Whoever first said, "if ¬(A→B) is true and B is false, A is true", the point is that it is unnecessarily cluttered by "and B is false".

"If A then B" is understood differently by different people in different contexts.

So any ambiguity in "It is not the case that if A then B" stems from "If A then B".

So specify what you mean by "If A then B", then you will have specified what you mean by "It is not the case that if A then B".
— TonesInDeepFreeze

Of course. — Lionino

Good, so we've taken care of your problem. Negation is not at issue.

Saying A→B is "if A then B" does not provide a solution to the matter of unambiguously converting A→B to English. — Lionino

'A -> B' is symbolic. In context of ordinary symbolic logic, it is unambiguous. What is ambiguous is everyday discourse. And, of course, many ordinary senses of "if then" don't fit 'A -> B' as 'A -> B' is used in ordinary symbolic logic. What you call an 'incongruity' stems from (1) "If then" has different sense in ordinary discourse. (2) The material conditional is not in accord with many (arguably, most) everyday senses of "if then".

"but no" — Lionino

Works idiomatically. And I edited anyway for even greater sharpness:

"There is rain but there is no wetness".

is idiomatically the equivalent with:

"There is rain but no wetness".

If you disagree, then so be it.

A→B is also true whenever A is false. — Lionino

"Every instance in which A is true is an instance in which B is true"

equivalent with:

"There is no instance in which A is true and B is false."

If A is false in an instance, then that is an instance in which it is not the case that A is true and B is false.

It is not strictly speaking a sentence, but idiomatically it is understood that it means "There is rain but there is no wetness".

if ¬(A→B) is true and B is false, A is true. — Lionino

It's been pointed out to you at least twice that B doesn't matter:

~(A -> B) -> A

we can't read ¬(A→B) as "A does not imply B". — Lionino

Then don't read it that way.

It is suggested to read it as: It is not the case that A implies B.

"There is rain without there is wetness". — Lionino

First, that is not idiomatic. I've never heard someone say "There is X without there is Y". Second, it could mean at least a few different things. Third, I don't know your point with the example. Fourth, the previous example at least had a nice haiku-like quality.

I don't think that "It is not the case that" is usually ambiguous. (It is not the case that "it is not the case that" is usually ambiguous.)

"If A then B" is understood differently by different people in different contexts.

So any ambiguity in "It is not the case that if A then B" stems from "If A then B".

So specify what you mean by "If A then B", then you will have specified what you mean by "It is not the case that if A then B".

The English phrase "A does not imply B" typically means "There are instances in which A is true but B is false". By your list, that does not mean the same as the material conditional. — Lionino

[EDIT: Dump the strikethrough potion]

Arguably, they are the equivalent:

(1) "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".(material conditional)

is equivalent with:

(2) "If A then B" if and only if "There are no instances in which A is true and B is false"

So:

(4) "It is not the case that every instance in which A is true is an instance in which B is true"

is equivalent with

(3) "It is not the case that there are no instances in which A is true and B is false"

is equivalent with:

(5) "There are instances in which A is true and B is false"

If 'it is not the case that if A then B' is to be understood as the third option, we are simply circling back. — Lionino

Circling back to what? Choose whichever option you like, or add options such as relevance, or state another option.

What is a phrase in English that unambiguously corresponds in meaning to ¬(A→B)? — Lionino

Choose which option you prefer for "If A then B", then prefix it with "it is not the case that".

It's only my guess as to what he might mean. I've never heard of second order recursion or what it might be, though it seems like something that might exist.

I don't opine as to what that other poster has in mind. But:

Rules themselves may be mathematical objects. Languages, axioms, rules, systems, theories, and proofs can be defined and named in set theory. Even informally, when, for example, we say "by the rule of modus ponens", the rule of modus ponens is a thing named by 'the rule of modus ponens'.

The albums 'Conversations With Myself' and 'Further Conversations With Myself'.

