an argument with two conditional premises should not be able to draw a simple or singular conclusion (because there is no simple claim among the premises). — Leontiskos

(1) There is no single everyday sense of "if then" or "implies". For that matter, I bet that in everyday discourse a lot of people would not even make sense of the original thread question.

(2) What senses people have in everyday conversation is an empirical question. It should not be presupposed that any particular sense is even the most common one without some basis. A reasonable conclusion about what people mean would have to take in natural languages world wide, not merely English.

(3) I think that language is not all that's in play, but rather also in play are the notions people have about things. We won't find in a dictionary what a person's response would be to "If snow is green then Winston Churchill was a Confederate general". Yes, it involves a person's meaning of "if then" but also that person's notions of what utterances are sensical, logical, or true.

/

(4) In what context is the criteria above supposed to be in? Everyday discourse? An alternative formal logic? Other?

(5) What is the definition of 'simple sentence'? A sentence with only one clause?

(6) Does the criteria pertain only to conditionals? And what a the sentence is written as an equivalent non-conditional?

(7) If not only conditionals, the what of single clause sentences such as "Alex is a dirty rotten scoundrel"? From that single clause sentence I should not infer "Aex is a rotten scoundrel"?

(7) What is the basis for the criteria? Without knowing more about it, I don't know a basis for disallowing the inference of a simple statement from compound statements.

(8) It might be the case throwing out the supposed bath water might be throwing out the baby that is the class of certain valued everyday, mathematical and scientific reasoning.

And to read Flannel Jesus' posts is to realize that he did not intend the OP in any special sense. I see no evidence that he was specifically speaking about material implication. — Leontiskos

It wouldn't be unreasonable to glean that he meant it not just in everyday senses. But, of course, it is open enough that everyday sense may be in play. And just to be clear, he did not indicate that material implication should be excluded.

Tones even mistakes natural language for his own system — Leontiskos

That is a flat out lie.

I said a number of times that ordinary classical logic (which is not just "my" system, especially since overwhelmingly it is not exclusive to me, and especially since I study and have perused logics other than classical logic).

I said at least three or four times that it is not the case that in all respects formal logic represent everyday reasoning and discourse. I even said explicitly that material implication is not an everyday sense of "if then".

Leontiskos would have seen my posts saying those things, and, if I recall, he even replied to some of them. So either Leontiskos ignored what a wrote or read what I wrote but stated a flat out lie about me anyway.

/

and normatively interprets natural language in terms of his system — Leontiskos

That's a second flat out lie.

(1) The question in this thread did not specify whether it should be answered as to everyday senses of "if then" or the ordinary sense in the study of the most ordinary logic. And the question was posed in the 'Logic and Philosophy of Mathematics' section of a philsophy forum. So, I (and others) chose to answer as to material conditional. And I have stated explicitly that answers may be different in contexts of other formal logics and in everyday use.

(2) I have never stated a normative claim that classical logic trumps all other approaches.

Moreover:

(3) It is not "my" system, since it is not exclusive to me, and since it is not the only system I study or have perused.

Lenontiskos should not lie about my posts.

I have no issue with being corrected or told new things. — Philosophim

Note that my corrections did not presuppose that only material implication can be countenanced.

he jumped into a conversation I was having with another poster without context — Philosophim

Oh please, everyone enters a thread by "jumping in" in media res. Maybe you noticed that there aren't 'hand waving' icons to click to be recognized like in a video meeting. And there are no "having a conversation with another poster" that doesn't admit of anyone "jumping in" to comment. And "out of context" would mean that my remarks were misleading or not worthwhile on account of them needing to be modified by context. That was not the case with my remarks.

and when I asked him to clarify his issue he came across as dismissive. — Philosophim

You didn't just ask me to clarify. You insulted personally with "Don't be a troll", and I did provide you with what you requested.

I encourage you not to do the same and jump into another conversation between two people. — Philosophim

I encourage anyone to jump into any conversation, among any posters, as much as they like.

You don't have an exclusive right to be the only one commenting on what other posters say, including what they say to you.

/

And as preemptory in case of complaints that I should just "let it go" with you, note that you are publicly making faulting me as a poster, not just as to what I've said on the topic but on a personal basis also. It is quite proper that I defend myself. I don't have to just let you run your posts running me down unanswered.

