"proposition" here refers to "(A → B) ^ (A → ¬B)", not to "imply ¬A"

Anyway we clarified that here https://thephilosophyforum.com/discussion/comment/916729

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.

↪Philosophim Wiki might suffice to show you the difference between material implication and strict implication. That might be what you have in mind.

Tones is correct. — Banno

If he is using the term of implication to mean, "could lead to", then that's fine. I've already written on that on the first page. I did not catch that 'material' conditional was anything different from the modal operator.

It is not trolling to point out an incorrect statement, and it not trolling nor handwaving to suggest that one can look in textbooks to see that the statement is incorrect. — TonesInDeepFreeze

Your attitude is hostile and condescending without backing up your claim clearly. I had no idea that there was a specific logical term called a material conditional. You spoke so tersely and dismissively, I didn't take your reply seriously. Give detail and respect, and it will be given back by good people.

I will state again, 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".

I already noted in the second case that A could lead to B and A could lead to not B are not contradictions. But they are contradictions in the sense of using the word 'implication' as necessary. Or to use Banno's link if we are going to use formal logical terminology, 'material conditional' vs a 'modal operator'.

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".

Related:

...but we can also usefalse propositions in good reasoning. Since a false conclusion cannot be logically proved from true premises, we can know that if the conclusion is false then one of the premises must also be false, in a logically valid argument. A logically valid argument is one in which the conclusion necessarily follows from its premises. In a logically valid argument, if the premises are true, then the conclusion must be true. In an invalid argument this is not so. "All men are mortal, and Socrates is a man, therefore Socrates is mortal" is a valid argument. "Dogs have four legs, and Lassie has four legs, therefore Lassie is a dog" is not a valid argument. The conclusion ("Lassie is a dog") may be true, but it has not been proved by this argument. It does not "follow" from the premises.

Now in Aristotelian logic, a true conclusion logically follows from, or is proved by, or is "implied" by, or is validly inferred from, only some premises and not others. The above argument about Lassie is not a valid argument according to Aristotelian logic. Its premises do not prove its conclusion. And common sense, or our innate logical sense, agrees. However, modern symbolic logic disagrees. One of its principles is that "if a statement is true, then that statement is implied by any statement whatever." Since it is true that Lassie is a dog, "dogs have four legs" implies that Lassie is a dog. In fact, "dogs do not have four legs" also implies that Lassie is a dog! Even false statements, even statements that are self-contradictory, like "Grass is not grass," validly imply any true conclusion in symbolic logic. And a second strange principle is that "if a statement is false, then it implies any statement whatever." "Dogs do not have four legs" implies that Lassie is a dog, and also that Lassie is not a dog, and that 2 plus 2 are 4, and that 2 plus 2 are not 4.

This principle is often called "the paradox of material implication." Ironically, "material implication" means exactly the opposite of what it seems to mean. It means that the matter, or content, of a statement is totally irrelevant to its logically implying or being implied by other statements. Common sense says that Lassie being a dog or not being a dog has nothing to do with 2+2 being 4 or not being 4, but that Lassie being a collie and collies being dogs does have something to do with Lassie being a dog. But not in the new logic, which departs from common sense here by totally sundering the rules for logical implication from the matter, or content, of the propositions involved. Thus, the paradox ought to be called "the paradox of wort-material implication. The paradox can be seen in the following imaginary conversation:

Logician: So, class, you see, if you begin with a false premise, anything follows.
Student: I just can't understand that.
Logician: Are you sure you don't understand that?
Student: If I understand that, I'm a monkey's uncle.
Logician: My point exactly. (Snickers.)
Student: What's so funny?
Logician: You just can't understand that.

The relationship between a premise and a conclusion is called "implication," and the process of reasoning from the premise to the conclusion is called "inference" In symbolic logic, the relation of implication is called "a tnith-functional connective," which means that the only factor that makes the inference valid or invalid, the only thing that makes it true or false to say that the premise or premises validly imply the conclusion, is not at all dependent on the content or matter of any of those propositions, but only whether the premise or premises are true or false and whether the conclusion is true or false.


...Logicians have an answer to the above charge, and the answer is perfectly tight and logically consistent. That is part of the problem! Consistency is not enough. Logic should be not just a mathematically consistent system but a human instrument for understanding reality, for dealing with real people and things and real arguments about the real world. That is the basic assumption of
the old logic.

