"So there is no permissible metalogical argument as follows:

(1 ^ ~1) → 2
∴ 2"

Agree, I think; correct me if I have this wrong: by metalogical I take you to mean a logical "move" (such as MP) that is not identical to its truth function.

Apparently, it is not just arguments with contradictions that are problematic.

If it is settled that any premise in an informal argument is demonstrably false, it is unclear whether such an argument's conclusion can be true and yet the argument still be valid, where a valid argument is signified only as an argument that operates with the material conditional. If all valid arguments use the material conditional, arguments with some false premises could seem to still have a true conclusion.

But this seems wrong, at least to me. If any premises are false, a valid argument will result in a conclusion that is necessarily false, according to my non-standard understanding of validity in an informal context.

You may agree. But if you do, then any argument that is valid will turn out to be, in the relation of premises to conclusion, either [true true], or [false, false]. But that is the truth function of equivalence. Indeed, were you to exclude [F, F] as a degenerate case, your resulting truth functionality for a valid argument [T, T] would be truth functionally equivalent to "conjunction." You may argue that either of those truth functionalities is the case, and yet that an argument is still structurally but metalogically MP, although what you meant by calling an argument structurally and metalogically MP would be unclear to me.

In any case, I am not sure I agree that an argument is MP in any formulation, as putting an argument in terms of MP would seem to lead to the result that every argument had an "infinite regress" of premises. What I mean is:

P
P→Q
Therefore Q

Is really..

(P^(P→Q))→Q
P→Q
P
Therefore Q

Is really...

((P^(P→Q)→Q)^(P→Q)^P)→Q
(P^(P→Q))→Q
P→Q
P
Therefore Q

Ad infinitum.

Thanks for the links. So then I think Gensler would say the argument I have is similar to the first of his two circular proofs for modus ponens. The circularity is, interestingly, a result of the structure of the argument, not because of any specific premise.

My version of the argument is missing the inductive element that would cause the argument to be justified, if still circular. It's like a track record argument for perceptual abilities.

Perhaps, in addition to an inductive argument for modus ponens, an argument from coherence can be made. For instance it seems that if modus ponens failed, then MT or RAA would also fail.

1. If MP could be false, then RAA could be false.
2. But RAA is not false.
Therefore neither is MP.

(MT isnt a premise, however the argument is structurally MT). That is to say if MT is veridical, and so is RAA, then that would guarantee the truth of MP.

"So my question about the Cogito was, Which sort of "thought" is it?"

For Descartes it may only be the former, for Sartre it may be both. Though for Sartre I would say that the latter is "cogito" only in a way that is mediate; that is, present but only through phenomenal "glasses." Not to say that such glasses are not needed for the rendering of the phenomenal in terms of thought (it (the phenomenal realm) contains a kind of solution to the problem that it (the phenomenal) posed in the first place when consciousness encountered otherness (read: the other, opposition, negation of self) and the phenomenal became "a reality" to consciousness.

In other words, when thought discovers someone as-they-are through phenomenal encounter, the phenomenal collapses into noumenality. But this is the same as the noumenal encountering the noumenal.

"It's very plausible that the thought "2+2 = 4", understood as content or proposition, is timeless, or at least not to be identified with any particular time-based instantiation." :chin: Maybe the thought exists outside of time even though it is co-instantiated by a phenomenal event that is conditioned by time. Thought is noumenal? Thought is direct access linking being-as-it-is and being as-it-appears.

Similarly, the resolution of an appearance by thought is thought contending with the contradictions inherent in its own systematic approach where understanding is the return of thought to itself, self-sameness, being-as-it-is.

Graham --"logic is a normative subject: it is supposed to provide an account of correct reasoning."

Agree.

This is tangential (in that it is about logic but doesnt really relate to the original post), but what would you say about this argument? Is it viciously circular? --

if modus ponens is logical then any argument of the form [P, P->Q] implies Q.
modus ponens is logical.
therefore, "any argument..."

Here are some of your quotes that I think are consistent and apropos to my remarks.

One thing I can infer from thinking "I think" is that I think. — Moliere

because I think "I think" that it does not follow that "I am" in some kind of logically deductive fashion. It's just something that makes sense: in order for me to do I must be. — Moliere

Sartre does not rely upon ourselves as a thinking thing: If we remove ourselves as a substance which thinks (and is not extended) then there is nothing for the "I think" to refer to -- though "I am" remains true, it's not through the indubitability of the cogito that we come to this. — Moliere

Whereas Sartre is trying to explicate the metaphysical structures of a being which can lie to itself, or find itself in bad faith. — Moliere

The cogito may be thought of as pre-ontological insofar as it is not a study of being-as-such and so lacks ontological dimensionality. Cogito is undetached thinking; it is thinking that has not yet thought itself; it is thought qua thought. It is un-transcendent. This is the mode of being called being-in-itself.

Cogito is still temporal but not understood as temporal; it merely resides within the architecture of temporality; only the process of doubt, a process of negation of cogito (ego) discloses the cogito by standing apart from itself; in other words, from the hill of certainty that has been climbed by “doubt” the cogito sees itself in a separate moment, and from that vantage point has a grasp of itself in time. Similarly, the “doubting” which is again temporal and is the negative mirror of cogito is engrained in this process.

Meanwhile, what is the conclusion of methodological doubt? It is being itself; “therefore, I am.” The assertion is contentless and that being the case it is also pre-reflective; unmediated awareness. And yet, it is an ontological claim; and in that regard it is full of content though perhaps it is undescriptive (being, but what is being?). The “I am” claim is the voice given to being by being itself; self consciousness.

And, the being there posited is instrumental. Not only is being in a sense externalized from itself, but it is instrumentalized as a means for acquiring knowledge; it is foundational. So, being is no longer just being-in-itself, but has become being-for-itself. Both in the sense of self-consciousness and in the sense of it’s use for itself. That’s what I mean by saying that “I think therefore I am” is not the culmination of cogito qua cogito but of the transcendence of itself viz. the externalization of being through the process of “doubting.” Thinking that thinks itself.

Sartre’s critique of Descartes is critique-as-exposition. That is, Satre critiques Descartes not by contradicting what Descartes said, but by saying what Descartes left unsaid.

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.

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

"(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.

To say that in a briefer manner: I think -> I doubt -> I am.

Bad faith. Hidden fullness. Sense-certainty. Ego. The other. Contradiction. Doubt. Clarity. Certainty. Thinghood is thought, thought is thinghood; being-in-itself; "I am." Being-for-another. Implication. Enlightenment. Reason. Authenticity. Absolute knowledge. The unfolding of the Absolute. Return to the beginning. Faith.

(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.