"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.

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".

"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.

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".

"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. — TonesInDeepFreeze

Ok, that is true.

Negation is not at issue. — TonesInDeepFreeze

Let's go with "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".

V: ¬(A→B)
X: It is not the case that if A then B.
Y: It is not the case that every instance in which A is true is an instance in which B is true.
Z: It is not the case there is no instance in which A is true and B is false.
W: There is an instance in which A is true and B is false.

Do we agree v, x, y, z, w are all the same?

If ¬(A→B) is correctly understood as "There is an instance in which A is true and B is false", that, in English, doesn't tell us anything about whether there is an instance in which A is true and B is true or all the other cases, only tells us that A=1 and B=0 returns 1.

By the truth table of ¬(A→B), every instance is false except when A is true and B is false. I think, if anything, that is better stated as "There is no instance in which A isn't true and B isn't false"; which, in English, I think is the same as "A is always true and B always false".

Someone else also gave the suggestion of A→B as "there is no A without B" and ¬(A→B) as "there is A without B".

At the end of the day the English sense of implication simply isn't truth functional. It is counterfactual in a way that material implication is not. — Leontiskos

Elaborate.

metabasis eis allo genos — Leontiskos

Passing to another kind? What kind?

In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values. One way to see this is to note that an English speaker will be chastised if they use the phrase to represent a correlation that is neither causative nor indicative, but in the logic of material implication there is nothing at all wrong with this. — Leontiskos

Yes, that sounds about reasonable.

Elaborate. — Lionino

For example:

The English has to do with a relation between P and Q that transcends their discrete truth values. One way to see this is to note that an English speaker will be chastised if they use the phrase to represent a correlation that is neither causative nor indicative, but in the logic of material implication there is nothing at all wrong with this. — Leontiskos

"If the Baltic sea is salty, then the Eiffel Tower stands." According to material implication this is a perfectly good statement, but according to English it is foolish. There is nothing which surpasses this sort of statement according to material implication: the antecedent is true, the consequent is true, and therefore the implication is true. What more could we ask? But for the natural speaker what is lacking is a relation between the two things. What is lacking is a relation between the saltiness of the Baltic Sea and the standing-ness of the Eiffel Tower.

Passing to another kind? What kind? — Lionino

Further, I am of the opinion that speech about contradictions is always a form of metabasis eis allo genos. Even in English when we say, "If you make that claim you will be contradicting yourself," we are shifting between two different registers: first-order claims and second-order rules of discourse (i.e. Thou shalt not contradict thyself). — Leontiskos

In the example I gave, "First-order claims and second-order rules of discourse."

A first order claim in propositional logic is something like, "P is true," or, "Q is false." Sentences consist of propositional affirmation, negation, and logical operators. Note, though, that, "You are contradicting yourself," or, "This is a contradiction," is a different genus, and deviates from first-order discourse, moving into the meta-language.

Take a look at these examples from Russell. ϑ ⊧ ϑ and ϑ & ϒ ⊧ ϑ might seem to be candidates for logical laws one might expect to have complete generality.

Identity: ϑ therefore ϑ;: a statement implies itself. But consider "this is the first time I have used this sentence in this paragraph, therefore this is the first time I have used this sentence in this paragraph"

Elimination: ϑ and ϒ implies ϑ; But consider "ϑ is true only if it is part of a conjunction". — Banno

:gasp:

It's acting as a tool to make clear what it is we are doing with our sentences. — Banno

I thought that was the job of syntax rather than logic.

Why?

:wink: They are neat little puzzles.

It's a mistake to think that there are laws of logic that have complete generality - and must be obeyed in all circumstances.

...

Logic sets up systems in which some things can be said and others are ruled out, but natural language is far broader than that, allowing for the breach of any such rule. — Banno

Yet if what Aristotle does in Metaphysics IV is correct, then there is a logical law that cannot be breached, namely the law of non-contradiction. Or in other words, "logic" is not a purely formal exercise. It was created for a reason and that reason has implications for reality/metaphysics.

But for the natural speaker what is lacking is a relation between the two things. — Leontiskos

The natural speaker assumes that there is somewhere some justification.

Formal languages may expect that too.

That is actually the main difference between classical logic and mathematical logic.

In mathematical logic, it is not just about truth tables. The goal is not limited to a bit of truth value calculus. The goal is proving entailment, i.e. (mathematical) justification.

"If the Baltic sea is salty, then the Eiffel Tower stands." According to material implication this is a perfectly good statement, but according to English it is foolish. There is nothing which surpasses this sort of statement according to material implication: the antecedent is true, the consequent is true, and therefore the implication is true. What more could we ask? But for the natural speaker what is lacking is a relation between the two things. What is lacking is a relation between the saltiness of the Baltic Sea and the standing-ness of the Eiffel Tower. — Leontiskos

Ok, I see. In the first page Flannel jesus brought up the same dilemma.

"First-order claims and second-order rules of discourse." — Leontiskos

But what is second-order rules of discourse?

From stackexchange:
"The phrase "vacuously true" is used informally for statements of the form ∀a∈X:P(a) that happen to be true because X is empty, or even for statements of the form ∀a∈X: P(a)→Q(a) that happen to be true because no a∈X satisfies P(a). In both cases, it is irrelevant what statement P(a) is."

