There are a lot of things to untangle in a discussion. I didn't purport to vindicate everything the poster has said. And I am not, at least at the present, interested in untangling whether the way you represented what he said is correct or not. In that post by me, I mentioned a particular thing he posted and I said it is correct.

You said

a terrible interpretation of the definition of validity. — Leontiskos

It's not an interpretation of the definition (as if I intepret it to be the definition of as if I interpret the defintion in some strange way. Rather, it is simple inference from the definition. That is a critical point not a "quibble".

READ exactly what I wrote:

The argument is valid; the conclusion follows from the premise. We can show this in four parts:

1. If "I am a man and I am not a man" is true then "I am a man" is true.
2. If "I am a man" is true then "I am a man or I am rich" is true.
3. If "I am a man and I am not a man" is true then "I am not a man" is true.
4. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.
— Michael

The difference between an argument from the definition of validity and an argument from explosion has been explained multiple times throughout this thread. Tones himself recognized it.
— Leontiskos

Michael's reasoning is correct there and doesn't contradict anything I've said. — TonesInDeepFreeze

I referred exactly to this:

"The argument is valid; the conclusion follows from the premise. We can show this in four parts:

1. If "I am a man and I am not a man" is true then "I am a man" is true.
2. If "I am a man" is true then "I am a man or I am rich" is true.
3. If "I am a man and I am not a man" is true then "I am not a man" is true.
4. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true."
— Michael

That is what I said is correct.

but most TPFers are able to recognize its truth. — Leontiskos

(1) You don't know that. (2) Even if it were true, it doesn't prove much. (3) I wrote again in my first post today that I am not arguing what the definition should be. But I have said what the ordinary definition is and what follows from it.

If the premises are inconsistent then the argument is valid by definition (and this does not presuppose the principle of explosion)," is just a terrible interpretation of the definition of validity. — Leontiskos

It's not a definition of validty! It's not supposed to be definition of validity!

You are terribly confused and not paying attention to what I've said over and over and to what I said in the last few posts.

Arguments are not valid in virtue of being inconsistent. — Leontiskos

Arguments are not consistent or inconsistent. Sentences or sets of sentences are consistent of inconsistent.

You are confused, as usual.

Got it. The last line of this post:

Explosion is related, but I didn't mention it or need to mention it for the purpose at hand.

There are both semantical and syntactical versions of principles. These are definitions I use. Different authors have variations among them, but they are basically equivalent, except certain authors use 'valid' to mean 'true in a given interpretation', which is an outlier usage. I mention only sentences here for purpose of sentential logic; for predicate logic we have to also consider formulas in general and some of the definitions are a bit more involved.

Semantical:

Valid sentence: A sentence is valid if and only if it is true in all interpretations. A sentence is invalid if and only if it is not valid.

Logically false sentence: A sentence is logically false if and only if it is false in all interpretations.

Contingent sentence: A sentence is contingent if and only if it is neither a validity nor a logical falsehood.

Satisfiable: A set of sentences is satisfiable if and only if there is an interpretation in which all the members are true.

Validity of an argument: An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.

Entailment: A set of sentences G entails a sentence P if and only if there is no interpretation in which all the members of G are true and P is false.

Sound argument (per an interpretation): An argument is sound (per an interpretation) if and only if it is valid and all the premises are true (per the interpretation). Note: When a certain interpretation is fixed in a certain context, we can drop 'per an interpretation' in that context. For example, if the interpretation is the standard interpretation of arithmetic. For example, informally, when the interpretation is a general agreement about common facts (such as that Kansas is a U.S state).

Explosion: For a set of sentences G, if there is no interpretation in which all the members of G are true, then G entails every sentence.

Syntactical:

Proof: A proof from a set of axioms per a set of inference rules is a finite sequence of sentences such that every entry is either an axiom or comes from previous entries by application of an inference rule. (And there are other equivalent ways to formulate the notion of proof, including natural deduction, but this definition keeps it simple.)

Theorem from a set of axioms: A sentence is a theorem from a set of axioms if and only if there is a proof of the sentence from the axioms.