I have some darned fine conversations with myself. — fishfry

So did Bill Evans.

real recursion — ssu

He might have meant something parallel to the distinction between first order induction and second order induction that he seemed to be mentioning, so that 'real induction' is second order and so too for 'real recursion'. But that would only be a guess.

recursive real numbers — ssu

That is not recursion over the reals. A recursive real r is such that there is a recursive function f on the naturals such that for each n, f(n) is the nth digit in the decimal expansion of r. That's still recursion over the naturals. I highly doubt that 'recursive real' is what he meant in this context. I think you're hearing hoofbeats in wild horse country and thinking zebras rather than horses.

¬(A→B) and ¬B entail A. I[f] A does not imply B and B is false, can we really infer that A is true? — Lionino

What do you mean by "A does not imply B"? Do you mean?:

"It is not the case that A implies B"
i.e., ~(A -> B)
which is true in any interpretation in which A is true and B is false.

or

"It is not the case that every interpretation in which A is true is an interpretation in which B is true".

P1: ¬(A→B)
P2: B is true
Concl.: A is true — Lionino

That should be ('M' here is an interpretation):

~(A -> B) is true in M
B is true in M
therefore, A is true in M

or, if M is tacit:

~(A -> B) is true
B is true
therefore, A is true

or, without 'true':

~(A -> B)
B
therefore, A

Rain without wetness
Wetness
Therefore rain. — Lionino

'rain without wetness', 'wetness', 'rain' are not sentences.

But it does have a nice haiku-like flavor.

"A does not imply B". In English that is ambiguous. It could mean:

There are instances in which A is true but B is false.

It is not the case that A entails B (same as above).

It is not the case that A implies B (where 'implies' means the material conditional).

It is not the case that A implies B (where 'implies' means a connective other than the material conditional).

Probably others.

The rest of this pertains to ordinary symbolic logic:

We have to be careful to distinguish between, on the one hand, mere implication and, on the other hand, and entailment or proof .

A -> B
is not generally equivalent with
A |= B or A |- B.

In ordinary symbolic logic, '->' does not mean 'entails' or 'proves':

A -> B is false in a given interpretation if and only if (A is true in the interpretation and B is false in the interpretation).

A |= B is true if and only if every interpretation in which A is true is an interpretation in which B is true.

A |- B iff and only if there is a derivation of B from A.

Example:

"If Grant was a Union general, then Grant was under Lincoln." True in the world of Civil War facts. But false in some other worlds in which Grant was a Union general but, for example, Lincoln was not president.

"Grant was a Union general" entails "Grant was under Lincoln". Not true, since there are worlds in which Grant was a Union general but, for example, Lincoln was not the president.

"Grant was a Union general" proves "Grant was under Lincoln". Not true, since there are not other premises along with "Grant was a Union general" to prove "Grant was under Lincoln".

/

Also, we need to be careful what we mean by letters such as 'A', 'B', 'P', 'Q', etc.

(In propositional logic, all formulas are sentences, but in predicate logic, some formulas are sentences and some formulas are not sentences.)

In different contexts, such letters are used to represent either:

(1) atomic formulas (atomic sentences)
or
(2) meta-variables ranging over formulas. (Sometimes logic books use Greek letters for this.)

In recent discussions, the letters are being used as meta-variables.

So, for example, when we mention 'A -> B', we understand that 'A' and 'B' range over all sentences, including ones of arbitrary complexity.

/

If you are asking what is the most accurate English translation of the intended meanings in ordinary symbolic logic, just put in:

"it is not the case that" where '~" occurs
"if ____ then ____" where '____ -> ____' occurs
"and" where '&' occurs
"or" where 'v' occurs

Don't understand that quote. But comments that might be on target:

(1) "given complete induction. Unfortunately Peano's axiom of induction is not fully reducible to a collection of first-order statements."

I guess by 'complete induction' he means induction over all properties (i.e. a second order theory). And, yes, the PA induction schema is over only formulas. But the induction schema does define a set of first order sentences.

But I guess 'Peano's axiom of induction' refers to a second order axiom, not the first order PA axiom schema.

(2) The theory of real closed fields doesn't define the predicate 'is a natural number' (I hope I've stated that correctly).