If the long reply made you feel better, that's fine. — Philosophim

So many things wrong packed into just that one sentence. (1) My post was hardly that long. (2) It's length was a function of the explanation it contains. (2) I don't begrudge posters making posts at any length they want. (3) What is the purpose of mentioning length if not as wedge to discredit? (4) The post carries a lot more message than being a way to "feel better". (5) You did not even address the points I made, but instead you tried to dismiss it with innuendo (if not outright implication) that the post is just a bunch of me trying to fell better.

You can't argue against how you come across to other people on a forum. — Philosophim

(1) I can't dispute that certain people feel certain ways. (2) I can dispute people's representations and characterizations of my posts and my interior, including feelings. (3) How you feel about me does not represent how all others feel about me. (4) And likewise, you can't argue against how you come across to me.

Hopefully we'll have a better encounter in another thread. — Philosophim

Hopefully so. And hopefully in this thread. The odds of that happening would be greatly improved by not sending your first post to me to include "Don't be a troll".

Good luck in explaining your side, I do agree with it. — Philosophim

Don't need luck. I've been explaining my thoughts well. I am glad for your agreement.

you're running into a mismatch between most people's general sense of seeing -> as a strict conditional. — Philosophim

(1) '->' is a symbol ordinarily used for material implication; while the strict conditional usually uses a different symbol or is written with '->' and a modal operator.

(2) Strict implication is what might be in mind in certain everyday contexts, but my guess is that relevance is crucial in average everyday contexts. If A is not relevant to B, then I bet most people would just take "if A then B" and "Necessarily if A then B" to be nonsense and, while some people would take it as false on account of being nonsense, others would just say it is plain nonsense.

(2) I have said at least a few times that I quite understand that there are many other notions of 'implies' other than material implication. There is relevance logic, strict implication, and I would bet there are other formal and/or philosophical approaches. And there ar everyday notions that may run from deliberate to not even having a worked out notion of what 'implies', especially to extent of untangling the original thread question (I would guess that you could ask many people walking around the original thread question and their best answer would be "huh?").

in your field or life 'material conditional' is a common phrase, but for most people who use logic, this is never introduced. For them, it's almost always seen as a strict conditional. Remember that this forum is populated by all types of people, and most of them are not logicians or philosophers themselves. — Philosophim

In a philosophy forum, in its 'Logic and Philosophy of Mathematics' section, a question posed that would hardly ever occur in everyday discourse, may be answered however one likes to answer it. Since in logic, the overwhelmingly most common sense is the material conditional, I answered in that context. I do not disallow anyone from answering in other contexts. (I did say that my answer is 'that simple" and regardless "digressions". But I also posted that I do not claim that material implication is the only context that can be countenanced).

Explaining and contrasting a strict conditional vs a material conditional should make the issue clear for most people. — Philosophim

That's fine. No one is stopping you from posting your explanation and contrast. But also, a hearty explanation would include other notions too, especially relevance.

There is nothing wrong with referring to truth in mathematics. (1) The everyday sense of 'truth' doesn't hurt even in mathematics. When we assert 'P' we assert 'P is true' or 'it is the case that P'. (2) There is a mathematical definition of 'true in a model'.

Just to be clear: Tarski did not disallow the notion of 'truth', but rather he sharpened it to 'true in a model'. The undefinability theorem doesn't vitiate the notion of truth, especially as formalized as 'true in a model'; rather the undefinability theorem is just that in certain interpreted languages there is no definition of a truth predicate.

As to describing what mathematicians do as "solving problems", that's fine as long as "solving problems" includes proving theorems, because mostly what mathematicians do is prove theorems.

And the evening star (which is the morning star) is not a star, it's a planet, and exists as a physical object.

And the crank adds to displaying his lack of intellectual capability by showing that he cannot comprehend intensionality and extensionality, which is not surprising since he is incapable of comprehending use and mention.

What do you mean by "put on"? I only said that Frege's system is one attempt to derive mathematics solely from logic, and the system is inconsistent.

/

I don't know what you have in mind about "throwing away everything finite"?

Frege's approach to even defining the number 0 from logic alone requires an infinite class.

