If it rains, then ground is wet.
The ground is not wet, so it doesn't rain.

This is the case of contraposition isn't it?
It looks like it is sometimes true or sometimes not true.
Would it not depend on the checking over with the reality outside?
Did you check the truth table for it? I am not sure off hand without making up the truth table for it.

If it's raining, then the ground is wet.
The ground is not wet, so it's not raining.

This doesn't seem like it depends on anything to me - if the first `if-then` is true, then the second `if-then` is true. In classical logic, given a statement of implication, contraposition is taken to be always true, not "it depends".

In classical logic, A -> B being true always means ~B -> ~A is also true.

They have the same truth tables as each other.

n classical logic, A -> B being true always means ~B -> ~A is also true.

They have the same truth tables as each other. — flannel jesus

Sure, if the truth table says so, then it must be it. But in the empirical cases, you can also compare the TF values with the reality events, in which case if it is FALSE, then it proves the starting premise were FALSE.

Sure, if the truth table says so, then it must be it — Corvus

So do you agree that, if one accepts a statement (A -> B), then according to classical logic one must always accept the contraposition, (~B -> ~A)?

If the truth table says so, yes. But compared with the reality event, it could be denied, in which case, the premise would be denied too. Proof process is for using the truth table rules, your inferences and also the reality events trying to prove that your argument was true. Some folks only cite truth table, and never allow you to make any inferences. Or they only check validity of the formulas which are not relevant. That is not logical proof.

This starts out sounding like a 'yes' but ends up sounding like a 'no'.

What's an example where (A implies B) is true, but (~B implies ~A) is not true?

There is no hard coded laws here. It depends on your inference and the case, and also checking with the real events in the world. The rules only says the principle. The application for finding out truth is flexible with other factors related.

(A implies B) is true, but (~B implies ~A) — flannel jesus

Even if it says True, when you compared with the real events, if they are false, then it is false. Also the premise is false.

There is no hard coded laws here. — Corvus

I think classical logic very much has hard coded laws. Basic logic very much has hard coded laws. Logical proofs are a sequence of steps using hard coded logical laws.

for example?

Of course there are general rules for truth table and syntax rules, but for proof process, you must reason yourself for brining in the relevant inferences to the process.

So the proof that you posted here then:

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

That's not based on logical laws, that's... what, then? Just some of your own personal reasonings based on your own personal relevant inferences?

That's from the truth table.

From what truth table? Would you mind posting the table here?

Yeah I am busy and in the middle of doing other stuff, and will have a look at it when I am free.

have fun

Yes, or in other words: denying the “antecedent” of a biconditional is not a fallacy. Yet denying the antecedent of a conditional is a well-known fallacy.

:roll: .

I would like to have a discussion with Corvus about if the logic used here is actually logical, or if it is perhaps fallacious — flannel jesus

You guys need to find a bedroom. I'm surprised we others are allowed to witness the proceedings.

Via three AI models:

"No, it is not logically sound to deny the antecedent in a bi-conditional proposition because it does not follow the rules of propositional logic. In a bi-conditional proposition, if both the antecedent and consequent are negated, the proposition as a whole is not necessarily true or false. It is important to consider both sides of the proposition in order to determine its truth value."
====
"No, denying the antecedent for a bi-conditional proposition is not a valid form of reasoning. In a bi-conditional proposition, if the antecedent is false, then the consequent must also be false. The only way to deny the bi-conditional proposition is to show that both the antecedent and the consequent are false."
=====
"Denying the antecedent of a bi-conditional proposition is logically unsound because it does not necessarily lead to the conclusion that the consequent is false. In a bi-conditional proposition, if-then both directions are linked together, meaning that if the antecedent is true, then the consequent must also be true, and vice versa.

For example, consider the bi-conditional proposition: "If it is raining, then the ground is wet." Denying the antecedent would result in saying "It is not raining, therefore the ground is not wet." However, the ground could still be wet for reasons other than rain, such as someone watering their lawn or a sprinkler system being turned on.

Therefore, denying the antecedent of a bi-conditional proposition does not provide valid grounds for concluding that the consequent is also false, making it logically unsound."


Creepily relevant.

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

It pleases me that the AI does not know what a biconditional is.

the statement can be expressed as "it is raining if and only if the ground is wet," making it a bi-conditional statement.

- What you have expressed is a different statement entirely, and yours is in fact biconditional.

Yes, or in other words: denying the “antecedent” of a biconditional is not a fallacy. Yet denying the antecedent of a conditional is a well-known fallacy. — Leontiskos

The post seem to be in an obvious case of internet info. snack gone down into wrong pipe.
In deductive syllogism it is fallacy, but in inductive case, it is not fallacy. Because the real life case can be contradiction to the premise.

You guys need to find a bedroom. I'm surprised we others are allowed to witness the proceedings. — jgill

Was just trying to be a help for the request from FJ for clarification.

This was the relevant AIs response.

So you would agree with the AI, and say that the falsity of the “first” half of a biconditional does not entail the falsity of the other half?

I wouldn't know. I'm leaving formal logic for institutional learning rather than as a hobbyist. For thoroughness though, The full response was:

"The bi-conditional statement "if it is raining, then the ground is wet" is true because it goes both ways. If it is raining, then the ground will be wet. And if the ground is wet, it must have rained at some point. Therefore, the statement can be expressed as "it is raining if and only if the ground is wet," making it a bi-conditional statement."

So do you think any time you have (a implies b) , it's always true that (not a implies not b) — flannel jesus

Yes, it is correct. — Corvus

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.

I was quoting this from your post:

For example, consider the bi-conditional proposition: "If it is raining, then the ground is wet." Denying the antecedent would result in saying "It is not raining, therefore the ground is not wet." However, the ground could still be wet for reasons other than rain, such as someone watering their lawn or a sprinkler system being turned on.

Therefore, denying the antecedent of a bi-conditional proposition does not provide valid grounds for concluding that the consequent is also false, making it logically unsound." — AmadeusD

As I noted, this is false because it mistakenly calls a conditional statement a biconditional statement. It also makes false claims about biconditional statements.

In the new quote that you provided in your last post, the reasoning is formally valid (except for the minor error in constructing the biconditional), but it is unsound given the fact that things other than rain can also make the ground wet, such as dew.