(3) What do you mean by 'recursion of the reals'? Recursion requires well ordering.

ZF-Inf is bi-interpretable with PA. And as I said, so is Z-Inf. — TonesInDeepFreeze

[EDIT CORRECTION: I think it is incorrect that (Z\I)+~I is bi-interpretable with PA. If every set is finite, then the axiom schema of replacement obtains, so (Z\I)+~I = (ZF\I)+~I. I was thinking that the negation of the axiom infinity implies that every set is finite. But I think that itself requires the axiom schema of replacement.]

better deep than shallow. — fishfry

Better deep in knowledge and shallow in misunderstanding. Better deep in love and shallow in hate.

I discussed this with myself and determined I'm right. — fishfry

Then you're discussing with the wrong person.

I prefer not to argue the point, if you'll forgive me. — fishfry

You may prefer whatever you want; there's no need for forgiveness for preferring whatever you like; meanwhile, I prefer to show how you are wrong in saying that 'true' is not defined entirely with the notion of interpretations and not the notion of axioms.

with the infinity axiom negated (ZF−inf)

Ah, that is not a notation I would have thought means "the axiom of infinity negated". I would have thought it means "ZF without the axiom of infinity". The notation with which I am familiar indicates (1) the axiom of infinity is dropped and (2) the negation of the axiom of infinity is adopted. But since the ZF-Inf is also used for that, of course, with that use, ZF-Inf is bi-interpretable with PA. And as I said, so is Z-Inf. [EDIT: cross out previous sentence.]

StackExchange also has a bad discussion design. And often some confused discussions, But at least as far as math and logic, I have found it to be far better than Quora, which is the pits.

Quora — jgill

Even worse than Wikipedia, which much too often is, at best, slop. Quora is close to the absolute lowest grade of discussion. It is a gutter of misinformation, disinformation, confusion and ignorance. Quora is just disgusting.

(1) The layout of the threads is quite illogical and very impractical. The illogical organization style of answers and comments makes discussions incoherent. Like the rest of the site, the design is not to facilitate reading but to add to click counts and ad views. A site is not to be faulted for having ads, but the entire design of Quora is egregiously manipulative. (2) Posters are allowed to delete their posts, which is okay, but deletion of one's posts includes deletion of replies to the deleted posts. Thereby, a poster can wipe out all your replies. (3) Mathematics and logic discussions at Quora are inundated with prolific, persistent, chronic, serial cranks who slater the threads with misinformation, disinformation, confusion and ignorance. It's a disgusting cesspool. It's the dark web of discussion.

The only thing worse is "AI", which can always be relied upon for absurd misinformation and computer generated lies, all under the imprimatur of computing.

"[...] as an introduction to a topic Wikipedia is very good."

I'll fix that:

as an introduction to a topic Wikipedia is very good lousy.

@Tarskian

And your claim about ZF\I is incorrect. ZF\I is not bi-interpretable with PA. Rather, it is (ZF\I)+~I that is bi-interpretable with PA. (Actually, we can simplify to (Z\I)+~I, which is bi-interpretable with (ZF\I)+~I and bi-interpretable with PA.)

[EDIT CORRECTION: I think it is incorrect that (Z\I)+~I is bi-interpretable with PA. This is correct: If every set is finite, then the axiom schema of replacement obtains and (Z\I)+~I = (ZF\I)+~I. But I don't think that works; I was thinking that the negation of the axiom infinity implies that every set is finite. But I think that itself requires the axiom schema of replacement.]

@Tarskian

Back to this matter:

Whether there are uncountably many truths or whether there are unexpressed truths depends on what is meant by 'a truth'.

In context of mathematical logic, I would take a truth to be a certain kind of sentence relative to a given model. So, for a countable language, there are only denumerably many truths (i.e. true sentences).

Or one might also say that truths are states-of-affairs, such as a certain tuple being in a certain relation is a truth, even if there is no sentence that states that fact about that certain tuple and certain relation. In that sense, yes, for an infinite universe, there are uncountably many truths.

