Consider: the metaphysical understanding of reality, R, entails both that a) there is a self and b) there is no self. — javra

Firstly, which metaphysical understanding of reality are you referring to? Those different entailments rely on different interpretations of what is meant by"self' so they are not speaking about the same things.

Those different entailments rely on different interpretations of what is meant by"self' so they are not speaking about the same things. — Janus

No, as per my previously given example, they are (or at least can be) speaking about, or else referencing, the exact same thing via the term "self" - but from two different perspectives and, hence, in two different respects (both of these nevertheless occurring at the same time). Again, one perspective being the mundane physical world of maya/illusion/magic-trick and the other being that of the ultimate, or else the only genuine, reality to be had: that of literal nondualistic being. The validity, or lack of, of this Buddhist metaphysics being here inconsequential. As logic goes, here A entails both B and notB at the same time but in different respects/ways.

Tones even mistakes natural language for his own system — Leontiskos

That is a flat out lie.

I said a number of times that ordinary classical logic (which is not just "my" system, especially since overwhelmingly it is not exclusive to me, and especially since I study and have perused logics other than classical logic).

I said at least three or four times that it is not the case that in all respects formal logic represent everyday reasoning and discourse. I even said explicitly that material implication is not an everyday sense of "if then".

Leontiskos would have seen my posts saying those things, and, if I recall, he even replied to some of them. So either Leontiskos ignored what a wrote or read what I wrote but stated a flat out lie about me anyway.

/

and normatively interprets natural language in terms of his system — Leontiskos

That's a second flat out lie.

(1) The question in this thread did not specify whether it should be answered as to everyday senses of "if then" or the ordinary sense in the study of the most ordinary logic. And the question was posed in the 'Logic and Philosophy of Mathematics' section of a philsophy forum. So, I (and others) chose to answer as to material conditional. And I have stated explicitly that answers may be different in contexts of other formal logics and in everyday use.

(2) I have never stated a normative claim that classical logic trumps all other approaches.

Moreover:

(3) It is not "my" system, since it is not exclusive to me, and since it is not the only system I study or have perused.

Lenontiskos should not lie about my posts.

And to read Flannel Jesus' posts is to realize that he did not intend the OP in any special sense. I see no evidence that he was specifically speaking about material implication. — Leontiskos

It wouldn't be unreasonable to glean that he meant it not just in everyday senses. But, of course, it is open enough that everyday sense may be in play. And just to be clear, he did not indicate that material implication should be excluded.

an argument with two conditional premises should not be able to draw a simple or singular conclusion (because there is no simple claim among the premises). — Leontiskos

(1) There is no single everyday sense of "if then" or "implies". For that matter, I bet that in everyday discourse a lot of people would not even make sense of the original thread question.

(2) What senses people have in everyday conversation is an empirical question. It should not be presupposed that any particular sense is even the most common one without some basis. A reasonable conclusion about what people mean would have to take in natural languages world wide, not merely English.

(3) I think that language is not all that's in play, but rather also in play are the notions people have about things. We won't find in a dictionary what a person's response would be to "If snow is green then Winston Churchill was a Confederate general". Yes, it involves a person's meaning of "if then" but also that person's notions of what utterances are sensical, logical, or true.

/

(4) In what context is the criteria above supposed to be in? Everyday discourse? An alternative formal logic? Other?

(5) What is the definition of 'simple sentence'? A sentence with only one clause?

(6) Does the criteria pertain only to conditionals? And what a the sentence is written as an equivalent non-conditional?

(7) If not only conditionals, the what of single clause sentences such as "Alex is a dirty rotten scoundrel"? From that single clause sentence I should not infer "Aex is a rotten scoundrel"?

(7) What is the basis for the criteria? Without knowing more about it, I don't know a basis for disallowing the inference of a simple statement from compound statements.

(8) It might be the case throwing out the supposed bath water might be throwing out the baby that is the class of certain valued everyday, mathematical and scientific reasoning.

No, as per my previously given example, they are (or at least can be) speaking about, or else referencing, the exact same thing via the term "self" - but from two different perspectives and, hence, in two different respects (both of these nevertheless occurring at the same time). — javra

No, "two different perspectives and, hence, in two different respects" just is two different interpretations of the concept or meaning of 'self'.

Deleted by me.