Contradiction: A sentence is a contradiction if and only if it is the conjunction of a sentence and its negation. (Sometimes we also say that a sentence is a contradiction when it proves a contradiction even if it is not itself a conjunction of a sentence and its negation.)

Inconsistent: A set of sentences is inconsistent if and only if it proves a contradiction. (Sometimes we say the set of sentences is contradictory)

Explosion as a sentence schema: For any sentences P and Q, (P & ~P) -> Q.

Explosion as an inference rule: For any sentences P and Q. From P & ~P infer Q.

/

So explosion and "any argument with an inconsistent set of premises is valid" are similar. — TonesInDeepFreeze

Earlier in the thread you said that the two are "similar," not that one presupposed the other: — Leontiskos

Yes, explosion is similar, in the context of what I posted preceding, with ""any argument with an inconsistent set of premises is valid". Call that (*).

(1) Michael mentioned a particular argument. It is a correct argument. (*) is consistent with that.

(2) The definition of validity and the principle of explosion are not equivalent. The latter follows from the former, but not vice versa.

(3) I've made no claim about "presupposes".

(4) You would do very well to reread that post.

I was editing my post, dropping the comment about a link, while you posted yours above.

We have a definition of validity. Then we show that that definition entails the principle of explosion.

It's not my concern to sort out what is in your mind about presuppostion.

you do disagree with Michael, who thinks that your construal of your definition is nothing other than a tacit appeal to the principle of explosion. — Leontiskos

Whether that is or is not a correct characterization of anything he said, all I said is that a certain argument he gave is correct.

I haven't said anything about 'presuppose'. Rather, I have shown that the definition of validity (semantic) entails the (semantic) principle of explosion. As for rules (syntactic) one can embody the principle of explosion as a rule without reference to semantics or a notion of 'validity'.

But, as I've told you at least a dozen times, we also go on to prove the soundness and completeness theorems, that is, an equivalence between entailment (semantic) and deducibility (syntactic).

the strange way you want to apply your definition is based on explosion — Leontiskos

I have not applied the definition in any strange way.

And the definition is not based on the principle of explosion. Rather, the definition implies the principle of explosion. You have what I said backward.

What you are apparently saying now is that someone who does not understand the principle of explosion cannot apply the definition in the way you prefer. — Leontiskos

I am saying no such thing.

And it's not just "the way I prefer". The definition of 'valid argument' entails the principle of explosion, no matter what I prefer.

If you think that your idiosyncratic application of your definition of validity — Leontiskos

I addressed the characterization that is is "my definition'.

And there is nothing idiosyncratic. I stated the standard definition and showed that it immediately entails explosion.

In any case, what I said (in whatever words) is that the (1) the definition of validity entails that (2) an inconsistent set of premises entails any conclusion. (1) and (2) are not equivalent.

And a while ago you claimed that I illegitimately claimed the equivalence of the two wordings of the definition based on using the material conditional in the meta-language. So I pointed out that of course ordinary logicians use the material conditional in both the object-languages and in the meta-languages. And I even quoted a text saying explicitly that the two wordings are equivalent.

use of a rule may not result in a contradiction — NotAristotle

I already explained that the only time a rule yields a contradiction is when it is applied to an inconsistent set of formulas. So, if you want to define 'valid argument' so that no valid argument has a contradictory conclusion, then stipulate that no valid argument has an inconsistent set of premises.

You ignore information given you.

A->not-A, when this rule — NotAristotle

A -> ~A is a sentence. It's not a rule.

First, you conflated connectives with rules. Now you conflate sentences with fules

You are hopelessly ignorant and confused about even the basic concepts:

connective
sentence
rule

The "following" of a rule versus it's being merely "present" can be illustrated by the following example:
A->B
B^C
Therefore, C.
In this example, the rule A-> B does not do any work, so even if it did result in a contradiction, the fact that it doesn't do any work in the argument and isn't followed or actually applied, means that the argument could still be valid. — NotAristotle

Again, A -> B is a sentence, in this case it's a premise. Again, A -> B is not a rule.

I'm guessing what you mean is that a valid argument has no premises that could be excluded and still have the set of premises entail the conclusion.

