Here's one:

Leontiskos insinuated rather than openly stating.
To insinuate instead of openly stating is dishonest.
Leontiskos has been dishonest.

That is logic that is not mere symbol manipulation.

Here's more logic that is not mere symbol manipulation:

Even just one counterexample refutes a universal generalization. So the argument above refutes that I regard logic to be mere symbol manipulation. And this argument does too!

Leontiskos: There are folk in these parts who drive Toyota Camrys.
Tones: I certainly don't drive a Toyota Camry!
Leontiskos: What kind of car do you drive?
Tones: — Leontiskos

Leontiskos: There are people here who think acetone is merely oxygen.

TonesInDeepFreeze: I don't think acetone is merely oxygen.

Leontiskos: What else do you think is in acetone? You must answer that for me to decide whether I meant you when I said that there are people here who think acetone is merely oxygen.

TonesInDeepFreeze: Whatever acetone is, I don't say it is merely oxygen, Who are you claiming thinks acetone is only oxygen.

Leontiskos: It is time for you to answer my question.

TonesInDeepFreeze: You first made the claim that there are people here who think acetone is merely oxygen. If you were undecided about me when you made that claim, then which of the people here, do you claim to think that acetone is just oxygen? It's a simple question, and would be honest to answer rather than being a sneaky insinuator.

Leontiskos: How dare you ask me to name the people I claim to think acetone is just oxygen, when you won't write a post about what your notion of logic is?

TonesInDeepFreeze: Well, this started with you making the claim. So, it is natural to first get clear who you meant. And, if you can't say a single person other than me, then that leaves only me.

Leontiskos: What car do you drive?!

Here's some informal logic that is not "mere symbol manipulation":

You said that there are some here who view logic as mere symbol manipulation. So, in such a small domain as this one, you could easily list them. And, since by your claim that there are "some", there is at least one that you have in mind. So, if you can't list any other than me, then we may infer that you meant me.

But you left it open, thus it is insinuation. But you don't have the integrity to say who you mean.

/

To maintain that I don't think logic is mere symbol manipulation, it is not required for me to say what logic is. To maintain that basketball is not mere players' statistics, I don't have to tell you what basketball is; whatever it is, I know that it is not mere players' statistics.

/

At some point, time and interest allowing, I may write a post with more about my own sense of the scope of logic. In any case, I use and recognize informal logic as well as formal logic, and I don't take formal logic to be mere symbol manipulation. That is apparent even by the fact that I have discussed, in your presence, certain English sentences vis-a-vis symbolization, as I even did a few posts ago.

Moreover, so many posts I have written about mathematics and logic, written mention a scope that is not at all confined to mere symbolization.

/

Who did you mean ? If you won't say, then I'll take it you don't have the guts to say, as you are sneaky insinuator. "Joe McCarthy" Leontiskos saying, "I have in my hand a list of posters who view logic as mere symbol manipulation".

Who did you think fits the bill?

It's a simple question.

When you wrote, "There are some logicians in these parts who view logic as mere symbol manipulation", who were you referring to?

(I meant 'posters' not 'photographers'.)

But you did order it.

You asked for a symbolization to see how the definition of 'valid argument' implies that explosion is a valid argument. I gave you exactly what you asked for.

When you wrote it, you were referring to unnamed posters. Was I one of them or not?

Your intent when you wrote it is not affected by anything I say retroactively.

But that the definition of validity implies that there being no interpretation with all true premises implies that the argument is valid - that I am not so sure about, because I do not see how a definition can imply anything. — NotAristotle

You said you asked for a symbolization of the definition for this reason:

I was trying to understand how the definition implies that in terms of symbolic logic. — NotAristotle

I gave you symbolizations of (1) the definition of 'valid argument' and (2) "if there is no interpretation in which all the premises are true, then such an argument is valid".

With those symbolizations it is merely an exercise to prove (2) from (1).

But your reply is to say that you don't see how a definition can imply anything! If you don't understand how definitions are used in proofs, then that is what should be discussed first, not belaboring symbolizations. But once I did give you the symbolizations, if you knew anything about basic symbolic logic, then you would be able to finish the exercise by showing the proof of (2) from (1) and thus not have to wonder how definitions imply things. Meanwhile you won't even look at a book to see how definitions and proofs work in first order logic.

You are oblivious to how very irrational you are.


