if you phrase "A -> B" as "from A follows B", then if A is false, you can say "A -> anything", from A anything follows — flannel jesus

It does not follow; it is moot. According to material implication (A → B) is true if A is false, but B does not follow given that A is false. We cannot derive B. That's why A is false in this case, because we cannot arrive at the contradiction of (B ^ ~B).

I don't know what you think I'm saying, but I feel like you're misunderstanding it.

Of course I agree that we can't conclude B and notB. The fact that you're saying that makes me think you've misunderstood what I said.

- On explosion the consequent "follows" in the sense that it can be affirmed as true. That is not the case here.

no - the consequent can only be affirmed as true IF the antecedent is first affirmed as true. It's THAT that is not the case here.

I'm not affirming the antecedent, so I'm not affirming the consequent.

no - the consequent can only be affirmed as true IF the antecedent is first affirmed as true. It's THAT that is not the case here. — flannel jesus

"Who are you, who are so wise in the ways of science?"

Put it together: ...therefore the consequent cannot be affirmed as true in this case. Therefore the consequent does not "follow."

The consequent follows from the premise in the implication, (A -> B)

You think when I use the word 'follow', and completely understandably, I mean "this thing is true". As in, I'm saying "B is true" period.

I'm not.

"follow" can also just be a synonym for implication. A -> B, From A follows B. If you assert A, B follows.

I can say "A -> B" without asserting B, and in the same vein, I can say "From A follows B" without asserting B. Because they're just different ways of phrasing the same thing.

I'm not asserting B. I'm asserting A -> B. You have to see the difference to understand. When you understand how I can assert A -> B without asserting B, you can understand how I can say "From A follows B", without me saying "B is true".

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.

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

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

There are two separate matters: negation and material implication. — TonesInDeepFreeze

The negation of a material conditional will be different from the negation of an if-then statement in natural language, and my post was highlighting that difference.

Or as I said earlier:

Given the way that common speech differs from material implication, in common speech the two speakers would generally be contradicting one another. — Leontiskos

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.

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

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.

I'm not saying "the material conditional in Boolean logic used for computing is nonsensical".

You changed the sentence. Here is what you wrote: — TonesInDeepFreeze

No, I was there giving an answer to the question at hand.

"If lizards were purple then they would be smarter" is not a contradiction — TonesInDeepFreeze

I give up. Go read Lionino's first post on the first page. He explains the two basic senses of contradiction operating in the thread.

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.

Your answer is incorrect. — TonesInDeepFreeze

You don't even understand what is being said. :roll:

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

I suspect you don't understand what you wrote.

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.

I take 'the contradictory statement is P' to mean that P is a contradiction, as a contradictory statement is a contradiction. — TonesInDeepFreeze

And I already corrected your misinterpretation in <this post>.

But maybe you mean it is a contradicting statement. — TonesInDeepFreeze

I'm glad you finally figured this out and even came up with your own fun way of describing it in English. Now you should go back and reread the original post, using what you have learned about your misinterpretations.

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?"

And if you think they do contradict each other, does that mean they can't both be true at the same time? — flannel jesus

Haven't read through all the posts so far. Still, to given one answer:

They'd contradict each other only if they are claimed to occur both at the same time and in the same respect. This irrespective of what "imply" might be taken to precisely specify.

For example, to the question "Is that song any good?" can validly be replied, "Yes and no," without any logical contradiction. Here, yes and no at the same time but in different respects: such that yes, the song is of substantial quality and thereby good and no, the song will be unable to make any revenue and is thereby bad.

Hence, here, song A implies (is taken to hold as a necessary consequence) quality B (here that of "goodness") and song A implies notB - this at the same time, but in different respects, and, hence, in perfectly non-contradictory manners.

Maybe this isn't the best use of the term "implies", but the same principle remains.

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'?

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.

It's not a misinterpretation. To say that P is contradictory is to say that P is a contradiction. — TonesInDeepFreeze

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

So you're wrong whether we interpret it as an adjective or a noun. You don't seem to have a great grasp of natural language.

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!"

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

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."

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.

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.

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.

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.

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

This is Kreeft and Dougherty's argument for the superiority of Aristotlean logic for many common uses (evaluating writing, rhetoric, scientific arguments, etc.)

See: https://thephilosophyforum.com/discussion/comment/916812

It seems to me though that the real benefit is that the "three acts of the mind," end up enforcing a sort of realism on the interpretation of syllogisms.

If the long reply made you feel better, that's fine. You can't argue against how you come across to other people on a forum. Hopefully we'll have a better encounter in another thread. Good luck in explaining your side, I do agree with it.

For what its worth, I think you're running into a mismatch between most people's general sense of seeing -> as a strict conditional. Perhaps in your field or life 'material conditional' is a common phrase, but for most people who use logic, this is never introduced. For them, its 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. Explaining and contrasting a strict conditional vs a material conditional should make the issue clear for most people.

This forum is populated by all kinds of people, yes. But I would ask you to remember that you're posting in the logic subforum, and "@TonesInDeepFreeze has responded with that in mind: and done so with precision and accuracy, so I'm grateful at least for their help.

"most people's general sense of seeing" -- I mean we all have places we come from and thoughts we start at, but if you walk into the chemistry department and start talking alchemy someone might correct you.

I mean we all have places we come from and thoughts we start at, but if you walk into the chemistry department and start talking alchemy someone might correct you. — Moliere

This is not a forum exclusive to collegiates, this is a general public forum. I have no issue with being corrected or told new things. While he may have responded well to you, he jumped into a conversation I was having with another poster without context, and when I asked him to clarify his issue he came across as dismissive. I encourage you not to do the same and jump into another conversation between two people.

We did have a conversation earlier though right? You asked my take on the barbershop paradox, and asked my clarification on what I meant by this being a language issue. Did you have any follow up on that?