It has been a long time since I learned some logic and I wasn't great at it, but I do know what truth tables are and I think how to use them.; I don't see how that implies a definition of "validity" using classical logic.

I mean take the definition of validity, and write it as an expression using symbols and logical operators; is that something that can be done?

I don't mean examples of valid arguments, I am referring to the definition itself.

Okay, I actually do get that the example I just gave has "an interpretation wherein all the premises are true and the conclusion is false" such that it is "not valid." " Would you care to formalize the validity definition as it concerns arguments and do so using logical operators? I was trying to apply De Morgan's laws to your definition but I don't think it worked. On a side note, Banno I can hear your laughter and it is most unwelcome at this time.

Or even if just one (but not all) of the premises is false and the conclusion is false (I am having trouble thinking of an example that meets this description).

A -> ~A makes sense whether A is true or A is false. — TonesInDeepFreeze

I am not clear on how A -> not-A "makes sense" if A is true.

Also, TonesInDeepFreeze, an argument where all the premises are false and the conclusion is false would necessarily be valid; is that correct?

I was thinking of:

P->not-Q
not-P
Therefore,
not-Q.

Assuming that all the premises are false and the conclusion is false, the argument must be valid. Is that correct?

Okay, correct me if I'm wrong, but you are saying that ordinary natural language is "mappable" onto formal classical logic because in formal logic a syntactic inconsistency viz., a negated sign that is present alongside the original sign, results in an argument that is "not derivable" whether the sign and negated sign are explicitly present or present by implication (A->not-A). So just as the ordinary natural language argument is meaningless, so the classical logic argument is underivable.

Can an underivable argument be valid? (I suppose you would say "yes" because the "underived" (unconditioned) constituents of the argument are mere possibilities).

I would think many people would apply a truth table to the argument (A->notA therefore not-A), as I did, and see based on that, that the premise is only true when "not-A." Maybe "infer" is too strong a word for the conclusion of not-A.

The conclusion does not seem to "follow" or be a "logical" conclusion when we apply the argument to ordinary natural language.

So I guess what I'm wondering is whether an underivable (or meaningless) argument may be regarded as logical? Or are soundness and validity insufficient for a logical argument? Or is meaning related to soundness?

It seems, to me, as though what is meant is critical to determining whether an argument is logical.

While you are proving what exactly is logical, you might as well prove that 2+2=4 and that there is an external world, but I don't want to hear any of that mathematical intuition or logical intuition or perceptual intuition nonsense.

Then I challenge you to prove that the following argument is logical:

P
P->Q
Therefore Q.

Or that this argument is logical:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

Or that this argument is logical:

If it rained yesterday then the lake is swollen today.
It rained yesterday.
Therefore, the lake is swollen today.

Well it seems to me that all we can rely on when it comes to logic is intuition. If logic is just a formal set of rules as to how symbols may relate then anything can be logical and in that case nothing really is "logical," though I take that to be the discussion in the logical nihilism thread.

Our logical intuitions are basic, or foundational for doing logic, much in the way that having a functional ear is foundational for making a musical symphony.

One could argue that P->Q and P together implies not-Q, but translating that into natural language with the conditional spoken as an "if...then..." (or A and not-A therefore A and not-A) will be very difficult and I would say impossible, and that's because logic relies on meaning maybe just as much as meaning relies on logic.

All that to say that, at least informally A->notA therefore not-A may not be valid after all if our starting point is a set of meaningful natural language propositions.

That doesn't imply that formal logic is merely "academic" because it clearly has application to fields like computer science and mathematics.

But it may imply that some definitions we use in formal logic may be reviseable or at least more fungible then we previously thought.

↪NotAristotle
Is it worth pointing out, again, that "P→~P" is not a contradiction? If P→~P is true, then P is false.

If that's been said once, it's been said a thousand times... which is not once. — Banno

I know Banno; I am not disagreeing with the formal validity of that argument.

there are ambiguities in the English use of "If... then...", "...or..." and various other terms that we must settle in order to examine the structure of our utterances in detail. — Banno

I don't disagree with that either. But the argument A → ~A ∴ ~A clearly does not translate into natural language very well (I don't think there is any way to translate it in a way that renders the translation sensible and "logical"). And yet, the argument is valid formally speaking.

