(I'm not sure if I'm right to equate pre-reflexion with being-as-such).

An instantaneous cogito implies the structure of doubt, that is, suspension of judgment. But the cogito is committed to more than mere suspension of judgement; it is by necessity interwoven within a time "architecture."

The architecture of doubt is directly mirroring the architecture of the cogito itself, in time, but as a negation.

This architecture is pre-ontological in the sense of not yet truly ontological. That is, it is prior to the formulation of an ontology. The movement from pre-ontological knowing, the cogito, to a pre-reflexive ontology of being-as-such (that is to actually study being), requires transcendence of the cogito, where "doubt" is understood as just the negation of the cogito, ego.

It may be strange for pre-reflective awareness to be after the cogito's pre-ontological mode, but this is just the path of consciousness. Whereas pre-reflection is wholly prior to the cogito, in consciousness it comes after, as it is from the perspective of the negation of the ego that pre-reflection is attainable in a self-conscious way. This is why the saying "I think, therefore I am" is concluded after Descartes' "doubt" meditation. The saying is not the culmination of cogito but its transcendence.

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.

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.

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

"Not playing your idiotic game"

Then I accept your unconditional surrender.

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

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

What textbook says that. If you can cite that statement I'll sell the farm.

I think that is right, it is arbitrary. Although I would say that an argument can have inconsistent premises and still be valid as long as those premises do not do any "work" in the argument, but I acknowledge that my definition of validity may be atypical. At least, I would guess that Tones regards it as unconventional.

Yeah, I don't get how you get Q from (P or Q) if P is true. And I understand the disjunctive syllogism. I get that your asserting not-P, but I don't see how that negates a proposition, P, that has been stipulated to be true, per the argument.

Alright, how might you render it in a simplified form, or can it not be so rendered?

Okay, Thanks for writing out that definition using quantifiers. So could I simplify your argument by saying

E↔A∧(B→¬(C∧D)) is the definition. I know that if we are being precise it is not, but thematically would this work for the definition of validity.



But that doesn't work if A and not-A are both true. That's my point. The proof doesn't work. The proof only works if you ignore that A is also true.

I can only guess, but I think that is what Tones meant by referring to that step as a "theorem."

"See the “⊢ Q” at the end? That means that Q follows from the bit before." Okay; can you spell it out for me? It's still not clicking.

P5. (P ∨ Q) ∧ ¬P ⊢ Q (disjunctive syllogism) I do not understand the move from P5 to C1 using disjunctive syllogism. Would you mind explaining?

" ¬∃x(P∧Q) "

where x is an interpretation, P is "all premises are true" and Q is "the conclusion is false."

Is there something problematic about writing the definition of validity that way?

By your own definition the argument is not valid.

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?

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

Ah, I see, then we will say as a shorthand "invalid" as a way of saying it does not follow, that is, that the conclusion cannot be derived using a priori reasoning.

My question is, if I use a priori reasoning, how can I conclude that "I live in Antartica" (assuming that is true) based on the premise "Pluto is a planet and Pluto is not a planet". How does the conclusion "follow?" I saw your reasoning from the earlier argument, I'm just wondering what rule of inference leads to this conclusion.

To be more specific, it seems to me that in the argument you stated, P1, P5, and C1 cannot all be true. That is, if C1 is true then P1 cannot be true. And if P1 is true then C1 cannot be.

"You can use the rules of inference to derive the conclusion "I am mortal" using a priori reasoning, but you cannot use the rules of inference to derive the conclusion "I am English" using a priori reasoning"

That is well said.

Perhaps we we disagree about what may be considered a rule of inference. Unless you think an argument that is invalid only coincidentally doesn't follow? Or is it invalid because it does not follow?

Okay I agree with you that only one of those two arguments is valid. Now, in a non-circular way, explain why the one follows but the other does not.

Why not? It satisfies the definition, does it not?

If I did live in Antartica it would have to be valid wouldn't it?

Your argument is that: If logicians have defined validity, then that definition is correct. Logicians have defined validity. Therefore, that definition is correct. This is a valid argument as far as I can tell. It is, however, unsound, as premise 1 is faulty.

