No, it isn't. Truth tables are easy enough to learn, and easy to do, if you don't have too many variables.
(p=>q)=>(~p=>~q) is false when p is false and q is true. — tim wood

Well spotted Tim. I am a bit too lazy to be staring at Truth tables, and was too busy at the time, and was guessing. It depends on the TF values of p q in the table. There is nothing always.

I'm not sure which two you're referring to, or what disconnect is being pointed out.

For example, consider the bi-conditional proposition: "If it is raining, then the ground is wet." — AmadeusD

"The bi-conditional statement "if it is raining, then the ground is wet" is true because it goes both ways. — AmadeusD

Are about the same quote. In any case, I don't defend or criticise it.

I've adjusted my query, and had another version of it's defence, though:

"Yes, the statement "if it is raining, then the ground is wet" is a biconditional statement because it can be written as "it is raining if and only if the ground is wet." This means that if it is raining, then the ground must be wet and if the ground is wet, then it must be raining. Both conditions are necessary for the statement to be true."

A good video for the thread.

A page that allows us to show truth tables:

https://truth-table.com/#(p→q)→(~p→~q)

Useful for those who are not too lazy to be staring at Truth tables.

Useful for those who are not too lazy to be staring at Truth tables — Banno

So, this entire discussion devolves to something I used to teach freshmen.

:wink:

Yeah, not the forum's finest hour.

The poor neck-beard can't afford heating. :worry:

Via three AI models: — AmadeusD

Interesting!

It is common for llms to provide incorrect answers when asked to provide a conclusion in advance of providing an explanation, or without allowing them to think step-by-step. Here is how I tripped GPT-4, and how it recovered on its own:

Pierre-Normand

Hi GPT-4!

Is it logically valid to deny the antecedent in a biconditional proposition?

GPT-4:

Denying the antecedent in a biconditional proposition is not logically valid, and here's why:

A biconditional proposition can be represented as "P if and only if Q" (P ↔ Q). It states that P is true if Q is true, and Q is true if P is true. This implies two things:

1) If P then Q (P → Q)

2) If Q then P (Q → P)

Denying the antecedent generally refers to a logical fallacy in a conditional statement (if P then Q), where one incorrectly infers that if P is not true, then Q is not true. In symbolic logic, this fallacy is represented as: Assuming "If P then Q" (P → Q) and "Not P" (~P), then concluding "Not Q" (~Q), which is invalid.

In the context of a biconditional, denying the antecedent would mean arguing that because P is not true, Q must also not be true. However, this doesn't hold because the biconditional also includes the reverse conditional (If Q then P). For example, consider the biconditional "You can drive if and only if you have a license." This means if you have a license, you can drive, and if you can drive, you have a license. Denying the antecedent here would be saying, "If you don't have a license, then you can't drive," which is true and follows logically. But this isn't a fallacy—it's just an application of the reverse conditional (If Q then P).

So, in a biconditional proposition, "denying the antecedent" can lead to a valid conclusion if you're applying the reverse conditional correctly. It's essential to differentiate between denying the antecedent in a simple conditional statement (which is a fallacy) and applying the logic of a biconditional proposition, where both directions of implication hold.

Pierre-Normand

Indeed! So, interestingly, you began your answer with denying the logical validity of this form of reasoning but then, after thinking it through, correctly concluded that it is, after all, valid.

GPT-4:

You're right, and I appreciate your patience as we navigated through the intricacies of this topic. Biconditional propositions indeed add a layer of complexity beyond simple conditional statements.

To clarify, when dealing with a biconditional proposition "P if and only if Q" (P ↔ Q), denying one component does logically allow us to conclude the denial of the other, due to the nature of the biconditional relationship. This is because a biconditional asserts that both P and Q are equivalent in terms of their truth values; they're either both true or both false. Therefore, denying one (saying it's false) does indeed allow us to conclude that the other is also false, which is a valid form of reasoning in this specific context.

Thank you for pointing out the need for clarification. The nuances of logical reasoning in different contexts can lead to interesting discussions!