Michael suggested that the argument is not sound in ordinary language. I think he may be right. However, even arguments that are not sound can still be valid such that we can understand how the speaker reached their conclusion (though we may point out to them that such-and-such premise is not true). For example, if someone argued:

1. P
2. P→Q
Therefore, Q.

We might correct them, "well, actually ~Q." "Your reasoning is spot on and logical, it just happens to be that ~P, so while your reasoning is valid, the argument you presented is unsound."

On the other hand, "If it is raining, then it is not raining, therefore it is not raining" sounds like an unwarranted leap that is not logical when we consider it in an informal way. The problem isn't just that the initial premise is unsound (within an informal context); the problem is that the argument just doesn't make sense and is not logical, so soundness aside, that is why I call it "not valid" informally.

In fact, I would say A->B does not "mean" B or not-A.

I think you mean to say that the one implies the other through logical equivalence. That is different than saying that the expressions mean different things.

"I disagree with regards to ordinary language" I'm not quite getting it, what is the disagreement you have concerning ordinary language? You think someone would make an inference from A->not-A to therefore not-A in ordinary language?

Can you explain how those meanings diverge?

"They probably wouldn't, because the grammar of ordinary language does not follow the rules of propositional logic.

In propositional logic, the following is a valid argument:

P → ¬P
∴ ¬P"

Exactly. And if someone wouldn't make such an inference, I am suggesting that that is a logical mistake of some sort, which is a way of saying the argument is not valid.

Michael, the argument is simply this:

If it is raining then it is not raining.
Therefore, it is not raining.

Who in there right mind would conclude the conclusion from the premises in a conversational setting?

(It is a different argument from the original argument in the first post).

I am referring to the "it is raining" example; the conclusion in that argument appears to be a logical leap. I get that the argument is formally valid, that's the entire point - while formally valid, the conclusion does not appear to "follow."

I may be using "equivocal" incorrectly; what I meant is that there may be two senses of the term "valid" in a logical context; one formal, the other informal and that evaluating an argument with either definition may cause different conclusions as to whether a given argument is valid.

I think you are right that material implication is a problem in the example I stated; the premise appears to not be true (and to never be true).

Still, it also appears that the conclusion is an unwarranted logical leap from the premises, so that is why I think there might be room to argue that the argument is not valid according to some informal definition of logical validity. That is to say, the conclusion doesn't follow or doesn't lead to the conclusion. I understand that this is not the definition of validity formally speaking.

I am not entirely sure what the original post is asking.

This comment does not directly answer the original post's question:

It seems that, if you accept the historicity of Jesus, and accounts given of the apostles such as the martyrdom of Peter, and if you also deny that Jesus rose after suffering a brutal execution at the hands of the Romans, then Peter would have to be just the least intelligent person imaginable to continue to preach about Jesus.

So either early Christians like Peter were all complete idiots, or he was an individual of heroic virtue. But where would such heroics originate? In other words, if someone considers as though they were Peter, why bother preaching if none of it is true?

Okay, but I can actually see how the edited conditional could be true. For instance, if Michael is a really great citizen, then maybe he would end up being President were he American, if so, then in the ordinary sense, the sentence can be "true" based on what it means.

I am not sure what you mean by saying "If I am American then I am the President" is true in propositional logic. But I do appreciate that that conditional is not true in ordinary language.

Yes, because I have a better understanding of how to define validity in a formal context. No, because in a non-formal ordinary sense, and in a natural language context, the argument still seems invalid.

If I uttered: "If it is raining then it is not raining." ... If formal logic is "mappable" onto ordinary language, then you should be able to infer "oh okay, it's not raining." But no one speaks like that and no one would make such an inference. At least, no one would consider such an "argument" "valid." That being so, while I would prefer there not to be equivocal definitions of validity, it appears that there are, one formal, the other informal.

and @Hanover, and @Banno, and @all participants to this thread,

I was hoping this thread would be a discussion investigating deduction, implication, and validity. I am thankful that that is what everyone is discussing and other topics. I wish I had more to add to the discussion, but I am not as well-versed in logic. I have learned what I think is a strong definition of validity, which TonesinDeepFreeze stated earlier in the thread. I encourage respectful discussion of these topics by all parties.

Insofar as the things you mentioned are objectively bad for the organism, I would argue that they are morally bad, or are at least morally worse than a situation wherein the organism was not so harmed.