But what you said above is actually the opposite of that, as you wrote "the argument could still be valid".

And note that your stipulation of not having unneeded premises would leave us without the monotonicity principle.

Informally not valid. — NotAristotle

You still have not defined 'informally valid'. You abandoned your first attempt after I finally got you to see the circularity in your attempt. Then your subsequent attempts have been nonsense even to the extent that you conflate the notion of 'sentence' with that of 'rule'.

The argument is valid; the conclusion follows from the premise. We can show this in four parts:

1. If "I am a man and I am not a man" is true then "I am a man" is true.
2. If "I am a man" is true then "I am a man or I am rich" is true.
3. If "I am a man and I am not a man" is true then "I am not a man" is true.
4. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true. — Michael

The difference between an argument from the definition of validity and an argument from explosion has been explained multiple times throughout this thread. Tones himself recognized it. — Leontiskos

Michael's reasoning is correct there and doesn't contradict anything I've said.

The definition of validity entails that the principle of explosion is valid.

Tones' definition — NotAristotle

It's not "my definition" in the sense that I am proposing it as the only acceptable definition or asserting that there can't be a better definition. Only that it is the standard definition, is clear, is understood by mathematicians, logicians, and philosophers, and has applications in those fields of study, and makes sense to me in certain formal contexts.

if an argument's conclusion follows from its premises using the rules of inference then they will name this type of argument "valid". — Michael

No mention of rules of inference is in the definition.



This is what I said:

(1) Two equivalent definitions:

(1a) Df. An argument is valid if and only if every interpretation in which all of the premises are true is an interpretation in which the conclusion is true.

(1b) Df. An argument is valid if and only if there is no interpretation in which all of the premises are true and the conclusion is false.

Therefore:

(2) Th. If there is no interpretation in which all of the premises are true, then the argument is valid. — TonesInDeepFreeze

And we can add:

(3) Th. If a set of sentences G is inconsistent, then for any P, <G P> is a valid argument. (i.e. explosion)

So, since {A -> ~A, A} is inconsistent,

A -> ~A
A
therefore ~A

is valid.

The wikipedia article you cited literally says the principle of explosion is "disastrous" and "trivializes truth and falsity." — NotAristotle

WRONG. You egregiously misrepresent the article.

That Wikipedia article does not say that the principle of explosion is disastrous. What it does say is that explosion makes any inconsistent axiomatization disastrous. And the point is that if you have an inconsistent theory and explosion then you have a trivial theory in the sense that every sentence is a theorem. One approach is to not have explosion but to allow inconsistent theorems. But in ordinary logic, we have the law of non-contradiction, so one would eschew inconsistent theories even if not for explosion.

Explosion is not incompatible with the law of non-contradiction. Rather, retaining explosion but eschewing inconsistency upholds non-contradiction. On the other hand, eschewing explosion but retaining inconsistency does not uphold non-contradiction.

All of your posted confusions and now a blatant misrepresentation of a cite. You are egregious.

a contradictory argument — NotAristotle

Arguments are not contradictory or not. Sentences or sets of sentences are contradictory or not.

The "principle" of explosion directly infringes the law of non-contradiction. — NotAristotle

That is directly false. You don't know what you're talking about. You're an ignoramus spouting misinformation and confusion while you won't even read a single page in a book or introductory article on the subject.

Posters are citing me as if to represent what I've said. That calls for, in my words, not those of other people, representing what I've said.

My comments concern ordinary formal logic unless stated otherwise:

The question was:

Is the following argument valid?

(1)

A -> ~A
A
therefore ~A

The answer is:

Yes, (1) is valid.

Df 1. An argument is valid if and only if there are no interpretations in which all the premises are true and the conclusion is false.

That is equivalent with:

Df 2. An argument is valid if and only if every interpretation in which all the premises are true is an interpretation in which the conclusion is true.

(By the way, there is no mention of inference rules in those definitions. The definition of 'valid argument' is couched only with regard to truth, falsehood and interpretations.)