Peter Kreeft & Trent Dougherty - Socratic Logic

I gave you the best advice anyone could ever give you regarding this subject: Look at a textbook. — TonesInDeepFreeze

You do me or no one else favors here. We're having a discussion, and if you want to make a point, link or note your point. "Look at a textbook" is dismissive and means you're removing yourself from the conversation.

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 — TonesInDeepFreeze

Like this?

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. You came on here with a chip on your shoulder to everyone. I gave you a chance to have a good conversation, but I didn't see a change in your attitude.

You misunderstand. You said that '->' means 'necessarily leads to'. And that is false. — TonesInDeepFreeze

If I'm clearly using it as a strict conditional, as I noted in prior posts as I was talking to someone other than yourself, then I'm correct. 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, you could have spent less then a minute citing a wiki post somewhere like Banno did. Instead, we have wasted time back and forth and your attitude didn't win you respect today.

This is not a place where we should banter back and forth for our egos. Its about spreading knowledge and wisdom with one another with good discussions. You seem to have knowledge, which is wonderful. Share it and teach. You'll earn respect. Don't bare it and preach. You'll get eyes rolled at you.

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.

Material implication is often written in natural language as:

If A, then B
Or
A implies B

But they aren't perfect translations because all sorts of shit that sounds very dumb in natural language flies in symbolic logic. E.g. "if Trump won the 2020 election then we would have colonized Mars by now."

Anything follows from a false antecedent, so anything would be "true" following the claim that Trump won the 2020 election, since he didn't. But obviously in natural language the if/then here is talking about a counterfactual claim, and to say that if x had happened then y and not-y is (normally) nonsense talk.

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.

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.

I can't think of any examples in natural language where "if A then B" and "if A then not B" do not contradict one another.If "proofs" do not accord with this fact, then so much the worse for the "proofs", given that formal logic is designed to illuminate natural language, not replace it.

But they aren't perfect translations because all sorts of shit that sounds very dumb in natural language flies in symbolic logic. E.g. "if Trump won the 2020 election then we would have colonized Mars by now."

Anything follows from a false antecedent, so anything would be "true" following the claim that Trump won the 2020 election, since he didn't. — Count Timothy von Icarus

I agree, I came across such things when studying logic as an undergraduate. I will just say that "if Trump won the 2020 election, then we would have colonized Mars by now" is neither true nor false (or at least cannot be determined to be true or false).

So I don't think it is true that "anything would be true following the claim that Trump won the 2020 election, since he didn't", because it could equally be said that "anything would be false following the claim that Trump won the 2020 election, since he didn't".

Eh, I'd say the formal logic is built upon natural language, but we can get by with specification of meaning -- and when the OP is in the Logic sub-forum it makes sense to default to trained logic, especially when it's using the language of "A", like a variable, so it's already abstract and not a natural language construction.

The way you would usually use it in any sort natural language statement would be to say: "Look, A implies both B and not-B, so clearly A cannot be true." You don't have a contradiction if you reject A, only if you affirm it.

This is a fairly common sort of argument. Something like: "if everything Tucker Carlson says about Joe Biden is true then it implies that Joe Biden is both demented/mentally incompetent and a criminal mastermind running a crime family (i.e., incompetent and competent, not-B and B) therefore he must be wrong somewhere."

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.

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.

ahhh yup. You're right.

I'll add a link to your post here to the original post.

The original question regarded '->', which ordinarily is taken as the material conditional. — TonesInDeepFreeze

Thanks, you obviously know much more about formal logic than I do. However I was not thinking in terms of formal logic, since the original question contains no symbols from formal logic

Do (A implies B) and (A implies notB) contradict each other? — flannel jesus

My point was that if you have two sentences 'if A then B' and 'if A then notB' they simply contradict one another regardless of whether A obtains. Of course we don't know what A is. 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.

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". — TonesInDeepFreeze

I think that is one way of interpreting it, separating the obviously false, or disconnected conditional "if snow if green" from the two contradicting statements about Macron, 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.