/

Frege's system was proven to be inconsistent.

First order PA has not been proven to be inconsistent. I don't see a reason to believe that first order PA is inconsistent. And accepting the premises of Gentzen's proof, first order PA is proven not to be inconsistent.

So I don't know why you would ask.

It's not a matter of whether I accept or reject. I said what I had to say in my post. If you wish not to address what I said it in, that is your right.

I'll retract it then, as an alternative to arguing the point. Or if you consider the second clause as adding fuel to the fire, I'll retract it. — fishfry

It is bizarre to suggest there's any arguing the point, when the point has been so profusely documented. Your retraction and your offer to retract the bizarre qualifier in the retraction are a self-serving and sneaky way to put the ball back in my court where it doesn't belong.

Is 1 + 1 = 2 a logical truth? — javi2541997

Frege proposed a way that it would be a logical truth. But his way was inconsistent.

1 + 1 = 2 is a 'definition'.
2 — javi2541997

That's often the case, and per the definition, '1+1 = 2' means that '1+1' names the name number as is named by '2'. 1+1 is 2.

First, take out 'would' since subjunctives unnecessarily complicate.
— TonesInDeepFreeze

It's like talking to a computer. "Get rid of that natural language, you're confusing our processes!" — Leontiskos

Oh please, that is a really dumb remark and a lame attempt at putdown. You gratuitously seize on my mere offer to simplify for clarity so that I could better address your question.

You're still involved in ambiguity. In order to know what Sue denied we must know what Bob affirmed. As noted in my original post, your interpretation will involve Sue in the implausible claims that attend the material logic of ~(A → B), such as the claim that A is true and B is false. Sue is obviously not claiming that (e.g. that lizards are purple). The negation (and contradictory) of Bob's assertion is not ~(A → B), it is, "Supposing A, B would not follow." — Leontiskos

I'm not ambiguous. The question is ambiguous, given that 'follows' is not defined. Or what sense of 'follows' Bob and Sue are using.

No matter what Bob's sense of 'follows' is, Sue is negating his claim.

The rest of your paragraph boils down to reiterating that the material conditional is not usually operative in such natural language situations. As I've said, now verging on a hundred times, no one disputes that the material conditional does not suit a wide range of natural language senses.

It was a pointless question and exercise. All you had to say is "in everyday conversation, people don't adopt material implication", though that had been agreed upon many posts ago.

And notice that the matter of material implication does not entail that negation is not usually in the sense I've used it. No matter what sense of 'if then', Sue's claim is the negation of Bob's claim. The unpacking of negating an if-then is different depending on the sense of if then but doesn't require adjusting the sense of negation.

But now I guess further as to your point. You were merely pointing out that the negation of "if A then B" is equivalent with "A and not B"? Of course that's true. But I have not said that one cannot have a different notion of the negation of an if-then statement. That equivalence is so obvious and so aside the point that I didn't get that you would bother to mention it in connection with the fact that (A -> B) & (A -> ~B) is not a contradiction.

Here's some help for you from the dictionary:

Merriam-Webster - Contradictory
(Adjective): involving, causing, or constituting a contradiction
| contradictory statements
| The witnesses gave contradictory accounts of the accident.
(Noun): a proposition so related to another that if either of the two is true the other is false and if either is false the other must be true — Leontiskos

That's no help for me, since I already know that definition. Here's some help for you:

(1) 'constituting a contradiction' is tantamount to 'being a contradiction'. My own point.

(2) implying a contradiction involves either a statement alone being a contraction itself, or with other statments, implying a contradiction, in which case there are at least two statements, each contradicting the other, just as in the examples. But you referred to a contradictory statement, not to some pair of statements. In accord with my own point.

(3) you used the adjective, not the noun, so you can leave out the clause about the noun.

I took what you wrote at face value, and in accord with that dictionary definition.

The reason we keep material implication is because we like truth functionality. — Leontiskos

It seems to me that that is true, and a very important point.

