* For argumentation, I suggest studying both informal and formal logic. Informal for practical guidance; formal for appreciation of rigor. I don't have particular texts in informal logic to recommend. For formal logic, I recommend 'Introduction To Logic' by Suppes, though I don't know whether it is available online.

* A -> ~A does not imply a contradiction.

You are not heeding the many explanations already in this thread.

(Frist though, if a sentence implies a contradiction, it does so no matter what interpretation is adduced.)

Next, having to repeat this again:

A contradiction is a sentence of the form P & ~P (or P and ~P both as lines in a proof).

A sentence or set of sentences is inconsistent (i.e. contradictory) if and only if it implies a contradiction.

A -> ~A is not contradictory. But {A, A -> ~A)} is contradictory.

* My guess is that for the most part, people do take the material conditional and "If it rains, then it does not rain" as counterintuitive. The point you make about this has been recognized over and over and over in this thread.

But the material conditional has use in mathematics and computing ('P->Q defined by '~(P & ~Q)'). It doesn't have to be intuitive to everyday speakers to be make sense to many (most, I surmise) mathematicians, logicians, and modern philosophers in the field of logic, and in other formal contexts. That people around town don't countenance the material conditional doesn't entail that it is not the case that it is useful and makes sense in other contexts.

Meanwhile, there are other alternative formal definitions for "if then".

And there are various competing notions of "if then" in everyday use.

* It's been explicated, in detail, over and over in this thread that, yes, there is a difference between a set of premises that has one or more false members but is not inconsistent and a set of premises that is inconsistent.

it fails to take into account the fact there are additional causes for a consequent to happen (any time really where correlation isn't causation). — Benkei

I don't know what that is supposed to mean.

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. — NotAristotle

Again, the distinction I adduced:

P -> Q
is true in a given interpretation if and only if either P is false in that interpretation or (inclusive 'or') Q is true in that interpretation.

and a different notion:

P entails Q
if and only if there is no interpretation in which P is true and Q is false.

It seems you take P -> Q to mean "P entails Q".

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

What are the "relevant rules"? What are your rules?

3. I refer to connectives as rules. — NotAristotle

That's not the usual notion, but let's see what we get from it:

As rules, what are these connectives?:

~
&
v
->
<->

4. Then we are out of luck. — NotAristotle

So, with your offer, we can't examine validity for a vast amount of everyday argumentation.

5. I drop the truth preservation condition for validity. — NotAristotle

You do now, after I showed you that your definition of validity in terms of 'relevant rules' in terms of truth-preservation is circular.

So, what now are your definitions of 'relevant rules' and 'valid argument'?

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.

Not period. Again, what is your defintion of 'relevant rule'?

I don't think it is necessary for me to stipulate that a rule be followed "correctly," just that it be followed. — NotAristotle

So, for you, an argument is valid if it follows some rule or another, irrespective of whether the rule is truth-preserving or correct in any aspect?

Anyone can state any rules they want, including ones that are not truth-preserving, and even ones that permit contradictions from a set of satisfiable premises. Pretty much logical anarchy.

I've said it maybe fifty times in this forum: Ordinary formal logic with its material conditional does not pertain to all contexts. But that is not a basis that one should not say how ordinary formal logic handles a question and not a basis that one should not explain ordinary formal logic to people who are talking about it without knowing about it. — TonesInDeepFreeze

This is not responsive to anything I've said. I know you want to keep saying over and over what formal logic dictates, but my post was referring to how ordinary language handled conditionals and how that was distinct from formal language. I didn't suggest we should jettison formal logic or that it lacked value because it was distinct from the ordinary way we speak.

* For argumentation, I suggest studying both informal and formal logic. Informal for practical guidance; formal for appreciation of rigor. I don't have particular texts in informal logic to recommend. — TonesInDeepFreeze

I really like this book for informal logic:

https://www.cambridge.org/core/books/arguments-about-arguments/27835C37D9CEDFA6BE9EFD11CA2DA5A3

It is exactly responsive to your post. You said:

This thread strikes me as more of a primer in formal logic nomenclature than in logic qua logic. — Hanover

So, I addressed that.

And I didn't say that you said that formal logic lacks value.

As to the difference between the material conditional and informal notions of the conditional, that point has been gone over and over and over. If there is something more you want to say about, no one is stopping you.

And I gave you information about modus ponens, consistency and arguments too, to clear things up for you after your confused comment about them.

That table of contents looks pretty good. But I wonder whether it is best as a very first book to read.

Yeah I'm not sure there. I can say I really enjoyed it though. Made me think about logic in totally new and different ways.

Very inviting.

As to the difference between the material conditional and informal notions of the conditional, that point has been gone over and over and over. If there is something more you want to say about, no one is stopping you. — TonesInDeepFreeze

Why are you telling me that no one is stopping me?

And I gave you information about modus ponens, consistency and arguments too, to clear things up for you after your confused comment about them. — TonesInDeepFreeze

Thank you for reminiscing, but that's not what my last post was about.

Why are you telling me that no one is stopping me? — Hanover

To emphasize that, for example, nothing I've said is a barrier to you adding whatever else to the subject you might have to add other than what has already been gone over.

Thank you for reminiscing — Hanover

Yes, I recall that you ignore good information you need to remedy your confusions.

but that's not what my last post was about. — Hanover

Whatever your post was "about", it included a confusion regarding modus ponens and consistency.

Hanover's confusions in this thread start in his very first post:

1. A -> not-A
2. A
Therefore,
3. not-A.
— NotAristotle

#1 is a contradiction, reducible to ~ A or ~A. Since it concludes A cannot be true, the antecedent (if A) is always false.
#2 is false and contradicts #1 that establishes ~A.
#3 is not a conclusion, but is a restatement of #1. — Hanover

All three of the above are incorrect.