Pierre-Normand

What interests me is the reason why you spontaneously gave a wrong answer in the beginning of your response. My conjecture is that the superficial similarity of my question with the very commonly stated explanation (in your training data) of the invalidity of denying the antecedent (of a conditional proposition) tripped up the next-token prediction algorithm, as it were. While you proceeded with generating the rest of your explanation, the richer context thereby generated (by the tokens already produced) enabled the underlying language model to latch on the correct rational pattern and to correctly articulate your response in a way that that is contextually sensitive to the fact that the topic at hand was biconditionals rather than simple conditionals.

GPT-4:

Your conjecture touches on some of the complexities and subtleties involved in how language models like mine generate responses. When I receive a prompt, I generate responses based on patterns, structures, and information present in the training data up to my last update in April 2023. Your observation about the initial incorrect direction in my response could indeed be influenced by the prevalence of discussions around the fallacy of denying the antecedent in conditional statements, which are more common than discussions specific to biconditional propositions.

As I generate text, each subsequent token (or word) is chosen based on the context of the preceding text, aiming to form a coherent and contextually relevant response. This process involves a complex interplay of probabilities and contextual cues. In the case of your question, the initial framing might have activated patterns related to the more common discussions around conditional statements. However, as the response progressed and the context became richer and more specific to biconditionals, the language model adjusted to better align with the correct logical principles pertaining to biconditional propositions.

This dynamic adjustment is a characteristic of how language models generate text, striving for coherence and relevance based on the evolving context of the response. Your insight into this process highlights the importance of context in natural language understanding and generation, as well as the adaptive nature of language model responses as they unfold.

In deductive syllogism it is fallacy — Corvus

Oh wow, that's amazing that you're saying that.

In this post you call it a deduction - you use the word "deduces".

https://thephilosophyforum.com/discussion/comment/889798

So you were incorrect about that when you said that?

And then earlier in this thread you agreed with the following:

"So do you think any time you have (a implies b) , it's always true that (not a implies not b), you just don't always need to bring that up. Is that what you're saying?"

But since now you're saying it's a Fallacy, then the above quote that you agreed with can't be true.

Well spotted Tim. — Corvus

But people have been telling you that for weeks already...

Anyway, I consider the thread resolved. Corvus sees that it's a Fallacy, he's agreed with Tim about the Truth table which illustrates that it's a Fallacy, so... thread resolved.

. Therefore, denying one (saying it's false) does indeed allow us to conclude that the other is also false, which is a valid form of reasoning in this specific context. — Pierre-Normand

This is clearly wrong. I'll leave it there.

The issue with this "it sprinkles water, the lawn is wet" is that Corvus confuses material implication with physical causation. Me, flannel and Banno spent over some 20 pages discussing this, we are veterans.
A better example to use is
I am running → I am moving
Does «I am not running» → «I am not moving»? The answer is evidently no. So from P→Q we don't have ¬P→¬Q. ¬P→¬Q is in fact the same as Q→P, which obviously is different from P→Q.
P→Q – Everytime there is P, there is also Q
Q→P – Everytime there is Q, there is also P
¬P→¬Q Everytime there is not P, there is also not Q; which is the same as saying that everytime there is Q, there is also P.

Edit: Sorry, I didn't see the thread was solved :-P

This is clearly wrong. I'll leave it there. — AmadeusD

How is it wrong? This was stated about a biconditional statement. In a biconditional statement 'P iff Q', the truth of either P or Q entails the truth of the other one while the falsity of either P or Q entails the falsity of the other one. In other words, 'P iff Q' entails that either P and Q are both true or P and Q are both false.

denying one component does logically allow us to conclude the denial of the other — Pierre-Normand

Therefore, denying one (saying it's false) does indeed allow us to conclude that the other is also false, which is a valid form of reasoning in this specific context. — Pierre-Normand

"This statement is logically incorrect because when one component of a biconditional statement is denied, it does not necessarily allow us to conclude the denial of the other component. In a biconditional statement "P if and only if Q" (P ↔ Q), denying one component does not automatically lead to the denial of the other component. This is because the truth values of P and Q are independent of each other in a biconditional statement, and denying one does not necessarily mean the denial of the other. So, it is not valid to conclude the denial of one component based on the denial of the other in a biconditional statement."