Speaking of pointless execises, you first post to me was one:

They imply ~A.
— TonesInDeepFreeze

Then give your proof. — Leontiskos

I gave two proofs. Your point in directing me to do that turned out to be ill-premised, as it was to say that I had overlooked an alternative notion:

Here is the alternative notion of contradiction that you are overlooking:

“opposite assertions cannot be true at the same time” (Metaph IV 6 1011b13–20)
— Aristotle on Non-contradiction | SEP — Leontiskos

That is in accord with the standard definition in modern logic, the definition I gave.

I already corrected your misinterpretation — Leontiskos

It's not a misinterpretation. To say that P is contradictory is to say that P is a contradiction. A statement is contradictory if it is a contradiction. A pair of statements are contradictory if they contradict each other. If you didn't mean that "lizard are purple and not smarter" is contradictory, then you would have done better to not write and not blame a reader for taking what you wrote in its plain meaning. And your other posts doesn't correct anyone.

I'm glad you finally figured this out — Leontiskos

I gave you the courtesy of allowing that what you wrote is not what you meant. I explained previously why the less smart purple lizard sentence is not a contradiction. It's not my fault that what you wrote is not what you meant.

And I did reread your post. And I addressed it again. You skip what I wrote about your remark in the sense of 'contradicting'.

To help you, Janus' point about natural language is something like this:

Supposing A, would B follow?
Bob: Yes
Sue: No

Now Sue has contradicted Bob. The question is, "What has Sue claimed?" — Leontiskos

You're not "helping" me. But I will help you, as I've been trying to help you from the start.

First, take out 'would' since subjunctives unnecessarily complicate.

So "Supposing A, does B follow?"

Sue claimed "It is not the case that B follows from the supposition A"

As I said, you're not "helping" me. You're just offering me a pointless exercise.

Now I'll ask you a question that is not pointless:

What is your definition of 'contradiction'?

Again, a contradiction is a statement and its negation. If there is a contradiction then you could show that both a statement and its negation are implied.

Again:

"if lizards are purple, then they would be smarter" and "if lizards are purple, then they would not be smarter" is not a contradiction.
— TonesInDeepFreeze

But the difficulties of material implication do not go away here. You are thinking of negation in terms of symbolic logic, in which case the contradictory proposition equates to, "Lizards are purple and they are not smarter." — Leontiskos

(1) I take 'the contradictory statement is P' to mean that P is a contradiction, as a contradictory statement is a contradiction.

(2) But maybe you mean it is a contradicting statement. Maybe you mean "Lizards are purple and they are smarter" contradicts some other statement? Well, it contradicts "It is not the case that lizards are purple and they are smarter". That doesn't vitiate anything I've said.

/

If you have some other definition of 'contradiction' then it would help to know it. Meanwhile, ordinarily, and not just in symbolic logic, a contradiction is a statement S and its negation. Or, less strictly but tantamount, a statement S and some other statement that implies the negation of S. At least along those lines. If that doesn't suit you, then so be it. But unless you provide a different definition, then I'll say that (A -> B) & (A -> ~B) is not a contradiction and does not imply one, whether regarding purple lizards and their intelligence or whatever sentential example you wish to mention.

You don't even understand what is being said. — Leontiskos

I suspect you don't understand what you wrote.

I was there giving an answer to the question at hand. — Leontiskos

Your answer is incorrect.

I give up. Go read Lionino's first post on the first page. — Leontiskos

You should give up, since Lionino's post is perfectly in accord with what I have said: the pair of statements are not a contradiction.

I don't think that's a coincidence at all. — flannel jesus

Right, it's not a coincidence. That doesn't entail anything about the material conditional in Boolean logic.

Explosion doesn't make the material conditional in Boolean logic used for computing nonsensical.

The question at hand is, "What is the contradiction of 'If lizards were purple then they would be smarter'?" — Leontiskos

(1) You changed the sentence. Here is what you wrote:

You are thinking of negation in terms of symbolic logic, in which case the contradictory proposition could be, "Lizards are purple and they are not smarter." — Leontiskos

(2) "If lizards were purple then they would be smarter" is not a contradiction, a fortiori not a contradiction in symbolic logic especially. And "Lizards are purple and they are not smarter" is not a contradiction.

The negation of a material conditional will be different from the negation of an if-then statement in natural language — Leontiskos

There are many senses of 'if then' in natural language. But, of course, material implication does not accord with many of the natural language senses of 'if then'. That's never been at issue.