Here's you at a restaurant:

Waiter: Welcome to TPF Bar & Grill, may I take your order?

NotAristotle: Yes, I'll have the ribeye steak, rare, cooked with an extra amount of salt.

cut to:

The waiter delivers the steak, cooked to order, rare and with extra salt, very nicely set on a beautiful china plate, with gleaming utensils.

Waiter: Here's your steak, sir, as you ordered it - a beautiful ribeye, rare, with plenty of salt. I hope you enjoy it.

NotAristotle: Take this back. I can't eat this. I'm a vegetarian. And I'm on a low-sodium diet.

Waiter: I don't understand, sir. It's what you ordered.

NotAristotle: Just take it back.

Waiter: Very well, sir. I'll bring you a menu to order something else.

NotAristotle: Yes, bring me a menu so that I can not read it. No, never mind, just bring me a ribeye steak, rare, extra salt.

When you claim, "There are some logicians in these parts who view logic as mere symbol manipulation", do you include me, thereby claiming that I view logic as mere symbol manipulation?

I'm sure you could find that in a textbook, but one must recognize that such textbooks presuppose that the premises are not inconsistent. — Leontiskos

By following the links to the posts, the poster referred to disjunctive syllogism.

In that context, what set of premises does Leontiskos think textbooks "presuppose" to be consistent?

If the poster meant explosion, then still, in that context, what set of premises does Leontiskos think textbooks "presuppose" to be consistent?

The principle of explosion is that from an inconsistent set of premises any conclusion follows. There's no "presupposition" that the set of premises in such an argument is consistent. It wouldn't even make sense otherwise.

There are some logicians in these parts who view logic as mere symbol manipulation — Leontiskos

Leontiskos does not name who he means, so it behooves me to speak for myself.

(1) I'm not a logician and (2) I do not regard logic as mere symbol manipulation.

As for professional logicians, I'd be interested to know of one who regards logic as mere symbol manipulation.

I'll use the notion of 'satisfiable' (there is an interpretation in which all the members the set are true, and 'unsatisfiable' denoting the negation of that) rather than 'consistent' (there is no deduction of a contradiction from the members of the set, and 'inconsistent' denoting the negation of that), to keep the matter all semantical, and as it is an obvious and easy to show theorem that if a set of sentences is inconsistent then it is not satisfiable.

Here I changed some variables from previously, to avoid using T as both a variable and relation symbol, and to make the role of others more clear. Hope I don't make any typos:

We already have (1) below:

(1) Definition of 'is a valid argument':

For all g(g is a valid argument
if and only if
(g is an argument
and
for all i(if i is an interpretation, then it is not the case that
((for all p(if p is a premise of g, then p is true per i))
and
for all c(if c is the conclusion of g then c is false per i)))))

Symbolized:

Let Vx stand for x is a valid argument
Let Bx stand for x is an argument
Let Dx stand for x is an interpretation
Let Rxy stand for x is a premise of y
Let Txy stand for x is true per y
Let Uxy stand for x is the conclusion of y
Let Fxy stand for x is false per y

Ag(Vg
<->
(Bg
&
Ai(Di ->
~((Ap(Rpg -> Tpi))
&
Ac(Ucg -> Fci)))))

(2) Then we want to show that, for any argument g, if there is no interpretation in which all the premises of g are true, then g is valid:

Ag((g is an argument
&
~Ei(Di & Ap(Rpg -> Tpi))) -> Vg)

It's merely a tedious, routine exercise to do the proof in a system of the first order predicate calculus.

(3) Also, we want to show that, for any argument g, if there is no interpretation in which the conclusion is false, then g is valid:

Ag((g is an argument
&
~Ei(Di & Ac(Ucg -> Fpi))) -> Vg)

It's merely a tedious, routine exercise to do the proof in a system of the first order predicate calculus.

(4) And you want to also show that, for any argument g, if there is no interpretation in which all the premises of g are true, and there is no interpretation in which the conclusion of g is false, then g is valid.

But that is implied, a fortiori, from (2) and (3), given this theorem of sentential logic:

((P -> Q) & (H -> Q)) -> ((P & H) -> Q)

Why resurrect this dead thread? — jgill

The BSer mentioned me in a misleading way, and posted more BS.

You asked me a question. I gave you a rigorous detailed answer. What was the purpose of your question?