It seems to me that science may not be very good at defining an act as right or wrong. On the other hand, I think it may be quite good at saying what is good or bad (for an organism). "Cigarettes cause cancer" is a scientifically established fact.

Perhaps it may be asked whether what is bad for an organism is morally bad, but, to my eyes, the answer seems to be "yes."

Follow-up question: when we say "ethics isn't a science" do we mean ethics does not require any kind of scientific knowledge and can be applied through a kind of a priori cognition/intuition? Or do we mean that the kind of knowledge that science supplies is either insufficient for ethics or does not apply to ethics at all?

I myself can be sympathetic to the view that scientific knowledge may be applicable-but-insufficient for ethics due to something like a normativity objection.

I also tend to think scientific knowledge is unnecessary for ethics even though it may be able to provide evidence concerning moral facts.

Is it a problem that "not-(A and not-A)" is also a valid conclusion of the argument? According to the definition proffered by Hanover, it would seem to be a problem given that "the negation of the conclusion flows from the premises."

Can you provide a citation for that criterion of validity? I did not find it in the wiki article.

Also, Hanover, thanks for articulating an argument against validity as I was not sure how to do so.

Are there any domains (I'm thinking of ethics) where you think methodological naturalism would not be instructive, or would you recommend it as a complete and holistic approach for understanding all of reality?

Nevermind, "A and B Therefore C" would be an invalid argument where the conclusion does not contradict the premises.

Can anyone think of an invalid argument where the conclusion does not contradict one of the premises?

It is an interesting problem to me because according to this website -- https://www.umsu.de/trees/#(A~1~3A)~5(A~1~3A) -- the argument is apparently valid. Even though you and I can plainly see that such an argument can never be true. It is an obviously bad argument in a way that:

not (P->Q)
Therefore, not-P

is not an obviously bad argument (bad argument though it is).

I would also note that the argument "A->not-A, Therefore not-A", though it is apparently a valid argument, does not make much sense in natural language; it would be like saying "if it is raining then it is not raining." Maybe someone could infer from that statement that it is not raining, but the statement seems more like a contradiction then a "valid" logical statement.

I can see why the premises imply G. I agree with Michael that there is a translation issue.

I think I meant to say:

1. not-G -> ( not (P->A) )
2. ( not (P->A) )
3. not-A
Therefore,
4. P

My understanding of a valid argument is that it is one such that if the premises are true, the conclusion must be true. Sounds like you are saying the initial argument is at least inconsistent; does that prevent it from being a valid argument? Or is it valid, just unsound?

Am I understanding you to be saying, similarly to unenlightened, that one of the premises must be false given that they are "inconsistent?" The argument is valid but unsound you are saying?

If so, can you say which premise is false and why?

I think adding content to the logical propositions definitely demonstrates the absurdity of the argument; but it seems to me that the absurdity is implicit in the structure of the argument itself - we don't really need the content to see that.

The argument seems a bit less problematic if the second premise were changed to: "Sue is not sitting" because then it seems to me that the argument can at least be true in some sense.

I agree that A -> not-A seems like a questionable premise; perhaps that is the premise you think is untrue. But what makes " A -> not-A " a premise that is not true? Does it have something to do with truth tables?

Okay, what about this argument -- https://www.umsu.de/trees/#((A~5~3A)~1A)~5~3A

A -> not-A
A
Therefore, not-A.

There must be a difference between implication and deduction, right?

not-G -> ( not- (P -> A) )
not - P

does not imply

G.

in fact, the premises do not actually tell us anything. On the other hand,

not- G -> ( not- (P -> A) )
not- A

does seem to imply..

P.

But again, it still does not imply G.

On the other hand,

not- G -> ( not- (P -> A) )
A

does seem to imply

G.

I am against abortions for religious reasons.

A lot of people seem to think consciousness is important to deciding whether abortions should be allowed. I guess my question is: are you proposing we have a "consciousness test" for fetuses? And will this test be 100% accurate, or will it sometimes mistake whether a fetus is conscious?

If the consciousness test is not 100% reliable, that would be a reason against allowing abortions. One could only still support abortions if one were willing to sacrifice an innocent person, a morally repugnant decision.

I think this reasoning defeats any condition stipulated that purportedly would have rendered abortions permissible.