But the difficulties of material implication do not go away here. — Leontiskos

I've not claimed that anything I've said dissolves any difficulties with material implication.

You are thinking of negation in terms of symbolic logic — Leontiskos

I'm thinking of it in context of symbolic logic, informal logic, and a primary everyday sense.

in which case the contradictory proposition could be, "Lizards are purple and they are not smarter." — Leontiskos

I know of no context in which that sentence is a contradiction.

Yet in natural language when we contradict or negate such a claim, we are in fact saying, "If lizards were purple, they would not be smarter." — Leontiskos

What is your basis for that claim? Your observation of what people mean when they say such sentences. I'm not privy to those observations.

The negation must depend on the sense of the proposition, and in actuality the sense of real life propositions is never the sense given by material implication. — Leontiskos

There are two separate matters: negation and material implication. I've addressed both in this thread. It is not disputed that material implication often does not accord with everyday senses of 'if then'. But such everyday senses are not explicated for definiteness usually don't submit to rigorous logical, mathematical or philosophical treatment. And though logic, mathematics, computing, and philosophy are not everyday, they aren't thereby relegated irrelevance nor is their profound relevance diminshed.

unsound — Janus

In the same vein as above, 'true', 'sound' and 'valid have definitions in logic. Of course, it's your prerogative to use any sense you like. But it behooves us to be clear which definition is at play in a given context. If you are merely disagreeing based on a different definition, then my reply is, "Okay, then there are two different discussions: One based in the ordinary definitions in logic, and the other based in whatever other definitions you stipulate."

both of the conditional statements are untrue, because being or not being smarter has no logical connection with being purple — Janus

Again, that is based on your notion of 'if then'. It is not based on the ordinary notion in logic. So, again, two different discussions: One based on the ordinary notion in logic and the other based on your notion (or better yet, based on relevance logic). Also, you ignored my point about using the term 'logical connection'.

I could say that the two statements are nonsensical because the antecedent has no relevance to the consequent. However, I cannot but see them as contradictory. — Janus

Ordinarily in logic, the expressions we study are not nonsensical. So the notion of contradiction would not apply. But, again, you'll use your own definitions, and that is a different discussion from a discussion in context of ordinary definitions in logic.

What about these two statements: 'if I was more educated in logic, I would be able to see that those two statements are contradictory" and "if I was more educated in logic I would not be able to see that those two statements are contradictory"—do those two statements contradict one another? — Janus

I don't know why you're asking me to comment on an example that is the same in form as the other examples.

Or what about 'if I was more educated in logic, I would be able to see that those two statements are contradictory" and "if I was more educated in logic I would be able to see that those two statements are not contradictory"? — Janus

I don't know what your point is, but to fulfill the exercise:

To save typing and copy/pasting:

L ... I know more logic

C ... I would see that the statements are contradictory

N ... I would see that the statements are not contradictory

-> ... if ___ then ___

~ ... it's not the case that

And I'll answer in the context of classical logic:

Note that in N, 'not' is in the scope of 'I would see that'. So, in mere sentential logic, N is an atomic statement, so I can't pull 'not' outside the scope.

(L -> C) & (L -> ~C) ... not a contradiction

(L -> C) & (L -> N) ... not a contradiction

I hope that's the only exercises I'll be doing here.

Do I understand what 'contradictory' means? I think so. — Janus

Perhaps you understand the sense you prefer to use. But it seems you don't understand the ordinary formal and informal sense I've explained.

I don't opine on what follows from your sense of 'contradiction', whatever your definition might be. But I do know what is the case with the ordinary formal sense that accords also with a primary informal sense. I hope that you don't hold that your use of your sense of 'contradiction' - however you might define it - trumps people talking about contradiction in the sense in the study of logic that accords with a primary everyday sense.

taken informally as statements, they contradict one another. — Janus

An informal sense of 'contradict' is 'to imply the opposite or a denial of'; and an informal sense of 'denial' is 'a proposition so related to another that though both may be false they cannot both be true'

And an informal sense of 'contradiction' is 'a proposition, statement, or phrase that asserts or implies both the truth and falsity of something'.

Both of those accord with the formal sense.

Of course there are many other informal senses.