And you would do well to re-read that Wikipedia article you cited about explosion, to see that you misrepresented what it says and to see that the passage you cited is itself based on a passage in a site about paraconsistency, which is a context that may, depending on the formulation of such a system, allow that contradictions can be other than false, which is the opposite of what you claim to hold. Indeed, explosion, which you reject, is the antithesis of paraconsistency.

This is not sophistry:

(1) define 'is an interpretation'

(2) define 'is true in an interpretation'

(3) define 'is an argument'

(4) define 'is a valid argument' (mentioning only truth, falsehood and interpretations)

(5) define 'is inconsistent'

(6) define 'is unsatisfiable'

(7) show that an inconsistent set is unsatisfiable

(8) define the principle of explosion

(9) show that the definition of 'valid argument' implies that the principle of explosion is valid

And note that, in this sequence, (4) precedes (8) and (9).

There is no sophistry in pointing out that that the standard definition of validity implies that any argument with an inconsistent set of premises is valid and that to state the definition it is not required to formulate the principle of explosion or to show the validity of the principle of explosion.

Are there any introductory textbooks that talk about the principle of explosion? — NotAristotle

Yes. But why do you ask, unless you'll read one?

I said "principle of explosion" not "disjunctive syllogism" — NotAristotle

You linked to my post about disjunctive syllogism.

Disjunctive syllogism is in lots of textbooks. You're ridiculous.

What I've said is correct, not merely because I said it.

Oh come on! Get a textbook that uses disjunctive syllogism. You won't even look at a textbook yet you are challenging me to cite one! Not playing your idiotic game.

I would guess that Tones regards it as unconventional. — NotAristotle

It's not a matter of what I "regard" to be the case.

The rule is completely unambiguous:

If P v Q is on a line, and ~P is on a line, then we may put Q on a new line.

Or better, without "we", "may" and "put", the rule may be stated:

A deduction from a set of formulas G is a sequence of formulas such that:

If a formula P appears on a line, then it is either a member of G or there are formulas on previous lines such that there is a rule such P comes from those formulas.

And among the rules is: P v Q, ~P |- Q.

What disjunctive syllogism is is settled by any ordinary textbook that has it as a rule. Whatever particular wording is used in an ordinary textbook, it amounts to the rule that from P v Q and ~P we may infer Q.

I don't get how you get Q from (P or Q) if P is true. — NotAristotle

In this case:

We have the premise P & ~P.

We want to get P v Q and we want to get ~P, so we can apply disjunctive syllogism to get Q.

We get P v Q by first getting P from P & ~P by conjunction elimination, then P v Q from P by disjunction introduction.

We get ~P from P & ~P by conjunction elimination.

/

The rule is: If P v Q is on a line, and ~P is on a line, then infer Q.

The rule is NOT: If P v Q is on a line, and ~P is on a line, and P is not on a line, then infer Q.

But that doesn't work if A and not-A are both true. — NotAristotle

(1) In no interpretation are both A and ~A true.

(2) Having A & ~A as a premise, thus being able to have A as a line and ~A as a line, does not vitiate use of disjunctive syllogism:

The rule is: If P v Q is on a line, and ~P is on a line, then infer Q.

The rule is NOT: If P v Q is on a line, and ~P is on a line, and P is not on a line, then infer Q.

The definition involves quantification, so I wouldn't reduce it to a merely sentential formula.

I adduced it as a previously proven theorem to help you see how the final step would by an application of modus ponens, so you'd have another way to look at it to see that the steps are correct. We could also do it this way:

Rule (disjunctive syllogism): If P v Q occurs on a line, and ~P occurs on a line, then infer Q.

1. P & ~P (premise)
2. P (from 1, conjunction elimination)
3. P v Q (from 2, disjunction introduction)
4. ~P (from 1, conjunction elimination)
Q (from 3, 4, disjunctive syllogism)

E↔A∧(B→¬(C∧D)) — NotAristotle

That is not a rendering of my formulation.

Here's how I would write it:

1. P & ~P (premise)
2. P (from 1, conjunction elimination)
3. P v Q (from 2, disjunction introduction)
4. ~P (from 1, conjunction elimination)
5. ((P v Q) & ~P) -> Q (theorem)
6. ((P v Q) & ~P (from 3, 4, conjunction introduction)
7. Q (from 5, 6, modus ponens)

Or, have explosion as either a primitive rule or derived rule in a natural deduction system:

1. P & ~P (premise) {1}
2. Q (explosion) {1}

First, why do you ask?

