1. Right, I mean P entails Q. The logical equivalence (not-P or Q) is an implication of the conditional, not having the same meaning as the conditional.

2. I take your question to be what would a rule be, how is it defined? I would define a rule as a member belonging to a set that exhausts all "truth possibilities." I would add that the following of a rule may not result in a contradiction.

A rule relating two different variables would have (I think) 15 possible truth configurations. The rules must at least enable all those possibilities to be instantiated (though perhaps it may exclude possibilities that are necessarily contradictory).

3. "Some proposition is not the case"
Both propositions must be true
Either proposition must be true
If the one proposition is true, so must the consequent proposition
Both propositions are either both true or both false.

5. Valid argument = following the rules, where rules are defined as those operations that enable each truth possibility to be instantiated but that do not result in a contradiction by following that rule.

8. Not logical anarchy; the rules must enable all truth possibilities to be instantiated except that the rule may not result in a contradiction if it is followed.

This way of defining validity may be preferable because it deals with cases such as A->not-A therefore Not-A that are intuitively illogical; such an argument does not involve the following of a rule, and so it is not valid.

Similarly, A, A->not-A therefore not-A another intuitively illogical seeming argument would not be valid because the following of the rule results in a contradiction.

1. I take a conditional to be saying: if the antecedent is true, it can't be the case (there is no circumstances such) that the consequent is false.

2. Rather than a correct conclusion, all we need are conclusions that follow the relevant rules, any and all such conclusions are legitimate.

3. I refer to connectives as rules.

4. Then we are out of luck.

5. I drop the truth preservation condition for validity.

8. If we drop the truth preservation part of the definition, it is not circular. An argument is valid where it follows the relevant rules. Period. I don't think it is necessary for me to stipulate that a rule be followed "correctly," just that it be followed.

1. The conditional means that in the event that the antecedent is true, the consequent must be true. It is one of the logical rules that must be followed for the argument to be valid.

2. Provided that a set of all conclusions follows the rules correctly and is exhaustive of all such conclusions, that set encompasses all legitimate conclusions.

3. logical operators.

4. We would have to ask the speaker to clarify.

5. Noted, let's set aside questions concerning meaning; the second definition may have more problems then I can resolve.

6. Okay.

7. So then "the truth of the premises guarantees the truth of the conclusion" is the same as "there is no interpretation (assignment of truth values) such that the premises are all true and the conclusion is false" ?

8. So I think what I am trying to say is that the definition of validity is following the rules correctly. And that following the rules correctly is defined by rule-following that results in truth preservation. Such that, truth preservation is a consequence of rule following, and it is the rule following itself that is responsible for the validity. In other words, the premises themselves don't guarantee the truth of the conclusion, rather the following of the rule(s), given that the premises are all true, is what guarantees the truth of the conclusion. Put another way, truth preservation does not make the argument rule-following, but rule-following is what makes the argument truth preserving. (Truth preservation does make the rule-following "correct.") Not sure if that totally makes sense.

So actually, I would say my definition of valid is different from the ordinary formal logic definition in that I am defining validity in terms of rule-following, not in terms of truth-preservation; truth-preservation is more like a consequence of the definition.

P->Q. P. Therefore, not-Q. would both flout the meaning of the conditional, and in such a way that it changes the conclusion. It's different than what the conclusion should be (namely Q).

I don't understand the second question.

Third question answered as correctly used rules is defined.

I don't know the difference between propositional logic and ordinary formal logic so I do not know how to answer this one.

The meaning of an expression depends on what the speaker intended by it - natural language I would think would go along way in dissolving confusion over what is meant.

Right, where there is disagreement over a meaning, that meaning is not well-formed and not suitable for logical operations. I would expect something like that to be true for your definition of validity as well.

I guess I agree with the ordinary definition of valid in formal logic. That is not the definition you cited earlier in the thread - the definition that I am suggesting an alternative to.

I do not see truth preservation as synonymous with validity; I defined validity as rule following; a rule is followed correctly if it preserves truth; I didn't define validity as truth-preserving. Truth preservation is a consequence of validity, namely, following the relevant rules correctly.

And a relevant rule is correctly followed just in case.. if it were the case that all the premises were true and the relevant rule is followed, then the conclusion must also be true.

Relevant rules like conditionals "And" "Or" operators-- when those are used correctly the rules are followed and the argument may be considered valid. Any rule that is such that if it weren't followed, the conclusion would be different, is a relevant rule. The rules would ideally be universal and based on logical intuition; if people use different sets of rules, then the rules must be clearly communicated so that that "logic" can be understood or followed.

The meaning of the premise and conclusion depends on the expressions used (I guess this definition isn't unequivocal as it would only apply to ordinary natural language, not to formal logic). I don't know any theories of meaning so I can't answer that. If the meanings differ, then I'm not really sure what the result would be, seems like communication is out the door let alone logic if we can't agree on the same meaning of words and sentences.

Here are two ideas for defining validity: (1) an argument is valid when all the relevant rules are followed. Or, (2) an argument is valid when the meaning of the premises leads to the meaning of the conclusion.

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.