Df. A set of sentences G entails a sentence P if and only if there is no interpretation in which all the members of G are true and P is false.

None of those mention derivation (proof, deducibility).

Df. A sentence P is derivable from a set of sentences G, per a set of axioms and a set of inference rules, if and only if there is a sequence of sentences such that each entry is either an axiom or is inferred from previous entries by an inference rule.

Then the key theorems:

Th. If a sentence is derivable from logical axioms alone, then it is valid. (soundness). Equivalently: If a sentence P is derivable from a set of sentences G, then G entails P. (soundness)

Th. If a sentence is valid, then it is derivable from logical axioms alone. (completeness) Equivalently: If a sentence P is entailed from a set of sentences G, then P is derivable from G. (soundness)

So:

Th. A sentence is derivable from logical axioms alone if and only if it is valid. (soundness and completeness). Equivalently: Th. A sentence P is derivable from a set of sentences G if and only if G entails P. (soundness and completeness)

That is what Enderton is referring to.

It is not merely about truth values. — Leontiskos

You, totally cluelessly misconstrue a central matter in logic.

Validity is semantic. It is usually defined with regard to truth values and interpretations. Such definitions do not mention deducibility. In sentential logic, truth tables represent the determination of validity.

Again, what Enderton refers to is the fact that validity and deducibility turn out to be equivalent. But still the definition itself of validity does not require mention of deducibility.

[EDIT: Leontiskos displays typical rank sophistry. He has never read Enderton's book, let alone studied it and understood it. He just cavalierly, unthinkingly picked a quote from it out of context to support his false claim. If he had actually read Enderton, he would see that Enderton's definition does NOT mention deducibility, indeed it is entirely semantic, and that Enderson's point is that it "turns out" that validity and deducibility are equivalent. Enderton didn't say that validity is "about" deducibility. Just as Leontiskos puts words in my mouth, he puts words in Enderton's mouth. Moreover, what Enderton mentioned is just a well known and central proven fact. Anyone familiar with the basics of this subject knows that validity is semantical, deducibility is syntactical, and they have separate definitions, but we prove an equivalence.

Leontiskos also says:

As Enderton notes, validity is about deducibility. It is not merely about truth values. It is about the inferential relationship between premises and conclusion. In order to show that Q follows from P, we have to show how Q is correctly inferred from P, and we need to have evidence that ~Q cannot also be inferred from P. — Leontiskos

That is not what Enderton wrote and not implied by anything Enderton wrote.

(1) Above I addressed the misrepresentation that Enderton wrote that validity is about deducibility.

(2) In order to show that Q is entailed by a set of sentences G (Enderton's terminology is 'logical consequence' rather than 'entailed') it is NOT required to show an inference, especially not an "inferential relationship" (whatever that would mean other than that there is a correct inference) and especially not a requirement to show that ~Q cannot be inferred from G. Rather, it suffices to show that there is no interpretation in which all the members of G are true and Q is false.

It is true that if we show that there is a deduction from G to Q, then Q is entailed by G (that is the soundness theorem). But it is not required that we use that method. We still may use the semantical consideration alone: showing there is no interpretation in which all the members of G are true and Q is false.

And it is true that if G proves Q and G is consistent, then G does not prove ~Q. But it is not true, contrary to Leontiskos's ignorance and tendentious mangling, that, to show that G entails Q, we are required to show that G does not prove ~Q.

Leontiskos is so often in really bad faith when talking about logic. It's fine that he has a different notion of logic, and fine even to critique what he doesn't like, but it is bad faith and destructive to reasoned dialogue that he misrepresents what he critiques and blatantly misrepresents other posters too and then blatantly misrepresents a cite he pulled blindly without reading and understanding the basics of logic to which the cite pertains.]

A key contention of mine is that I am representing the notion of validity in formal logic better than Tones is. — Leontiskos

Hilarious!

It is not valid since there are interpretations in which the premises are true but the conclusion is false.

It is not valid since there are interpretations in which the premises are true but the conclusion is false. — TonesInDeepFreeze

I guess you mean there are interpretations where the sentences are uttered in a context where they could be true. Thanks for your help.

A sad thread, this one. A low point in the history of the forums. — Banno

I don't think so. My experience with logic is with the logic gates that make up a computer's microprocessor. If it's an or-gate, either input goes through, that kind of thing. I never had to worry about validity. :lol:

I guess you mean there are interpretations where the sentences are uttered in a context where they could be true. — frank

No, that is not what I mean.

In a post, I spelled out in detail what an interpretation is.

Ok. Thank you!

"respectful discussion"

Respect includes not intentionally or carelessly putting words in the mouth of a poster, especially after the poster has dropped a flag on it and more than once. Respect includes not intentionally or carelessly seriously mischaracterizing a poster's main point, even to the point of reversing it.

to a lesser extent Michael — Banno

Awww. Do you feel bad now @Michael?

Awww. Do you feel bad now Michael? — Srap Tasmaner

I took it to refer to the word count. :smirk:

Let's just say it was pleasing to learn that at least one of the mods knows a bit about logic.

I don't know what you mean. Example? — TonesInDeepFreeze

https://en.wikipedia.org/wiki/Proof_by_contradiction — Hanover

Not sure how this fits in with the OP. I've used this approach from time to time, but never to the extent of assuming ~P is true and showing P follows. Usually one shows a logical contradiction of sorts short of P being true. But I digress from the conversation, which has long ago become absurd. @Tones clarified the issue way back imo.