One informal sense of 'contradiction' is 'incongruity'. That might be what you have in mind.

It is not at issue that people may use different senses. It is senseless to argue with two incompatible senses both at work.

In context of the study of modern logic, in both philosophy and mathematics, 'contradiction' ordinarily means 'a statement that is the conjunction of a statement and its negation', from which follows 'a statement that asserts the both the truth and falsity of something'. And that is also an informal sense.

When someone says, "You contradicted yourself when you said you didn't visit the store", they mean "You said you you didn't visit the store but the day before you said that you did visit the store". Or, "You said you didn't visit the store, but you also said you saw Ted yesterday at 1:00. But at 1:00 Ted was at the store." That is the informal sense of 'contradiction' I refer to, and it accords with the formal sense.

No one can dispute that you find your example incongruous in some personal way, while what is incongruous to one person is not incongruous to another. But my point is that your example is not a contradiction in the ordinary sense in modern logic or in an everyday sense such as when someone says, "You contradicted yourself" to mean "You said you didn't visit the store but also the other day you said you did visit the store" they don't mean that it was merely odd or incongruous. Rather, they mean that you claimed both a statement and its negation.'

Of course, no one should deny you using whatever sense of 'contradiction' you like. Better yet, would be for you to define it. Meanwhile though, in logic, 'contradiction' has a precise definition and it accords with a natural everyday sense too.

you don't see those two sentences as contradicting one another [?] — Janus

Again, a contradiction is a statement and its negation. If there is a contradiction then you could show that both a statement and its negation are implied.

Again:

"if lizards are purple, then they would be smarter" and "if lizards are purple, then they would not be smarter" is not a contradiction.

"if lizards are purple, then they would be smarter" and "if lizards are purple, then they would not be smarter" and "lizards are purple" does imply a contradiction.

"if lizards are purple, then they would be smarter" and "if lizards are purple, then they would not be smarter" and "lizards are not purple" does not imply a contradiction.

/

I'm not sure, but maybe you should check whether you are conflating "not intuitive" with "contradictory".

Why is it incorrect informally? — Janus

Because even informally, the statements don't entail a both statement and its negation.

relevance is another way of saying logical connection — Janus

I wouldn't use the word 'logical' since that has a certain meaning in the study of logic that is not the same as 'relevance'.

I asked you if any were nonsensical, I didn't say they were. — Janus

Oh, I thought your question was rhetorical. I thought you meant that it is a genuine philosophical question about computing.

I don't know enough to answer your question. So I would turn it around to ask: Is there an argument to be made that they are nonsensical?

In informal language if the antecendent has no relevance to the consequent then I would say that counts as nonsensicality. — Janus

You asked if the logic paths are nonsensical. I thought 'logic paths' related to the logic gates you mentioned in your previous sentence.

I was not thinking in terms of formal logic — Janus

As someone pointed out, the use of variables suggest formality. But, of course, we may address the question in both formal and informal contexts. And I've done that.

If the two sentences were 'if monkeys had wings, then they could fly to the moon' and 'if monkeys had wings, they could not fly to the moon' the two sentences contradict one another regardless of whether it is true that monkeys have wings or whether it is true that if they had wings they either could or could not fly to the moon. — Janus

That's incorrect, formally or informally. I explained why it's not correct.

but assuming that there would be some logical connection between the conditional and the implications (and why would we even bother thinking about statements where there is no such logical connection) then the two statements do contradict one another. — Janus

I don't think so. Or I would like to know of a system or approach that supports it.

I sense that it is not logical connection you have in mind, but rather, what is called in logic, 'relevance'. If the relation between the antecedent and consequent is not relevant, then that does not accord with many everyday uses of 'if then'. But that doesn't entail that the two statements contradict each other. It only entails the statements don't accord with certain everyday uses. Same for green snow and Macron's nationality. But, of course, the material conditional also does not accord with relevance logic. Though I'd have to do a bit of reading to see whether even in relevance logic (A -> B) & (A -> ~B) is a contradiction when the implications are not relevant.

Are any of those useful logic paths nonsensical? Genuine question... — Janus

How are they nonsensical?

There's a mistake in the last row. The value of ~A v ~B is T. So there are two rows, not just one, where (A -> B & (A -> ~B) is true.