The statement you seem to have in mind is:

(1) It is not the case that there is an interpretation in which all the premises are true and the conclusion is false.

One way to write that:

Let Px stand for "x is an interpretation in which all the premises are true".

Let Qx stand for "x is an interpretation in which the conclusion is false".

Then the statement is:

(2) ~Ex(Px & Qx)

But neither (1) nor (2) capture the definition of 'is a valid argument', which, if we spell out the quantifiers is:

For all T(T is a valid argument
if and only if
(T is an argument
and
for all N(if N is an interpretation, then it is not the case that
((for all p(if p is a premise of T, then p is true per N))
and
for all c(if c is the conclusion of T then c is false per N)))))

Symbolized:

Let Vx stand for x is a valid argument (the definiendum)
Let Bx stand for x is an argument
Let Dx stand for x is an interpretation
Let Rxy stand for x is a premise of y
Let Txy stand for x is true per y
Let Uxy stand for x is the conclusion of y
Let Fxy stand for x is false per y

AT(VT
<->
(BT
&
AN(DN ->
~((Ap(RpT -> TpN))
&
Ac(UcT -> FcN)))))

(I hope I got that all correctly, including the parentheses.)

Again, below is what obtains in ordinary formal logic, regardless of any inclinations I might have or not have toward what the definitions or results "should be or should not be". That point stands throughout this discussion.

Semantic: Explosion as an argument form is valid. It is easy to show that explosion as an argument form is valid by noting that the definition of 'valid argument' implies that explosion is a valid argument form.

Syntactic: Explosion as an inference rule, depending on the system, can be either a primitive rule or a derived rule.

(1) Contrary to Leontiskos, I am not "disingenous" in this discussion. No statement of mine, about Michael or anything else, has been shown to be dishonest and none to be materially incorrect.

(2) As to validity, I said that the standard definition of 'valid argument' implies that any argument with an inconsistent set of premises is valid. That it is correct: The standard definition implies that any argument with an inconsistent of premises is valid.

(3) I said that the standard definition of 'valid argument' does not mention the principle of explosion, and I listed the definitions and points that the definition of 'valid argument' does rely on. Then from the standard definition of 'valid argument' we easily show that the principle of explosion is valid. That is correct.

(4) There is no "contrarianism" in any of that.

I don't see how this cannot be understood.

From the definition of validity, we show that the principle of explosion is valid.

Then, if an argument has an unsatisfiable set of premises, we thereby show that it is valid, no matter what the conclusion is. (i.e. an instance of the principle of explosion).

Also, from the definition of validity, we show that modus ponens is a valid argument form.

I pointed out that the particular argument in the original post happens to be an instance of modus ponens, so the argument is valid on that basis too. That doesn't contradict showing that the argument is valid on account of explosion. Both are correct: it is valid on account of modus ponens and it is valid on account of explosion. And it valid on account of both of those because both of those are valid on account of the definition of validity.

That modus ponens is a valid argument form is shown from the definition of 'valid argument'; and that explosion is a valid argument form is also shown from the definition of 'valid argument'. The validity of modus ponens and the validity of explosion are both consequences of the definition of 'valid argument'.

And that is the case no matter what I say, no matter what Michael says, and no matter how you characterize any relationship between what Michael says and what I say.

/

Again, since people say things like "Tones's view":

When I report how it goes in ordinary formal logic with the standard definition of 'valid argument', I have not thereby claimed that ordinary formal logic (with the usual definition of 'valid') is the only reasonable logic, or the best one in all situations, or the best one for a philosophical understanding of logic and validity, or that it accords with all of everyday reasoning, or even that is more than hardly found in everyday reasoning, or that is the only one worthy of study or adopting. Alternative formal logics abound and are, in my opinion, worthy of study and adoption in the contexts they are suited for. But ordinary formal logic is appreciated by, studied, applied and discussed greatly among mathematicians, logicians, computer scientists, and philosophers. It is at the heart of mathematical logic that axiomatizes the branches of mathematics, including computability, including the very invention of, and improvements to, the digital computer. But, meanwhile, ordinary formal logic is fair game to critique. But critiques of any logic are mindless, and not even close to philosophy, when they are premised in ignorance and confusion about how it actually goes in the logic. And corrections given to such ignorance and confusion are not themselves a form of endorsement of the logic nor do they constitute claims that it is "right", unless one does go on to endorse the logic. For one to explain what actually happens in the logic is not in and of itself to say that people should adopt the logic - but rather, at least that they should not attack it on false bases. By analogy, one doesn't have to agree with the constitution of a country just to study it and report what it actually says. Moreover, one may rebut arguments against a logic without endorsing the logic, as one's rebuttals may be merely that the arguments are not good, without opining on whether the conclusions are correct or incorrect.

