We've talked about the equivalence of P -> Q to ~P v Q, but it's often more intuitive I think to use another equivalence ~(P & ~Q), and to read this as "no P without Q" .

The short-circuiting is still the same. Once you have ~P in hand, you know that ~P v anything is true; you also know that P & anything is false. When you have ~P, no P without Q is true for every Q, because there are no P; you'll never find a P unaccompanied by Q, because you'll never find a P.

We've talked about the equivalence of P -> Q to ~P v Q, but it's often more intuitive I think to use another equivalence ~(P & ~Q), and to read this as "no P without Q" . — Srap Tasmaner

We have a whole thread on this idea:

A→B means not(A without B). — bongo fury

...This is why I would prefer "No A without B." — Leontiskos

So you think it is literally impossible to give argument 2 without implying argument 1? — Leontiskos

Yes. I'll rephrase the argument in propositional logic:

P1. P ∧ ¬P
P2. P ∧ ¬P ⊢ P (conjunction elimination)
P3. P ⊢ P ∨ Q (disjunction introduction)
P4. P ∧ ¬P ⊢ ¬P (conjunction elimination)
P5. (P ∨ Q) ∧ ¬P ⊢ Q (disjunctive syllogism)
C1. Q

Notice that P2 - P5 are all rules of inference; they are implicit in every argument (along with every rule I didn't write out) and so don't need to be said.

Yes. — Michael

I am going to limit myself to serious interlocutors.

I am going to limit myself to serious interlocutors. — Leontiskos

I am being serious. Read up on the principle of explosion.

- This has already been explained to you.

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

(Your contention that argument 2 cannot ever exist without argument 1 is magical, ad hoc thinking. There is nothing serious about it.)

This has already been explained to you. — Leontiskos

He's talking about something else. I'm taking about this:

P1. P ∧ ¬P
C1. Q

This is literally the principle of explosion:

In symbolic logic, the principle of explosion can be expressed schematically in the following way:

P ∧ ¬P ⊢ Q For any statements P and Q, if P and not-P are both true, then it logically follows that Q is true.

---

Your contention that argument 2 cannot ever exist without argument 1 is magical, ad hoc thinking. There is nothing serious about it. — Leontiskos

See the section titled "Proof" which includes all my additional steps P2 - P5. They don't need to be made explicit because they are inherent rules of inference.

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

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

It says "the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous", which it is.

Given that P ∧ ¬P entails Q, we must be sure that we never allow for P ∧ ¬P to be true.

Forget "formal axiomatic system," a contradictory argument is always a problem. The "principle" of explosion directly infringes the law of non-contradiction. It's silly to even call it a principle.

The "principle" of explosion directly infringes the law of non-contradiction. It's silly to even call it a principle. — NotAristotle

The principle of explosion is simply the acknowledgement that if we apply the rules of inference to a contradiction then we can derive any conclusion we like. That is simply an a priori fact about propositional logic.

Obviously almost nobody will accept that a contradiction can be true.

Although there are dialetheists like Graham Priest who argue that they can.

If the first premise were agreed to, that would mean the disjunctive elimination leading to C1 would not work. If P and not-P are accepted, I take it that they are accepted propositions throught the entire proof. Unless P is suddenly not accepted in P5?

By your own definition the argument is not valid.

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.

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.

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. — TonesInDeepFreeze

That's fine if you want to say that the strange way you want to apply your definition is based on explosion, but this is a new claim. Earlier in the thread you said that the two are "similar," not that one presupposed the other:

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

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.

My point here has been that validity has to do with "follows from." If you think that your idiosyncratic application of your definition of validity is permissible because "anything follows from a contradiction" (i.e. explosion), then you have not disagreed with me that validity has to do with "follows from."

(I won't belabor the point of how strange it is to count on explosion enthymeme to understand a foundational definition.)

(On Tone's account) ...your idiosyncratic application of your definition of validity... — Leontiskos

Tones is the one being idiosyncratic... :grin:

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.

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

Your interpretation of your definition presupposes explosion.

I am saying no such thing. — TonesInDeepFreeze

So you would say that someone who does not understand the principle of explosion can apply the definition in the way you prefer?

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).

I haven't said anything about 'presuppose'. Rather, I have shown that the definition of validity entails the principle of explosion. — TonesInDeepFreeze

It only "entails" it because it has presupposed it. Else 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.

But you are stuck in your quibbles again. When you figure out how exactly the principle of explosion relates to your definition, feel free to get back to me.

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.

If you link to my quote, time permitting I will address it. — TonesInDeepFreeze

Did you see the colon at the end of that sentence?

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. — TonesInDeepFreeze

Well, if your strange interpretation of your definition is not based on explosion, then we are back to square one, and it is simply wrong. "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.

Arguments are not valid in virtue of being inconsistent (lol). You obviously won't admit this, but most TPFers are able to recognize its truth. Indeed, there is only one person who has agreed with you in this.

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.

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.

Here is what Michael said:

That ((P→Q)∧Q), therefore P is not valid, whereas ((A∧¬A)∧(P→Q)∧Q), therefore P is valid, does seem strange to me. Inconsistent premises don't seem to have anything to do with whether the argument "follows." Although I have a feeling that Tones will have something to say about that. — NotAristotle

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

• NotAristotle: Tones' claim that inconsistent premises make an argument valid by definition seems strange to me.
• Michael: It is valid because of explosion.
• Leontiskos: Tones is giving a different explanation for why it is valid.

You came in and said, "Michael's argument is valid," but you haven't at all reckoned with what was really said. I invite you to do that. NotAristotle made an observation about your construal of validity, and Michael defended your construal of validity with recourse to the principle of explosion. Reckon with that.

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

"If the premises are inconsistent then the argument is valid by definition," does not mean that the definition of validity is equivalent to the premises being inconsistent. It only means that any argument whose premises are inconsistent is valid in virtue of the definition of validity. Your boat is being swamped by your quibbles. :roll:

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.