I wrote:

"There are sentences that are like this. There are sentences that are like that. Both could exist." — Tarskian

You said something similar to that. But later you said something very different. It's not the reader's job to suppose you don't mean what you write. Moreover, even if one did figure out that you meant something different from what you wrote, then it is still appropriate to point out that what you wrote is incorrect as it stands no matter that in your mind you meant something different.

You should not say 'logic sentences' in general, since the theorem pertains to sentences in certain languages for certain theories.
— TonesInDeepFreeze

It's related to PA or similar. That is always implied. — Tarskian

No, a person doesn't know that you're not making the generalization that you stated. Instead of saying "logic statements" (and what is a "logic statement"?) it would have been correct to say "sentences in the languages of said theories" or something like that. Or say, "For theories of a certain kind". People are not supposed to guess that you don't mean what you say.

it is about properties that have formulas in PA. — Tarskian

Then you need to say that. People are not required to not take you to mean what you say when you say "all properties".

Actually, all you needed to say is, "all predicates in the language" rather than "all properties" which is not only mathematically wrong but philosophically wrong.

Mentioning all of that, including the above, will make the entire explanation impenetrable. — Tarskian

Actually, it's inpenetrable when you don't specify what you mean but instead put the burden on the reader to suss out what you might mean.

I just refer to a link that contains all these details — Tarskian

Another way of putting it on the reader to then wade through an article to figure out what you mean in a post, when you don't at least say what particular passages in the article you have in mind, or at least say that something like "my thesis depends on the bulk of this article that needs to be understood first". And linking to Wikipedia about mathematics is rank. Wikipedia articles about mathematics are too often incorrect, inaccurate, poorly organized or poorly edited. It's often to the subject matter what fast food is to nutrition.

My rendition is not suitable for a mathematical forum, but I had hoped that it would be for a philosophical one. — Tarskian

Your writing about mathematics is so often incorrect and ill-formed That it is in a philosophy forum and not a mathematics forum doesn't alter that it is so often incorrect and ill-formed.

You leave out crucial points because you think they are too technical. But then people who don't know that there are such crucially needed points are liable to be misled by your bad oversimplifications.

What you call "noise" is actually needed clarity of the signal. What you think is your signal is the sound of a blown woofer.

Also, look at it this way:

Given a set of axioms G and a different set of axioms H, it may be the case that the class of models for G (thus for all the theorems from G) is different from the class of models for H (thus for all theorems of H). So let's say S is a member of G or a theorem from G, and S is inconsistent with H. Then, yes, of course, S is true in every model of G and S is false in every model of H.

But, given an arbitrary model M, whether S is true in M is determined only by M.

That is a deep misunderstanding.

An interpretation for a language determines the truth or falsehood of each sentence in the language.

Different axiom sets induce different theorems, hence different theories, but for a given interpretation, what axioms are in a given axiomatization has no bearing on that interpretation and no bearing on which sentences are true in that interpretation.

Again:

'sentence S is true in a model' is semantical.

'sentence S is an axiom for a system' is syntactical.

I suspect that what you have in mind may be put this way:

Given a set of axioms, every model in which the axioms are true is a model in which the theorems from the axioms are true.*

That is the case. But it is not a definition. It is a theorem (the soundness theorem). The definition of 'S is true in model M' does not mention axioms. Again: Given an interpretation (a model) M, the truth or falsehood of every sentence in the language is determined irrespective of what axiom sets there are for different systems.

* We also keep in mind that there are axiom sets such that there are sentences such that neither the sentence nor its negation is a theorem from the axioms, so, if the axioms are consistent, then there are models for the axioms in which a given independent sentence S is true and there are models for the axioms in which S is false.

For example, most trivially:

Let 'P' and 'Q' be sentence letters. Let the only axiom be 'P'. There are two models in which 'P' is true:

P is true, Q is true
P is true, Q is false

So the axiom P does not determine the truth or falsehood of Q.

Mathematical truth is always:

Axioms plus an interpretation. — fishfry

It is not the case that 'mathematical truth' means ''axioms plus an interpretation'.

The definition is:

sentence S is true in model M if and only if [fill in the definiens here]

and 'axiom' is not mentioned.