Thanks :).



This is a meaty post.

Almost too much for me :D -- one thing that's interesting is your reduction of material implication to set theory. I'm not sure how to understand that, really -- if the moon is made of green cheese then 2 + 2 = 4. That's the paradox, and we have to accept that the implication is true. How is it that the empirical falsehood, which seems to rely upon probablity rather than deductive inference, is contained in "2 + 2 = 4"?

I'm intentionally throwing wrenches/spanners here so kindly tell me to 'ef off if it's uninteresting or simply misinformed. I'm starting to feel the tread in this conversation where I'm in too deep over my head.

Of course, reduction ad absurdum. But how is that "checking the validity of one argument using another"?

https://web.stanford.edu/class/cs103/tools/truth-table-tool/

More information and explanation [aka 'increasing the word count']

I specified exactly what a sentential logic interpretation is. To add to that, here is what is meant by "true (or false) per an interpretation" or "true (or false) in an interpretation":

First we define 'is a sentence' by induction:

Every sentence letter is a sentence (drop parentheses when not needed):

If P is a sentence, then ~P is a sentence.

If P and Q are sentences, then (P & Q) is a sentence.

If P and Q are sentences, then (P v Q) is a sentence.

If P and Q are sentences, then (P -> Q) is a sentence.

If P and Q are sentences, then (P <-> Q) is a sentence.

It is in "stages" ('P' and 'Q' here range over sentences):

Then, an interpretation assigns a truth value to each sentence letter. So a sentence letter alone has, per that interpretation, the truth value assigned by that interpretation ('P' and 'Q' here range over sentences):

If P is just a sentence letter, then P is true per the interpretation if the interpretation assigns true to P; otherwise P is false per the interpretation.

~P is true per the interpretation if P is false per the interpretation; ~P is false per the interpretation otherwise.

P & Q is true per the interpretation if both P and Q are true per the interpretation; P & Q is false per the interpretation otherwise.

P v Q is true per the interpretation if at least one of P or Q is true per the interpretation; P v Q is false per the interpretation otherwise.

P -> Q is true per the interpretation if either P is false per the interpretation or Q is true per the interpretation; P -> Q is false per the interpretation otherwise.

P -> Q is true per the interpretation if either both P and Q are true per the interpretation or both P and Q are false per the interpretation; P <-> Q is false per the interpretation otherwise.

An example:

(P -> Q) v (R & Q)

Suppose the interpretation is:

P ... true
Q ... true
R ... false.

Then:

P -> Q is true per the interpretation
R & Q is false per the interpretation
so, abracadabra, voila, and drumroll please ...
(P -> Q) v (R & Q) is true per the interpretation

Similarly, in stages like that, for arbitrarily complicated sentences.

That's what is meant by 'true (or false) per an interpretation' or 'true (or false) in an interpretation'.

/

Various definitions we've seen mention things like 'cases', 'circumstances'.

Those can be taken to mean 'interpretations'.

And sometimes 'possible' and 'impossible' are used.

Those can be taken to mean 'true in at least one interpretation' and 'true in no interpretation', respectively.

/

This is what I say is the common interpretation of your sources on validity:

1. Assume all the premises are true
2. See if it is inferentially possible to make the conclusion false, given the true premises
3. If it is not possible, then the argument is valid — Leontiskos

Whatever is meant by "See if it is inferentially possible to make the conclusion false, given the true premises", here instead is one* common method for checking for the validity of a sentence in sentential logic. *There are some more efficient ways, but they are harder to specify in a post.

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

To check for the validity of an argument (with finitely many premises):

1. Write the conjunction of the premises. Follow that with '->'. Follow that with the conclusion.

2. Write the truth table for the above formed sentence.

2. If there is a row in which the antecedent is true and the conclusion is false, then the argument is invalid, and it is valid otherwise.

Indeed, this highlights a connection between arguments and conditionals.

No "assuming". No seeing "if it is inferentially possible to make the conclusion false, given the true premises" whatever that means. No messing with the modality of possibility. Indeed, just a simple, utterly clear, step by step mechanical method.

Note: There is no mechanical procedure to check for the validity of arbitrary formulas of predicate logic.

/

I mentioned that I don't mention 'inconsistency' when defining 'valid argument'. There is good reason for that, which is:

The notion of consistency requires the notion of deducibility and deducibility is a whole subject in itself.

Df. A set of sentences is consistent if and only if there is no deduction of a contradiction from the set.

But that requires having a deduction system from which to define 'is a deduction'.

But we may wish to consider validity without having first done all stuff we have to do to set up a deduction system, which we can do later.

Indeed, often textbooks in logic devote early chapters to semantics (truth/falsehood, interpretation, entailment, validity, etc.) and then separate chapters to deduction. And then, chapters in which we prove meta-theorems about the connection between semantics and deduction. Such, as I recently mentioned, the central theorems of soundness and completeness. That is a conceptually elegant approach. Indeed, this engenders two branches of study in logic: model theory (interpretations) and proof theory (deductions).

/

Another definition of 'valid argument' to add to the list:

"it is impossible that all the premises should be true and the conclusion false" (Intermediate Logic - Bostock)

But how is that "checking the validity of one argument using another"? — TonesInDeepFreeze

Sorry. Wrong @Hanover post.