It is not entailed that hte denial of one requires the denial of hte other. I should have been clearer in my objection. It was clearly inadequate.

What he is trying to say there is that for the biconditional to be true, the truth values of P and Q must always match. Of course if you just put the biconditional on a truth table and leave it there, nothing will follow.

denying one component does not automatically lead to the denial of the other component. This is because the truth values of P and Q are independent of each other in a biconditional statement, and denying one does not necessarily mean the denial of the other. — AmadeusD

I haven't checked the logic yet, but I'm pretty sure you're exactly incorrect about this. P <-> Q does not mean they're independent, it means they always have the same truth value. Denying one always means the denial of the other. Affirming one always means affirming the other. Happy to prove it if necessary - with truth tables, or just using basic Modus Tollens.

"This statement is logically incorrect because when one component of a biconditional statement is denied, it does not necessarily allow us to conclude the denial of the other component. In a biconditional statement "P if and only if Q" (P ↔ Q), denying one component does not automatically lead to the denial of the other component. This is because the truth values of P and Q are independent of each other in a biconditional statement, and denying one does not necessarily mean the denial of the other. So, it is not valid to conclude the denial of one component based on the denial of the other in a biconditional statement."

It is not entailed that hte denial of one requires the denial of hte other. I should have been clearer in my objection. It was clearly inadequate. — AmadeusD

The two initial snippets that you quoted and attributed to me were written by GPT-4, although I am agreeing with them.

Who is the author of the long paragraph between quotes that you posted above ("This statement...")? Was that authored by the free version of ChatGPT (hence GPT-3.5 turbo)? What was the context and the prompt? It rather sounds like GPT-3.5 when it is in the throws of a bad case of hallucination and has lost coherence after having done a reasoning mistake and thereafter attempts to justify its initial claim. (Note that many human beings behave like that too.)

Yes, and that much is clear - Largely why I didn't say anything until this last couple of pages: You lot had it well-covered.

But that's not what I was saying was wrong. I entirely misspoke in my objection.

It is not entailed that hte denial of one requires the denial of hte other. I should have been clearer in my objection. It was clearly inadequate. — AmadeusD

This seems to me something I can claim without knowing much formal logic (at least, not conversant).

"It has rained, and so my lawn is wet"

It is plainly true that denying your Lawn is wet does not entail denying that it has rained and vice verse. They are conditionally distinct. The only thing that requires the truth values to align in P and Q here is so. It's possible that what I'm outlining, without hte requisite knowledge to discuss it adequately, is that bi-conditional statements aren't a logically sound tool because we can look at the logic of a bi-conditional statement that appears sound, and it can be wrong.

This is not my utterance. Here's an interesting further response I got:

"I apologize for the confusion. Let me clarify:

In a biconditional statement "P if and only if Q" (P ↔ Q), the truth values of P and Q must be the same. This means that if P is true, then Q must also be true, and if P is false, then Q must also be false. So, in the context of affirming the components, the truth values are indeed dependent on each other.

However, when we talk about denying one component not necessarily leading to the denial of the other, what is meant is that if we deny the truth of P, it does not automatically mean that Q is false (and vice versa). The components are logically connected in a biconditional statement in the sense that they must have the same truth value, but denying one component does not automatically imply the denial of the other."

I believe the difference I'm seeing is between assertion and assent. It seems fully wrong to me that you can assent as a result of assertion, without something (other than triangle of dots) between them.

Who is the author of the long paragraph between quotes that you posted above — Pierre-Normand

An AI. Which also, when i pushed further, said this:

"So, in a biconditional statement, if you affirm Q, then you must also affirm P, and if you deny P, then you must also deny Q in order for the statement to be logically sound. The truth values of P and Q are interconnected in a biconditional statement, so they cannot be affirmed or denied independently of each other. Thank you for pointing out the inconsistency in my previous responses."