/

'axiom' is a syntactic notion, not semantical.

A system T is comprised of

a language L
a set G of sentences in the language L (the axioms)
a set of inference rules

That induces a set T of theorems.

That is all syntactical.

'true' is a semantical notion.

a sentence S is true in a model M if and only if [fill in definiens here]

It doesn't matter whether S is an axiom or not. The definition doesn't mention 'axiom'.

By the way, every sentence is an axiom of uncountably many axiomatizations.

So you didn't write what you meant regarding S and F.

And still no recognition of these:

You should not say 'logic sentences' in general, since the theorem pertains to sentences in certain languages for certain theories.

You should generalize over formulas in those languages and not over properties (since there are properties not expressed by formulas).

You wrote:

"(S is true and F is false) or (S is false and F is true) or both."

Which is:

(S is true and F is false) or (S is false and F is true) or ((S is true and F is false) and (S is false and F is true))

That is false.

[EDIT CORRECTION: It is not false. I should have said the third disjunct is false, thus otiose as added to the two other disjuncts.]

If you meant something different, involving 'there exists', then you need to write it.

And just to be clear: The theorem is of the form: For all formulas P(x), there exists a sentence S, such that ....

The consequence is that nobody reads what I just wrote. I could as well not write it at all ... — Tarskian

I'm not nobody. You have a problem with quantifiers.

You could fix the interpretation and change the axioms to show that truth depends on the axioms plus the interpretation. — fishfry

I don't know what that means.

The definition is:

sentence S is true in model M if and only if [fill in the definiens here]

and that definiens doesn't mention 'axiom'.

I have made the point that whether there are uncountably many truths or whether there are unexpressed truths depends on what is meant by 'a truth'.

In context of mathematical logic, I would take a truth to be a certain kind of sentence relative to a given model. So, for a countable language, there are only denumerably many truths (i.e. true sentences).

One might also say that truths are states-of-affairs, such as a certain tuple being in a certain relation is a truth, even though no sentence asserts that fact. In that sense, yes, for an infinite universe, there are uncountably many truths.

You still resist recognizing these points:

You should not say 'logic sentences' in general, since the theorem pertains to sentences in certain languages for certain theories.

You should generalize over formulas in those languages and not over properties (since there are properties not expressed by formulas).

Disjunction is inclusive, but it is never the case that both of these are true: "P is true and Q is false" and "P is false and Q is true".

In general, a disjunction 'phi or psi' might not allow 'phi and psi', depending on the content in phi and the content in psi.

/

People can decide for themselves what is too technical or not. Symbolic logic, some mathematical logic and set theory are included in some philosophy department programs. Symbolic logic, even in community colleges. And philosophy of mathematics is based on understanding the mathematics being philosophized about.

And my point was not that people wouldn't be interested or that there are not enough technically minded readers (we don't know who is reading now or who might read in the future), but rather that the subject deserves to not be mangled by oversimplification.

It is possible to precisely state all the conditions that apply, but in that case, the explanation becomes impenetrable. Nobody would be interested in a multidisciplinary forum. In order to keep it readable, there is no other alternative than to leave things out. — Tarskian

First, I made a mistake as I misread your disjunction for conjunction. I made edit notes for that in my posts now.

You don't know what every person is interested in. As far as posts thus far, the only person to comment on your remark is me, and I am interested in seeing the subject properly represented. Better not to post terribly poor renderings of technical matters than to mangle the subject. The fact that this is a philosophy forum doesn't entail that it is good to oversimplify to the point of foggy vagueness and/or substantive misstatement. And I don't know why you would suppose that people would care about your synopsis of Carnap if they didn't also grasp the mathematical basis. You think many (if any) people are going to read your one liner about the theorem and grasp anything about it without a clearly stated mathematical basis? Moreover, your one-liner is incorrect.

Your synopsis is poor:

It does not make clear that it pertains specifically to the mathematical theorem.

It should not say 'logic sentences' in general, since the theorem pertains to sentences in certain languages for certain theories.