A -> ~A is not contradictory.

A is true or false depending on a given interpretation.

~A in that argument is the conclusion.

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. 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. — NotAristotle

What on earth would the meaning of P -> Q be such that it differs from not-P or Q? It's not like P -> Q is in any way a natural language sentence. We're already relying upon substitution (with the variables P and Q) so what prevents us for substituting "not-P or Q" for "P-> Q" when they are logically equivalent?

You can absolutely substitute them logically, however I do not think they mean the same thing. P->Q either means just that "P->Q" or it doesn't have a meaning at all, either way P->Q does not, in my opinion, mean the same thing as its logical equivalent.

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. — NotAristotle

It has the same meaning in truth-functional terms, but it does not have the same meaning qua modus ponens (or modus tollens). See, for example:

I suppose it is worth asking whether these are the same two inferences, and whether the first is any more "directional" than the second:

(A→B)
A
∴ B

¬A∨B
A
∴ B — Leontiskos

It seems to me that the disjunctive equivalent does not capture the full meaning of P->Q.

"If A then not A" means "not A or not A". — Michael

"If P then Q" means "not P or Q". — Michael

Only if "or" is used inclusively. The ordinary language use is often exclusive, or ambiguous.

So this doesn't help.




...except perhaps to provide an example of how formal logic works out the ambiguities of natural language. Which is part of what it is for.

A ⊻ ¬A ↔ A ∨ ¬A

Right.

So I'm asking -- what else is there? What does the full meaning say?

I'm sympathetic to informal logic taking priority over formalization, though it gets difficult because of the inherent ambiguity in informal logic. When I say "If I touch the stove then my hand will burn" I'm not talking in terms of material implication or disjunction at all, but a causal relationship between action and event.

But cashing out that causal relationship into the formalization is not in any way easy. Which is why we usually rely upon the informal.

But once we start introducing symbols and substitution I tend to believe we're not really talking about this imprecise, though more frequently utilized, informal reasoning.

When I say "If I touch the stove then my hand will burn" I'm not talking in terms of material implication or disjunction at all, but a causal relationship between action and event. — Moliere

You are talking in terms of the first premise of a modus ponens, and that is what the material conditional is in many logics. If there is a difference between modus ponens and disjunctive syllogism, then there is a difference between A→B and ¬A∨B. You can of course say that there is no difference and that "informal logic" and formal logic are infinitely separated, but I think that is to put the cart before the horse. Something which has nothing to do with human reasoning is not logic, and so I would say that if someone is talking about something which is wholly separate from human reasoning then they are not talking about logic (and besides that, they are not paying any attention to the historical development and motivations behind formal logics).

You are talking in terms of the first premise of a modus ponens, and that is what the material conditional is in many logics. If there is a difference between modus ponens and disjunctive syllogism, then there is a difference between A→B and ¬A∨B. — Leontiskos

The difference is the shape of the little squiggly marks.

You can of course say that there is no difference and that "informal logic" and formal logic are infinitely separated, but I think that is to put the cart before the horse. Something which has nothing to do with human reasoning is not logic, and so I would say that if someone is talking about something which is wholly separate from human reasoning then they are not talking about logic (and besides that, they are not paying any attention to the historical development and motivations behind formal logics). — Leontiskos

I think there's a difference and I've committed to indications for the difference -- in the recent posts substitution has been the criteria I've been using.

I don't think that makes them entirely separable.

I think there's a difference and I've committed to indications for the difference -- in the recent posts substitution has been the criteria I've been using. — Moliere

There is no difference in terms of substitution.

(The thread already linked goes into these questions in detail. <This post> would be one example of that.)

Cool. We agree there.

So when we start using things like "P" to represent any proposition whatsoever this relies upon substitution. Without a notion of substitution we'd not be able to make sense of variables.


Or, what's probably a better way of stating this, informal substitution is subject to more criteria than formal substitution is. Informally substitution is usually reserved for mathematics and statements of that form -- which is what formal logic is very much like.

But since A -> ~A uses symbols it's more appropriate to call this a formal construction of material implication, which we can write the truth-tables out for and easily conclude it's valid, but unsound, as said.

But since A -> ~A uses symbols it's more appropriate to call this a formal construction of material implication, which we can write the truth-tables out for and easily conclude it's valid, but unsound, as ↪unenlightened said. — Moliere

I opined on some of this earlier in the thread*, but I think you are talking about the argument of the OP, not merely the sentence (A→~A).

* It depends on what we mean by "valid" and it depends on the argument for why it is or is not valid.

Oh, yes. Very much. The topics are related but I'm trying to remain on target with @NotAristotle here.

I get mixed up with this, but I think the disjunction (not-P or Q) can still be true even if P does not imply Q. So the "meaning" of the disjunctive is not specific enough.

I get mixed up this, but I think the disjunction (not-P or Q) can still be true even if P does not imply Q. — NotAristotle

You could say that, but you would end up having to admit that "P does not imply Q" cannot be formalized in any way whatsoever, at least in propositional logic. See the thread that I have been referencing for this topic.

Whether or not the two expressions are semantically equivalent in a meta-logical sense depends on how one is using them. But I think your intuition is correct and defensible, namely that they are not semantically equivalent in a meta-logical sense. Trouble is, you can never prove that at the level of the object language.

My mistake. Not "If A then not A" means "not A or not A" but "If P then Q" means "not P or Q".

"If P then Q" means "not P or Q" presumes an inclusive OR.

Fair, so I suppose I should "if P then Q" means "not-P or Q or both".