There is repetition in the above, but it is there to drive these points that keep getting overlooked no matter how many times they are stated and especially to, hopefully, foreclose against someone yet again putting words in my mouth to make it appear that I've adopted a position that in fact I have not adopted.

Your interpretation is irrational without recourse to the principle of explosion. — Leontiskos

I didn't give an "interpretation" of the definition. I stated the standard definition. The definition doesn't make "recourse" to the principle of explosion. Rather, from the definition, it is simple to show that the principle of explosion is valid.

That can't be more clear: (1) State the definition. (2) From the definition, infer the principle of explosion. In that order: (1) then (2). How cannot someone not understand that?

Including the prior steps:

(1) we define an 'interpretation'

(2) we define 'true in an interpretation'

(3) we define 'argument'

(4) we define 'valid argument' (mentioning only truth, falsehood and interpretations)

(5) we define 'inconsistent'

(6) we define 'unsatisfiable'

(7) we easily show that an inconsistent set is unsatisfiable

(8) we easily show that the definition of 'valid argument' implies the principle of explosion (that is, that the principle of explosion is valid)

The principle of explosion was not mentioned, assumed, invoked or recoursed to. Only that it was proven to be valid from the definition of 'valid argument'.

I gave details in an earlier post.

the reason NotAristotle is so confused is because Michael is failing to recognize that he is justifying validity in a different way than you are; and you are aiding and abetting his failure. NotAristotle made an argument against your view — Leontiskos

Again, I'm not interested, at this juncture, in untangling Michael's role. I am especially not interested in commenting on his posts vis-a-vis you in between with your characterizations of his posts and your characterizations of my posts and your characterizations of how they compare.

NotAristotle is confused because he knows virtually nothing about the subject, not even chapter one of any book or first material in an article, whether standard or alternative - doesn't even know the differences among connectives, sentences and arguments. And talk about cherry picking, that one by NotAristotle from Wikipedia is a doozy! It was a terrible misrepresentation and failure to even read the context cited by the article, resulting in him flirting with the very point of view he claims to oppose! Talk about what is NOT philosophy!

If you don't want to be honest and reckon with the actual object of the conversation, I'm sure no one will be surprised. — Leontiskos

I've addressed the subject of this thread in detail.

My intellectual credibility does not require that I sort through your own disagreements with a poster, nor even that I sort through your notions in and of themselves.

you and Banno attempted to agree with Michael in order to disagree with me, despite the fact that you ultimately disagree with Michael. — Leontiskos

I don't know what passages you're referring to. You've not shown that anything I've said is incorrect. Your picture of this thread as some kind of tag team match doesn't interest me.

I didn't "cherry pick" anything. And I didn't "misrepresent" anything. YOU picked out a quote by him, and cited ME in connection with that quote by him. So I quite rightly exercised my prerogative to make clear that what he wrote is correct and is not in contradiction with anything I've written.

I laid it all out here; go look: https://thephilosophyforum.com/discussion/comment/948275

Oh, for Pete's sake. I just wrote that I am not, at least at this time, interested in sorting out the disagreement between you and him. Rather, at that juncture, I posted to make clear that a particular quote of him (quoted by you followed by mention of me and something I wrote) is correct and not inconsistent with anything I've said. That's it. That's all I posted. Get it now? [Edit: Also, I said, "The definition of validity entails that the principle of explosion is valid." And that is correct.]

No, YOU quoted a certain argument by him and followed with a comment about me, as if that argument is not compatible with what I said. So I made clear that that argument is not incompatible with what I said.

You cited me in your disputes (and without linking my name). So I exercised the prerogative to make clear that that particular argument is not incompatible with anything I've said.