Because you said in the post I quoted above: "It's acting as a tool to make clear what it is we are doing with our sentences."

And, that's what syntax is about. The arrangement of words and phrases in a specific order to make clear what we are doing with our sentences. Transposing them could change the meaning. So, syntax is the specific tool to make our phrases clear or let's say, 'understandable'. I don't attempt to deny the value of logic in all of this. I simply think that this is a subject of linguistics rather than logic.

Well, sentences are built by syntax, so syntax can't be a tool for what we do with those sentences. Like the coupling of the wagons can't be guiding the train.

If syntax is not a tool for working with sentences, what is the main point of syntax then?
Does logic make clear what we do with sentencing as Banno suggested?

I can’t see how ‘P(a)→Q(a)’ helps me to properly write: ‘the cute dog ate the bone’ for example.

what is the main point of syntax then? — javi2541997

Connecting words.

Does logic make clear what we do with sentencing as Banno suggested? — javi2541997

Sometimes it does for sure. I have seen folks here using proof checkers to show an argument given in English is fallacious.

Ah, I see the problem, and I carelessly extended it.

I'm dumping this:

"If A then B" if and only if "Every instance in which A is true is an instance in which B is true".

That is wrong.

As I mentioned before, there are two different notions:

(1) "If A then B"

and

(2) "A entails B"


(1) in the sense of material implication means "(A is true and B is true) or (A is false and B is true) or (A is false and B is false)". And that reduces to "A is false or B is true".

(2) means "Every instance in which A is true is an instance in which B is true".


(1) is symbolized as 'A -> B'

(2) is symbolized as A |= B


Indeed, they are not equivalent.

...if what Aristotle does in Metaphysics IV is correct, then there is a logical law that cannot be breached, namely the law of non-contradiction. — Leontiskos

To which the dialetheist may simply say "so much for Aristotle".

Since Aristotle, the assumption that consistency is a requirement for truth, validity, meaning, and rationality, has gone largely unchallenged. Modern investigations into dialetheism, in pressing the possibility of inconsistent theories that are nevertheless meaningful, valid, rational, and true, call that assumption into question. If consistency does turn out to be a necessary condition for any of these notions, dialetheism prompts us to articulate why; just by pushing philosophers to find arguments for what previously were undisputed beliefs it renders a valuable service... And if consistency turns out not to be an essential requirement for all theories, then the way is open for the rational exploration of areas in philosophy and the sciences that have traditionally been closed off. — Dialetheism, SEP

Not following you here - there is more to clarity, and to logic, than just syntax.

"In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q."

I speak English, and I don't take "if P then Q" (whether in the sense of material implication or in everyday senses, including necessity or relevance) to be about presences. Indeed, where 'P' and 'Q' are sentences, I would take "the sentence P is present", etc., to be nonsense unless it meant that the sentence P was being displayed in some way, such as on a page or screen. Indeed, I've never heard an English speaker in everyday conversation say something like "The sentence P is present". Moreover, let P stand for a sentence such as "The world is big", then I've never heard any English speaker say anything like ""The world is big" is present". Indeed, if an English said "If the world is big, then the sun is huge" then I don't know any English speaker who would say, "Yes, the presence of "The world is big" indicates the presence of "The sun is huge". Not only is that dialogue not idiomatic, but it registers as nonsense.

I don't think there are laws of logic that cannot be broken, but that there are laws of thought that can't be broken (for obvious reasons). Some laws of logic may express those laws of thought. But that is just a semantic contention.

What do you mean by "cannot be broken"? Do you mean "cannot break without being in error"?

After the semantic contention, a syntactic contention:

"If X, then Y" is incorrect.
"If X, Y" or "X, therefore Y", not both.

If relevance is required between the antecedent and consequent for meaningfulness, then we don't know whether a given conditional is meaningful until we've settled whether there is relevance between the antecedent and the consequent. So if the question of relevance is unsettled, we have to wait before taking the conditional to be meaningful or not. "If Jackie has blue hair then London is noiser this year than last year". We don't know whether the antecedent is relevant to the consequent without knowing more. Maybe Jackie having blue hair causes a big fashion trend in which people go to London to be seen having blue hair or many other possibilities. For that matter, when would we ever be certain that there is no relevance between two sentences? A butterfly flapping its wings in Tierra del Fuego, so to speak.

After the semantic contention, a syntactic contention:

"If X, then Y" is incorrect.
"If X, Y" or "X, therefore Y", not both. — Lionino

What does that mean?

That is actually the main difference between classical logic and mathematical logic. — Tarskian

Usually, mathematical logic is studied by means of classical logic. Indeed, mathematical logic is formulated by classical set theory. The theorems of mathematical logic, if formalized, are themselves theorems of set theory.

Not following you here - there is more to clarity, and to logic, than just syntax. — Banno

I agree. I just wanted to point out that syntax is a tool to make clear sense of our sentences. Not the only one, for sure. But it is one of the main tools in linguistics at least.

For example: sometimes logic formulas or axioms are not clear, but thanks to syntax we can get a better approach to understand it.