So, my objection isn't to the logic, it seems. It's to the application. The logic clearly fails in many cases.

This is not my utterance. — AmadeusD

Ah my mistake, it was in quotes. But you still ended your post with

It is not entailed that hte denial of one requires the denial of hte other. — AmadeusD

Do you still believe that? I believe those were your words.

I didn't believe it then, or now. I have been quite clear I'm not across formal logic enough to make those types of claims. |

The only one I've been able to stand behind is that bi-conditional logic isn't relevant to real life, in most cases.

So, my objection isn't to the logic, it seems. It's to the application. The logic clearly fails in many cases. — AmadeusD

If you're saying that the logic fails in many cases, then you are objecting to the logic. If your issue is with the application of the logic to a particular case, then you may be claiming that the logic (e.g. the affirmation of the consequent in the context of a biconditional statement) doesn't adequately capture the form of the argument that someone made in this particular case.

Also, if someone were to claim that the lawn is wet if and only if it rained and that therefore, since the lawn is wet, it must be the case that it rained, you can object to this conclusion on the ground that there are other reasons why the lawn might be wet. In that case, you are not objecting to the validity of affirming the consequent in the context of the (true) biconditional premise "the lawn is wet if and only if it rained." Rather, you are challenging the soundness of the argument. That is, you are taking issue with the truth of the biconditional premise. You are saying that "the lawn is wet if and only if it rained" is a false biconditional statement.

The only one I've been able to stand behind is that bi-conditional logic isn't relevant to real life, in most cases. — AmadeusD

I think I tentatively agree with that, sure.

doesn't adequately capture the form of the argument that someone made in this particular case. — Pierre-Normand

= fails.

You are saying that "the lawn is wet if and only if it rained" is a false biconditional statement. — Pierre-Normand

It is. They are obviously, patently, inadequate. They neither capture the nuance of reality, or justify their relation. It is a nonsense.

They are obviously, patently, inadequate. They neither capture the nuance of reality, or justify their relation. It is a nonsense. — AmadeusD

It's not nonsense. It's just false. Logic deals just fine with false statements. Someone who would have made the wrong inference that it must have rained could have overlooked the fact that the sprinklers could also have made the lawn wet. (Overlooking that is not nonsense either.) They did not commit a logical fallacy like affirming the consequent of a conditional premise. They merely reasoned (validly) on the basis of a false biconditional premise. Incidentally, do you know the difference between the soundness and validity of an argument? Understanding this difference is quite relevant in the context of the present discussion about formal logic and its applications.

They did not commit a logical fallacy like affirming the antecedent of a conditional premise. — Pierre-Normand

Is that a fallacy?

Is that a fallacy? — flannel jesus

No, it's not. I misspoke. Thanks for catching that! I meant to say that affirming the consequent of a conditional statement is a fallacy. My brain's next-token prediction algorithm must have misfired. (I've edited the post to correct this.)

got it

So you were incorrect about that when you said that?

And then earlier in this thread you agreed with the following: — flannel jesus

I seems to be the case your whole point was not trying to find and learn something in logic, but trying to assert my one post was wrong. Is it that meaningful to you and to the rest of the world?

But since now you're saying it's a Fallacy, then the above quote that you agreed with can't be true. — flannel jesus

I can say anything I feel correct. Whether you agree to it or not, that doesn't make anything different. After all the whole of you point seems to be dependent on the internet searches and some other folks agreeing with you. My point is more trying to learn something in Logic. I am not really bothered if what who said was wrong or right, as long as I learn something. I am not here to prove you are right, and I am wrong, or I am right, or you are wrong. If that was your ultimate motive for privately messaging me with the point, then I am a bit disappointed with your stance. Give it a rest, and move on. There are even some folks complaining that the posts that you spew out are hurting his eyes.