You should generalize over formulas in those languages and not over properties (since there are properties not expressed by formulas).

Disjunction is inclusive, but that does not entail that it is ever the case that "P is true and Q is false" and "P is false and Q is true".

You conflate a predicate with a sentence.

(S ∧ ¬F(r(#S)) ∨ (¬S ∧ F(r(#S))

Meaning:
(S is true and F is false) or (S is false and F is true)

Meaning:
A true sentence that does not have the property, or a false sentence that has the property, or both. — Tarskian

(1) You skipped that I pointed out that:

(S is true and F is false) and (S is false and F is true)

is never the case.

(2)

"S ∧ ¬F(r(#S)" is not the same as "S & ~F".
"¬S ∧ F(r(#S)" is not the same as "~S & F".

F

does not have a truth value. What has a truth value is

F(r(#S))

Saying "F is false" is nonsense.

(3) I agree with this:

C entails that there is a sentence S such that T proves:

(S & ~F(r(#S))) v (~S & F(r(#S))).

F expresses a property. F(r(#S)) is true if and only if S has the property expressed by F.

But:

[EDIT CORRECTION: I misread the quote. The quote was a disjunction not a conjunction. Mine is not a counterexample. No true sentence is not equivalent with itself, but every false sentence is equivalent with itself.]

(1) Your quoted characterization did not have the specifications you are giving now. Your quoted characterization was a broad generalization about properties and sentences.

(2) PA doesn't say 'true' and 'false'.

(3) Inclusive 'or' allows 'both' but not regarding 'true' and 'false'.

We may have:

P is true and Q is false
or
P is false and Q is true

But we cannot have:

P is true and Q is false
and
P is false and Q is true

(4) There are properties not expressed by formulas, so the generalization should be over formulas, not properties.

(5) I did not say: "P(S) := S <-> S." I said: "with the arithmetization of syntax, both 'is a sentence' and 'is equivalent with itself' are expressible in PA." What is not expressible in PA are 'true' and 'false'.

('r' for 'the numeral for' and '#' for 'the Godel number of')

Let C be this theorem:

For certain theories T, for every formula F(x) there is a sentence S such that T |- S <-> F(r(#S)).

Let K be:

"For any property of logic sentences, there always exists a true sentence that does not have it, or a false sentence that has it, or both."

C is not correctly rendered as K.

(1) K doesn't qualify as to certain kinds of theories.

(2) C generalizes over formulas, not over properties.

(3) C doesn't say anything about 'true'.

(4) C doesn't say that for every property of sentences there is a true sentence that does not have the property. C doesn't say that for every property of sentences that there is a false sentence that does have the property.

[EDIT CORRECTION: I misread the quote. The quote was a disjunction not a conjunction. Mine is not a counterexample. No true sentence is not equivalent with itself, but every false sentence is equivalent with itself.]

Moreover:

(5) I showed a counterexample to both prongs of K.

[EDIT CORRECTION: I misread the quote. The quote was a disjunction not a conjunction. Mine is not a counterexample. No true sentence is not equivalent with itself, but every false sentence is equivalent with itself.]

You said that my counterexample is not in PA. So what? It doesn't have to be in PA, it merely needs to be a counterexample to K. And, by the way, K is not in PA, especially since PA doesn't have a predicate 'true'. And C includes PA as one of the T's, but C itself is not in PA.

(6) And with the arithmetization of syntax, both 'is a sentence' and 'is equivalent with itself' are expressible in PA. But I didn't do that, because K doesn't specify any language or kinds of theories.

/

For certain theories T, for every formula F(x) there is a sentence S such that T |- S <-> F(r(#S)).

is not remotely anything like:

For any property of logic sentences, there always exists a true sentence that does not have it, or a false sentence that has it, or both.

For any property of logic sentences, there always exists a true sentence that does not have it, or a false sentence that has it, or both. — TonesInDeepFreeze

You wrote:

"For any property of logic sentences, there always exists a true sentence that does not have it, or a false sentence that has it, or both."

That doesn't mention PA. Rather, it a universal generalization over properties and sentences.