'=' is interpreted:

For any terms 'T' and 'S'

T = S

is true

if and only if

the denotation of 'T' is the denotation of 'S'. — TonesInDeepFreeze

The crank claims that we may look in a textbook in mathematics to see that mathematics doesn't agree. What textbooks are those?

And notice that the crank has shifted his argument. Previously he recognized that mathematics regards '=' as 'is the same as' and that mathematics is wrong to do that, but now he's claiming mathematics doesn't but that it is only mathematical logic that does, modulo his latest tact of blaming only pure mathematics. The crank is always greased for easy shifting.

Most simply, when we say "1+1 = 2" we mean that '1+1' and '2' name the same number. Or we are to believe they name different numbers? What numbers are those?

Clerk: Okay, we have one plus one plus one plus one. Okay, that is four. At $2 each, that's $8.

The Crank: No, one plus one plus one plus one is not four. I only pay for one plus one plus one plus one, not for four

Clerk [into mic]: We have a problem at register ten.

Mathematical logic formalizes the logic used in other mathematics. The explication of '=' in mathematical logic conforms to the use in mathematics.

The crank says that we may look in a textbook in mathematics to see that the definition of '=' differs from mathematical logic. What specific textbook does the crank refer to?

"two values are the same"

Indeed, in both mathematical logic and in other mathematics:

'1+1 = 2' means that the value of the expression '1+1' is the same as the value of the expression '2'.

In ordinary formal logic and classical mathematics, the material conditional obtains. But, of course, there are other natural language senses.

In everyday speech, one would not ordinarily say "If snow is green then Emmanuel Macron is an American, and if snow is green then Emmanuel Macron is not an American". But even then, the assertion is not inconsistent unless we also assert "snow is green".

A contradiction is a pair of statements of the form P and ~P.

Such as "Emmanuel Macron is an American" and "Emmanuel Macron is not an American".

Notice that "If snow is green then Emmanuel Macron is an American" and "If snow is green then Emmanuel Macron is not an American" is not of that form and together they don't imply the contradiction "Emmanuel Macron is an American" and "Emmanuel Macron is not an American". They only imply that contraction along with the statement "Snow is green".

An example where we do say both "A then B" and "A then not B":

Background: A lawyer is defending Ruth. We know that Ruth wore a blue dress on the night of the crime. And we know that the assailant did not wear a blue dress on the night of the crime. The lawyer says:

"If Ruth is the assailant, then the assailant wore a blue dress" and "If Ruth is the assailant then the assailant did not wear a blue dress. So, the assertion that Ruth is the assailant implies a contradiction, so Ruth is not the assailant."

That's an awkward and verbose way of saying "The assailant wore a blue dress, but Ruth did not, so Ruth is not the assailant". But despite it being awkard and verbose, it is correct English and logical.

/

As to what the purpose of formal logic is, there are different purposes. For logic that pertains to mathematics and computability, ordinarily the material conditional is used. For example, the computer you're using now is based on logic paths in which "if then" is the material conditional.

Does (A implies B) mean that 'if A then B'? Does (A implies notB) mean that 'if A then not B'? If the answer is 'yes' to both, then they contradict one another. — Janus

The answer is 'yes' and they do not contradict each other. You can read the several differently arranged proofs of that in this thread.

if you want to make a point, link or note your point — Philosophim

There's no note or link needed. You can find out about material implication all over the place. I'm not your linking service.

"Look at a textbook" is dismissive and means you're removing yourself from the conversation. — Philosophim

You skipped what I said about "dismissiveness". You merely reiterate your claim without addressing my response to it. That is dismissive. And of course, by giving my best advice to look at a textbook, I am not thereby opting out of any conversation. Interesting that you continue to attack me rather than take my offer for some recommendations of books or other resources.

Then give your proof.
— Leontiskos

Are you serious? You don't know how to prove it yourself?
— TonesInDeepFreeze

Not exactly the model of a sage and wise poster. — Philosophim

You leave out that I went on to give a proof in two versions. And it is appropriate to ask whether a poster is really serious asking for something that is, as far logic is concerned, as simple as showing that 4 is an even number. If in a thread about number theory someone happened to write "4 is even", and then another said "Prove it", you think that would not be remarkable enough to reply "Are you serious? You don't know how how to prove it?", let alone to then go on to prove it anyway.