For example, the computer you're using now is based on logic paths in which "if then" is the material conditional. — TonesInDeepFreeze

Right, I do have some familiarity with logic gates. Are any of those useful logic paths nonsensical? Genuine question...

The way you would usually use it in any sort natural language statement would be to say: "Look, A implies both B and not-B, so clearly A cannot be true." You don't have a contradiction if you reject A, only if you affirm it. — Count Timothy von Icarus

It seems to me that saying you don't have a contradiction is one way of interpreting it; that is, I don't think there is any fact of the matter. Consider "If lizards were all purple then they'd be a hell of a lot smarter" and "if lizards were all purple, they would not be a hell of a lot smarter": you don't see those two sentences as contradicting one another despite the fact that lizards are not all purple?

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?

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

Why is it incorrect informally?

I sense that it is not logical connection you have in mind, but rather, what is called in logic, 'relevance'. — TonesInDeepFreeze

Yes, relevance is another way of saying logical connection in the context.

How are they nonsensical? — TonesInDeepFreeze

I asked you if any were nonsensical, I didn't say they were. In informal language if the antecendent has no relevance to the consequent then I would say that counts as nonsensicality.

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.

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".

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

I'm not sure what you mean: I was considering the two statements separately and it still seems to me, that regardless of the soundness or relevance of their content, that, taken informally as statements, they contradict one another.

"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

I see those two statements as saying contradictory things about what lizards being purple would entail.

"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. — TonesInDeepFreeze

I see two of those statements, as above, as being contradictory and the third as being unsound. And I see the two contradictory statements as saying nothing about whether lizards are purple. I mean, I think it's fair to say that both of the conditional statements are untrue, because being or not being smarter has no logical connection with being purple. Or 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.

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?

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"?

I'm not sure, but maybe you want to check whether you are conflating "not intuitive" with "contradictory". — TonesInDeepFreeze

I don't believe I am conflating "not intuitive" with "contradictory", but of course I admit I could be wrong. Do I understand what 'contradictory' means? I think so.

What about "the present king of France is bald" and "the present king of France is not bald" do they contradict one another?

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." 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." We say, "No, they would not (be smarter in that case)." 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.

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

I'm not sure what you mean: I was considering the two statements separately and it still seems to me, that regardless of the soundness or relevance of their content, that, taken informally as statements, they contradict one another. — Janus

Note 's testimony.

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.

Right, I do have some familiarity with logic gates. Are any of those useful logic paths nonsensical? Genuine question... — Janus

Well... yes, kind of.

From falsehood, anything follows. Have you ever heard of this? This example before us is a great example of that.

You think if a then b, and if a then not b contradict each other.

Many other posters think they don't contradict each other, BUT with the caveat that if they're both valid statements, A must be false.

If a is false, and "from falsehood, anything follows", then (if a, then anything) fits. Replace anything with b, replace it with not b, replace it with a snail with a tophat, replace it with à̶̙̦͔́̀b̴͈̼̞̓͘y̵̝̣̳̲̟̤͑̏̈́͝ş̷̭̼͖͓̼̈̿̈́͐͐̃̕ș̶̡̲̘̯́́̋̄͘͜ä̵̉̓͊̋͜l̸̯͛̀̒̕ ̷̞͎͔̱͛̕d̴̪̬̻̠͕̋̃͗̾̉ẹ̴̪̭̌̒͝ś̷̢̢̢͔͉͎̄̿p̸̨͎̘̼̬̼͇̓͌̊ā̵̢̤̗͌i̴̡̤̹̘̰̿͜͝r̴͉͙̣̍̂̈́̓̄̚...

As long as A is false, "if a then anything" obtains. You can verify this with the truth table. And this is where your sense of nonsense comes in. Do you see?

From falsehood, anything follows. Have you ever heard of this? — flannel jesus

I don't think the principle of explosion is quite the same as material implication. It's kind of the opposite. We are running from a contradiction, not running on a contradiction. See and .

You're certainly not alone in thinking that,

But I personally think it's not a coincidence that "from falsehood, anything follows" perfectly mirrors how, if you phrase "A -> B" as "from A follows B", then if A is false, you can say "A -> anything", from A anything follows.

I don't think that's a coincidence at all. I think the principle of explosion is actually really relevant here. But I understand that not everyone sees it that way.

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.