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

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

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

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.

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

Sure it is, unless you are the Source of Truth Itself.

I thought not. Wierd that such an important principle would be neglected from a foundational book.

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

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

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

This is metalogic, but note that validity is meant to show how conclusions rightfully follow from premises. It is meant to provide us with a way to think correctly, and increase our knowledge.

Anything follows from an explosive system, and yet not anything follows with respect to correct thinking. This means that explosion is an aberration (along with the contradiction that it flows from). In propositional logic contradictions are supposed to be eliminated (via reductio), not utilized.

So is an explosive argument valid? In one sense it is, and in one sense it is not. It does not provide us with the thing that the notion of validity is meant to provide, but it is nevertheless valid in a certain (arguably degenerative) sense.

There are some logicians in these parts who view logic as mere symbol manipulation, without any relation to correct reasoning. For these logicians an explosive argument is uncontroversially valid.

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

"Not playing your idiotic game"

Then I accept your unconditional surrender.

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

You linked to my post about disjunctive syllogism.

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

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

So is an explosive argument valid? In one sense it is, and in one sense it is not. — Leontiskos

So we have something like tiers of sophistry:

• Michael: An argument explicitly leveraging explosive inferences is valid. (light sophistry)
• Michael: Argument 1 and argument 2 are the same argument. (medium sophistry)
• Tones: An argument with inconsistent premises is valid, irrespective of explosion. (heavy sophistry)

What's interesting about the "medium sophistry" is that Michael has detached logic from humans in a remarkably thoroughgoing way. He is basically saying, "If a conclusion is inferentially reachable from the premises, then the argument is valid, even if the argument does not present the necessary inferences." He collapses an argument and an enthymeme into one thing, which doesn't make any sense in the end. Pace Michael, inferences (or lack thereof) are part of the argument itself.

(@NotAristotle)

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.

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).

If someone were asked to "explain the reasoning" for a conclusion, then the inferential steps definitely matter.

Although, I would say there's a "logical floor" where no further arguments or definitions can settle whether an inferential rule is "necessary;" that is why I refer to "logical intuition" - so I would say that while modus poenens fits into a set of rules, I am skeptical that the move itself can be justified using argument. That it really is logical is basically a matter of faith that the way we're thinking is "correct."

Maybe there's an evolutionary argument to support "correct thinking" although that would assume that passing on genes correlates with correct thinking or something like that, which would still leave an open question of whether the thinking is "normatively" right. Maybe we might be able to conclude that it's "logical enough." or something along those lines. But it would be interesting if logical thinking could somehow be proven scientifically, and yet that would seem to also be a very circular argument.

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.

- :up:

enthymeme — Leontiskos

Hmmm.

Do you make any distinction between premises and inference rules?

He is basically saying, "If a conclusion is inferentially reachable from the premises, then the argument is valid, even if the argument does not present the necessary inferences." — Leontiskos

I'm trying to understand this. Are you arguing against the cut rule?

In practice, we show only the inferential steps we don't assume the audience can fill in for themselves. The overwhelming majority of mathematical proofs are not "complete", don't show every single step. For good reason.

As a practical matter, if your audience can't fill in the missing steps, they may not find your argument persuasive. But if you can show them the missing steps on demand, you should be on the same page.

Do you make any distinction between premises and inference rules? — Srap Tasmaner

You could, but I am not.

I'm trying to understand this. Are you arguing against the cut rule? — Srap Tasmaner

My point is that argument 1 and argument 2 are different arguments. Argument 2 could be an enthymeme form of argument 1, but it need not be. Michael somehow thinks it needs to be.

This is important because if we cannot speak about argument 2 apart from argument 1, then we cannot even understand the difference between Michael and Tones (and at this point in the thread Michael looks to be actively suppressing the emergence of that difference in a very strange way).

Edit: On my view no argument is demonstrably an enthymeme just in virtue of its material constituents. On Michael's view this is apparently not right, and therefore validity has to do with possible inferences, not documented inferences. Frege's judgment-stroke and the difference between inference and consequence seems relevant here, although Michael is pushing consequence even further than Frege's opponents would.

You said "(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."

I was trying to understand how the definition implies that in terms of symbolic logic. I think I understand how the definition could imply that an argument with inconsistent premises must be valid according to the definition you stated, and I think you will agree with me that if the conclusion is necessarily true, then the argument must be valid, according to the definition you stated. And, if the premises are inconsistent and the conclusion is necessarily true, then such an argument must again be valid according to the definition you stated.

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)

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'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.

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

If you say that logic is not merely symbol manipulation, then what do you say it is?

"(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."

"(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)"

This might not make too much of a difference, but it seems to me that (if we use the definition of validity you stated)... that there being no interpretation s.t. all premises are true does imply an argument is "valid."
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.

I asked a while back, but can anyone think of an example where at least one premise of an argument is necessarily false and yet the conclusion will not follow as an inference?

It seems like two definitions are often included in for validity, and often both are presented side by side:
1. An argument is valid when it is impossible for the premises to be true and the conclusion false; and
2a. The conclusion follows from the premises.
2b. The conclusion is contained in the premises.*

Normally, these overlap. If we accept our inconsistent premises as true we can move to the conclusion (but of course they aren't true). We can use disjunctive syllogism and disjunction introduction to move from our inconsistent premises to whatever we'd like. But this relies on a certain notion of implication and explosion, both of which have been controversial in the history of logic (if nonetheless mainstream), precisely because they seem counterintuitive and don't seem to capture natural language reasoning or notions of "good reasoning."

My thoughts were that a combination of relevance conditions for implication and changes to avoid explosion could perhaps get us to a case where an argument is valid under definition 1 but not 2? That is, we'd have inconsistent premises but no inferences connecting them to our conclusion even if we did affirm all the premises.

Can there be such a counter example where the two diverge?

I mentioned quia demonstrations vs. propter quid demonstrations earlier. Supposing that the two definitions do rightly overlap, it would seem like 1 would be a quia demonstration (going from effects backwards), while 2 actually gives us the "why." But in natural language, with our penchant for equivocal and analagous predication, fuzzy terms, and lack of clarity, I can see why people would like to hold to 2 over 1 even if they thought they properly overlapped. We might say, "1 is simply a consequence of 2."


*I am not sure how 2b works with explosion. I have seen Floridi and D'Agastino argue in the context of the "Scandal of Deduction," from an information theoretic lens that there is a certain sense in which some conclusions aren't contained in their premises, with some forms of inference injecting new information.

You mean this: ((A∧¬A)∧(P→Q)∧Q), therefore P?

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?

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.

Waiter: yes sir, of course, here it is.

NotAristotle: Was that so hard? ... thank yo-- what the hell is this?

Waiter: it's the ribeye sir, rare, with extra salt.

TonesinDeepFreeze: it's what you ordered NotAristotle, just eat it.

NotAristotle: I don't even like steak, why would I order it?

Tones: don't ask me.

Michael: stop making a scene NotAristotle, you do this every time!

NotAristotle: Leontiskos, do you want the ribeye?

Leontiskos: Not really, no.

Banno: check please.