Then note that there are no interpretations in which both the premises A -> ~A and A are true (see the truth table), perforce there are no interpretations in which both the premises A -> ~A and A are true and the conclusion ~A is false.

However, we show: For any inference rule, and for any interpretation, there is no application of the rule that allows deducing a falsehood from true premises. One such inference rule is modus ponens, so the validity of (1) can also by shown by this proof:

1. A -> ~A
2. A
3. ~A (1, 2 modus ponens)

But the definition of validity provides that if a set of premises is inconsistent, then any argument with that set of premises is valid (call that 'explosion concerning arguments'). So, since the set of premises (A -> ~A, A} is inconsistent (see the truth table), the argument with ~A as conclusion is valid.

just meta-logically different — NotAristotle

What, according to you, is the difference? Yes, they are different formulas, but equivalent according to your own definition. Just waving "meta-logic" like some kind of magic wand is nothing.

The ratio of my substantive typos to good information and good arguments is 1:10000.

The ratio of your confusions, circularity, and ignorance to good information and good arguments is 1:1.

Down the slippery slope of formalized illogicality. — NotAristotle

You've not shown any illogic I've committed. Meanwhile, you are slippery mess of informal illogic.

You're slipping Tones. — NotAristotle

While you are slipshod.

Yes, my typo. Thank you.

A v B were the only rule we applied — NotAristotle

A v B
is a sentence, not a rule.

I just think you're disregarding the proviso I stated, namely that a rule must actually have been followed, not merely be present in an argument. — NotAristotle

What is the difference between "followed" and "present"? And, according to you, how does it vitiate what I said about contradiction.

what I mean is that the possibilities for what is true and what is false are arrayed across a truth table. — NotAristotle

The ordinary rules do not eschew any rows in any truth table.

So for the expression A v B, the truth table is T, T, T, F. On the other hand, T, F, F, F, is A ^ B. Every possibility wherein T is present must be uniquely accounted for by the rules. — NotAristotle

You didn't read a word I wrote about realizability.

If the conditional is construed as only being true when A and B are true — NotAristotle

Is that your offering? So, the conditional is false when A is true and B is false (that is ordinary), when A is false and B is true, and when A is false and B is false?

Then we would have A -> B is true if and only if A & B is true, So "if then" to you is just conjunction?

I have tried to formalize it and can't seem to do so; this is an approximation:

(A v ~A) → (~B v ~A) — NotAristotle

That's very incorrect.

Why don't you just read one chapter in an intro textbook? Is there some reason you won't read even a few pages of a book or article to inform yourself on the subject? Are you allergic to books and articles or something? Have a phobia of them or something?

You still have not stated any rules.

The argument:

A -> ~A
therefore A

makes use of the interpretative clauses for '->' and '~'.

But I have not mentioned a rule, since the above is merely an argument and not a proof.

With the argument A->not-A, A, therefore not-A, the following of the rule, namely the conditional in that argument, leads to a contradiction between A and not-A — NotAristotle

You didn't read a word I wrote about that.

What are the "full meanings" of "If P then Q" and "P does not imply Q", according to you?

And what is the difference, according to you, between "the meaning" and "the full meaning"?

I was disputed that the following definitions are equivalent versions of the ordinary textbook definition:

(1) An argument is valid if and only if there are no interpretations in which all of the premises are true and the conclusion is false

(2) An argument is valid if and only if every interpretation in which all of the premises are true is an interpretation in which the conclusion is true

The dispute comes down to a claim that, without justification I had applied the material conditional in the meta-language.

I have explained that in ordinary formal logic, the material conditional is used for both the object language and meta-language, especially since the meta-language itself is formalizable and logicians don't eschew the material conditional merely on account of moving to the meta-language.

I cited numerous definitions of 'valid', some using (1) and some using (2). They are equivalent, though a writer may choose one or the other and not mention the other or the equivalence, since the material conditional is indeed the sense of "if then" in ordinary formal logic, whether object-language or meta-language.

But I happened to come across a text that does mention the equivalence:

"A set of sentences G implies or has a consequence the sentence D if and only if there is no interpretation that makes every sentence in G true, but makes D false. This is the same as saying that every interpretation that makes every sentence in G true makes D true." - Computability And Logic - Boolos, Burgess and Jeffrey.