You came on here with a chip on your shoulder to everyone. — Philosophim

Where is here? This thread? I came with no shoulder chip, not to anyone, let alone "everyone". If I permitted myself to do as you do - to posit a false claim about interior states - I would say that you do so from your own umbrage at having been corrected.

And my point stands that I did not insult you, whereupon you insulted me.

I gave you a chance to have a good conversation — Philosophim

By saying "don't be a troll".

You can converse as you please. I'm not stopping you. And I have read your subsequent posts, even after your insulting "don't be a troll" and have given you even more information and explanation. I have not shut down any conversation.

Now if you had an issue with my use of -> or wanted to teach me the difference between a modal and material implication, something I did not know before today — Philosophim

The first step is to at least point out that '->' does not mean "necessarily leads to". And, I'm glad that you do know that there is a difference now, and glad that my posting the correction has led to you knowing that there is a difference.

citing a wiki post — Philosophim

(1) I don't usually reference Wikipedia or similar sites. They often have misinformation and poor explanations. It is not my job to find a site for you, read it through to vet it for accuracy, then fashion a link for you.

(2) I gave you even better advice anyway.

(3) And if you had asked for more, then I would have recommended specific texts or suggest search terms for you, though you could fashion your own search.

we have wasted time back and forth — Philosophim

I'll judge for myself what is or isn't my time wasted. But your time wasn't wasted since at least my correction led to you learning something about logic, and not just an incidental detail but rather a key fundamental aspect of logic.

Share it and teach. — Philosophim

I don't presume to be a teacher. But I have shared a tremendous amount of information and explanation over years in this forum alone.

If he is using the term of implication to mean, "could lead to" — Philosophim

I am not. I'm treating '->' as standing for material implication as is ordinary.

I did not catch that 'material' conditional was anything different from the modal operator. — Philosophim

They are very different.

Your attitude is hostile and condescending — Philosophim

Actually, you insulted me. I hadn't written anything "hostile" or "condescending" but then insultingly you wrote:

Don't be a troll. — Philosophim

without backing up your claim clearly — Philosophim

I gave you the best advice anyone could ever give you regarding this subject: Look at a textbook. Then you could read a full explanation in full context, thus to inform yourself on the subject properly. And there is no better "proof" that '->' ordinarily means material implication than to read for yourself in an authoritative and widely referenced textbook.

The fact that '->' is ordinarily understood as the material conditional is ubiquitous. It's not my call to provide you references that you could easily find yourself by just looking at basic texts.

You spoke so tersely and dismissively — Philosophim

Terse in the sense of "devoid of superfluity" not in the sense of "brusque". To suggest looking in a textbook is the best advice I can give you; It should not needed be needed for me to further elaborate on that advice. However, if you asked me to recommend textbooks, then I would be happy to do that. And to be really dismissive I could have simply ignored you; instead I gave you the very best advice I can give.

you misunderstood what I was stating earlier. I'm replying to someone specifically in which I covered both types of meanings of the words 'imply', as the OP did not specify what they meant. One where "Imply" means "necessary" and one where imply means "Could lead to". — Philosophim

You misunderstand. You said that '->' means 'necessarily leads to'. And that is false. And "necessary" and "could lead to" are not the two meanings of '->', as material implication is the ordinary meaning and does not mean "necessarily leads to" nor "could lead to".

and "imply ¬A" as the proposition being True means A is False
— Lionino

Yes, this was my concern. Tones requires the assumption, as I thought he must. — Leontiskos

That is what I replied to.

"imply ¬A" as the proposition — Leontiskos

'imply ~A' is not a proposition, and I didn't say that it is, so I don't require it as an assumption.

I don't claim that one may not discuss all kinds of non-formal, formal, alternative formal, or philsophically formal or informal, or mystical New Age incantational informal senses.

and "imply ¬A" as the proposition being True means A is False
— Lionino

Yes, this was my concern. Tones requires the assumption, as I thought he must. — Leontiskos

I don't require such an assumption. "imply ~A" is not even a proposition.