Besides, if someone gave the argument you gave -- "I am a man and I am not a man. Therefore I am rich" that is a nonsensical argument; the conclusion just has nothing to do with the premises, you might as well argue "I am a human and it might snow this week, therefore I live in Antartica." Even if conclusion and premise are all true i.e. the argument is sound, what kind of argument is that?

It seems that that argument would be valid, but only if one accepts that an argument is valid iff there is no interpretation s.t. all premises are true and the conclusion is false per Tones' definition.

If it turned out that validity required more than what that definition suggests (I think it does), then the argument you stated may well turn out to not be valid, as I think is the case.

Maybe another way of coming at this is as follows - the conclusion is true. Period. Under that understanding, "there is no interpretation where the conclusion is false" ergo there is no interpretation s.t. all the premises are true and the conclusion is false. But the conclusion being true does not seem to guarantee that the argument is valid. But with Tones' definition, it would. Similarly, inconsistent premises also guarantee the validity of the argument according to Tones' definition, but that also seems problematic.

"Validity has to do with the conclusion following from the premises, and inconsistency is not evidence that the conclusion follows from the premises." — Leontiskos

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.

One of the main takeaways from this discussion, for me, is that while some formal arguments may be valid, they are not necessarily valid in an informal setting.

To wit,

B
Therefore A→B
Formally valid.

Water was added to the lake.
Therefore,
If it is cloudy out, then water was added to the lake.
Informally not valid.

as well as -

A ^ B
Therefore, (A→B).
Formally valid.

Kangaroos are marsupials and Paris is the capital of France.
Therefore,
If kangaroos are marsupials, then Paris is the capital of France.
Informally not valid.

If I am referring to the right quotation, you said:

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. — TonesInDeepFreeze

What I responded with --a rule must have been "followed" not merely be "present" and the use of a rule may not result in a contradiction means that the use of a rule, or I guess you would call it an operator or connective, whatever you call it, must not result in a contradiction. A->not-A, when this rule is applied and followed, that is, when it is true that "A" and the rule "A->not-A" is actually applied, a contradiction results, specifically "A and not-A."

By "actually applied" I mean that the rule, or connective, does work in leading to the conclusion.

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.

Down the slippery slope of formalized illogicality.

You're slipping Tones.

Then note:

P -> Q |= ~P v Q
and
~P v Q |= ~P v Q — TonesInDeepFreeze

I think you meant:

P -> Q |= ~P v Q
and
~P v Q |= P -> Q

?

Not just conjunction, no, but having the same truth functionality as conjunction yes, just meta-logically different (if I am using that terminology correctly).

No, I read it, 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.

As for the instantiation of truth possibilities by the rules, what I mean is that the possibilities for what is true and what is false are arrayed across a truth table. The rules must account for all the ways that those truth possibilities can be instantiated. 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. So T, F, F, F, and F, T, F, F, and F, F, T, F, and F, F, F, T, must all be "achievable instantiations" based on the rules we bring to the variables. If A v B were the only rule we applied, then not all of the truth possibilities could be instantiated, does that answer?

By "the following of a rule" I mean a literal rule such as a connective is actually used to reach a conclusion. The argument A->not-A therefore not-A does not, in my opinion, make any use of the conditional such that any rule has been followed. 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, as such, it is disqualified from being a valid argument according to my definition.

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

I may have mispoken, but to me the full meaning of "If P then Q" captures the fact that "P does not imply Q" can still be true even though not-P v Q can still be true. But then I now think P->Q is a meaningless expression so saying it "means" the same think as not-P or Q is unsubstantiated.

Whether or not the two expressions are semantically equivalent in a meta-logical sense depends on how one is using them. — Leontiskos

Hmm interesting, I think my position is that the formal conditional is meaningless then, insofar as it is just symbol manipulation.

You could say that, but you would end up having to admit that "P does not imply Q" cannot be formalized in any way whatsoever, at least in propositional logic. — Leontiskos

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

(A v ~A) → (~B v ~A)

When (B and A) are both true, the expression seems to be false. On the other hand, the negation of that expression seems to imply that (A and B) must both be true. If the conditional is construed as only being true when A and B are true, then the negation of the initial expression maps onto A→B. Perhaps that could be written as, it is not the case that A does not imply B therefore A implies B. (Though if that were the case then A→B would be logically equivalent to A^B, although not meta-logically equivalent).

But then I don't mind saying "P does not Imply Q" can't be formalized.

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

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

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