(Very minor and not material technical disclaimer: The quote above is followed by a technical exception, but also an explanation of how that technical exception dissolves upon understanding that 'every interpretation' may be taken to mean 'every interpretation that assigns denotations to all the nonlogical symbols in whatever sentences we are considering'.)

the "meaning" of the disjunctive is not specific enough — NotAristotle

The ordinary clause is:

P v Q is true if and only if either P is true or Q is true. ('or' inclusive)

P -> Q is true if and only if either P is false or Q is true ('or' inclusive)

What specificity is lacking?

the full meaning of P->Q — NotAristotle

According to you, what is the full meaning of P -> Q?

"P does not imply Q".

Depends on what 'implies' means.

It is not the case that if P then Q
is formalized
~(P -> Q)
that's in the object language

It is not the case that P entails Q
is formalized
P |/= Q
that's in the meta-language

A -> B
A
therefore, B

is not the same argument as

~A v B
A
therefore, B

That doesn't vitiate that A -> B and ~A v B are 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

Where G is a set of sentences and Q is a sentence, "G entails Q" is symbolized:

G |= Q

I.e, there is no interpretation in which all the members of G are true and P is false.

If G is a singleton {P}, then we sometimes write:

P |= Q

I.e., there is no interpretation in which P is true and Q is false.

Then note:

P -> Q |= ~P v Q
and
~P v Q |= P -> Q [corrected in edit]

Whatever you mean by "meaning", the sentences are equivalent in the sense above.

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. — NotAristotle

See later in this post for my reply to your supposed rules given lately.

Member of what sets? What sets are you talking about?

And what does "exhausts all truth possibilities" mean?

And an application of rule may not result in a contradiction? You said previously that you don't define 'correct' for rules. So consider the rule of conjunction-intro:

From P and Q infer P&Q.

If P is A and Q is ~A, then apply the rule to get A & ~A.

Conjunction-intro is not a rule for you now?

A rule relating two different variables would have (I think) 15 possible truth configurations. — NotAristotle

There are 16 2-place Boolean functions.

The rules must at least enable all those possibilities to be instantiated (though perhaps it may exclude possibilities that are necessarily contradictory). — NotAristotle

What does it mean to "instantiate" in that regard?

What we do have is this:

All 16 2-place Boolean functions are realized by certain sets of connectives. And if all 2-place Boolean functions are realized, then all n-place Boolean functions are realized for all n.

That is not regarding inference rules, but merely the definitions of the connectives and the truth evaluation of compound sentences.

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. — NotAristotle

Those are not rules for arguments. Those are just the standard clauses in the ordinary definition for the truth value of compound sentences. (Except your use of "must", and note that you left out "must" for the first and fifth.)

And it's not clear what you mean. Do you mean "must be true" as "is necessarily true"? With modal operator n for 'necessarily', taken at face value, your formulations seem to be:

~P is true if and only if P is false. (So far, so good!)

P & Q is true if and only if both nP and nQ.

P v Q is true if and only if either nP or nQ. (And, unless you tell me otherwise, I take 'or' as inclusive).

P -> Q is true if and only if P necessarily implies Q. (?)

P <-> Q if and only if either both P and Q are true or both P and Q are false. (Good!)

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. — NotAristotle

What does "enable truth possibilities to be instantiated" mean?

And what if a rule allows a contradictory sentence (but that itself is not a contradiction) to be derived? Note that in predicate logic there is no mechanical procedure to determine whether any given sentence is or is not contradictory. So, with your offer, there would not be a mechanical procedure to determine whether an argument did use only your rules. But, of course, we may consider such a logic.

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. — NotAristotle

That might be helpful if you define "enable all truth possibilities to be instantiated" vis-a-vis rules.

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. — NotAristotle

It doesn't derive a contradiction. So in what way does it fail to "enable all truth possibilities to be instantiated"?

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. — NotAristotle

No, it doesn't result in a contradiction. The conclusion is ~A, which is not a contradiction. Yes, the premises are inconsistent, but your definition of "rule" doesn't disallow inconsistent sets of premises, only required is that application of the rule doesn't allow a conclusion that is a contradiction. The particular application you mentioned doesn't derive a contradiction. But other applications do derive contradictions. I know of no rule that disallows deriving contradictions, since rules don't disallow inconsistent sets of premises. However, as mentioned previously in this thread, one might require rules to not have inconsistent sets of premises. But the catch ... for predicate logic, it would not be algorithmically checkable to see whether a rule was applied, since it is not algorithmically checkable to see whether a set of sentences is inconsistent. Though, I suppose you mean for all your rules to be informal anyway.

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.

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.

Very inviting.

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

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.

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.

* 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.

Meanwhile, I hope you're looking up the method of truth tables.

(1) How is your meaning of the conditional different from the ordinary meaning in formal logic?

You use "must"; is that in addition to "is"? Example:

If Bob is smart then Bob knows English history.

In ordinary sentential logic, that is false in an interpretation if and only if Bob is smart and it is not the case that Bob knows English history.

What is your sense of "must"?

Is it that there are no possible circumstances in which Bob is smart and Bob does not know English history? That is, there are no interpretations in which Bob is smart and Bob does not know English history? Or does "must" mean something else for you?

As I shared already, note the difference in ordinary formal logic:

Here 'P' and 'Q' are variables ranging over sentences:

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.

Symbolized :

{P} |= Q

(2) Do you mean, "as long as the rules are followed (applied) correctly" or do you mean "as long as the conclusion follows correctly from the rules"?

And I asked you already, what is "the correct conclusion" when there may be many correct conclusions? And you skipped answering

And "exhausts"? A lot of different conclusions may follow from a given set of premises. How would you know that you exhausted them? I asked that already, and you skipped answering.

(3) Connectives are not rules. Rather, we have rules for connectives. What are your rules for the connectives?

(4) What if the speaker is not around to clarify? What if the speaker is too confused himself? What if there's not a particular speaker but rather the statement is a general public statement? I asked you about that already, and you skipped answering.

(5). Okay, so take out meaning. Now, how is your offer different from ordinary formal logic?

(7) Yes, they are equivalent. The former is a less rigorous way of saying the latter.

(8) You say, "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."

And, as I've shown, that is circular. You have not given an answer to that other than eliding that validity and truth-preservation are the same.

(9) Whether a rule is truth-preserving or not is not based on whether it happens to be a rule, but rather on the fact that it is truth-preserving. One can make any rule one wants to make, and it will or will not be truth-preserving not on the basis that one says, "It's a rule" but rather on the basis that any application of the rule is truth-preserving.

So, still in your situation:

To define 'valid' (truth-preserving) you appeal to correct rules. But what is a correct rule? Well, it's one that is valid (truth-preserving). That's circular.

Let's compare with a non-circular approach in ordinary formal logic:

First, define, 'is an interpretation'.

Second, define 'is true in an interpretation' and 'is false in an interpretation'.

Third, define 'valid argument' without mentioning inference rules, but only mentioning interpretations and true and false.

Fourth, we state rules that are correct in the sense that they provide for only valid arguments.

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. — NotAristotle

That is equivalent to saying: A rule is correctly used only if application of it never leads from true premises to a false conclusion.

And "never leads from true premises to a false conclusion" is validity.

You've managed to define validity in terms of correct rules and correct rules in terms of validity.

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. — NotAristotle

Nope. You say that your notion of validity is based on proper use of rules, but your notion of proper use of rules goes through the notion of rules being truth-preserving. But truth-preserving is validity, even as you defined yourself "If the premises are true then the conclusion is true".

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). — NotAristotle

Notice that you didn't say anything about the meanings of P and Q, even if they were translated to a natural language.

Rather, you mentioned only the meaning of the conditional (and I would mention also the meaning of 'not'.)

That's an example of formal logic. We stipulate how the connectives determine the truth or falsehood of compound statements depending on all the assignments of the sentence letters or atomic statements, without having to consult otherwise as to the meanings of those atomic statements.