This is a fairly common sort of argument. Something like: "if everything Tucker Carlson says about Joe Biden is true then it implies that Joe Biden is both demented/mentally incompetent and a criminal mastermind running a crime family (i.e., incompetent and competent, not-B and B) therefore he must be wrong somewhere."
— Count Timothy von Icarus

This actually runs head-on into the problem that I spelled out <here>. Your consequent is simply not a contradiction in the sense that ↪Moliere gave (i.e. the second clear sense of "contradiction" operating in the thread). — Leontiskos

It's close enough for purposes of an informal illustration. Obviously, it is implicit in this particular example that 'incompetent' and 'mastermind' are to be regarded as mutually exclusive.

No, "two different perspectives and, hence, in two different respects" just is two different interpretations of the concept or meaning of 'self'. — Janus

As in the concept/meaning of self as "that which is purple and square" vs. "that which is orange and circular" or any some such? And this in relation to "there both is and is not a self"?

I don’t see how your answer can rationally follow. Nor have you put in any effort in justifying your contention via any reasoning or examples. Nor have you evidenced how the examples and reasoning I have repeatedly provided to support my own contention cannot feasibly, rationally, work.

But I’ll leave you to it.

I have always had difficulty with argument by supposition. What does it mean to suppose A and then show that ~A follows? — Leontiskos

Without seeing a definition, I would take 'argument by supposition' to mean arguing from a premise or conditional:

Suppose A. Infer B. Infer If A then B.

Then, with the supposition A, and the inference If A then B, we infer B.

That is ubiquitous everyday reasoning. "Suppose Bob is an orchestra player. Every orchestra player can read music, so Bob in particular can read music. So if Bob is an orchestra player thenBob can read music." Then later, "We established yesterday that if Bob is an orchestra player then can read music, and today we confirmed that Bob is an orchestra player, so Bob can read music.''

Of course, everyday reasoning is not so stilted nor belabored. Such inferences occur virtually instantaneously, but such are the reasoning forms if we or the reasoner were to spell them explicit.

We suppose a proposition A and go through some more reasoning to get B, then we conclude that A implies B. Then we claim A and deploy that A implies B to claim B.

Also, we have two two forms of prove with negation:

(1) Suppose A. Infer a contradiction. Infer ~A

(2) Suppose ~P. Infer a contradiction. Infer A.

Those also are common in everyday reasoning.

What does it mean to do that? I don't know what is meant by "what does it mean to do it" other than the obvious:

There are no circumstances in which a contradiction is true. And if a statement P implies another statement Q, then any circumstances in which the P is true are circumstances in which Q is true. But if Q is a contradiction, then there are no circumstances in which P is true. So, since Q is a contradiction, there are no circumstances in which P is true.

Tones gave an argument for ~A in which he attempted to prove it directly — Leontiskos

I didn't merely attempt, I proved. And by reductio ad absurdum.

What does it mean to suppose A and then show that ~A follows? This gets into the nature of supposition, how it relates to assertion, and the LEM. — Leontiskos

LEM is not needed for my proof.

It also gets into the difference between a reductio and a proof proper. — Leontiskos

Anyone is welcome to another context, but if another context is not stated, and since your remarks were related to my proof, I'll suppose that material implication is the context.

First, what is a definition of 'proof proper' in any context?

What is not "proper" about reductio ad absurdum?

How else would one prove a negation if not by reductio ad absurdum, or modus tollens (which is equivalent with reductio ad absurdum) or some other equivalent in either a natural deduction or logical axiom system?

And those are formalizations of everyday reasoning forms: Notice that vacuousness (which probably the main objection to material implication) is not involved

There are two forms:

(1)

Suppose P, infer a contradiction, infer ~P.

Or, suppose that P is true, infer a falsehood, infer that P is false.

That is intuitionistically (thus also classically) acceptable and common everyday reasoning.

"The plumber said he fixed the pipe. But, suppose the pipe was fixed. Then, since there are no other leaks around, we would't see leaking water. So it shouldn't be leaking, but it is leaking. So he didn't fix the pipe."

In everyday life, it may compressed:

"The plumber said he fixed the pipe. But if he fixed the pipe, then, since there no other leaks around, we wouldn't see leaking water. So he didn't fix the pipe."


(2)

Suppose ~P, infer a contradiction, infer P.

Of, suppose that ~P is true, infer a falsehood, infer that P is true. l

That is classical acceptable and common everyday reasoning, but not intuitionistically acceptable.

"Ralph said he didn't the candy bar. But, suppose he didn't eat it, then, since no one else was home, it would be where I left it on the table. But it's not there. So he did eat the peach."

In everyday life, it may compressed:

"Ralph said he didn't the candy bar. But, suppose he didn't eat it, then, since no one else was home, it would be where I left it on the table. But it's not there. So he did eat the peach."

The point is one I had already made in a post that Tones was responding to, "You think the two propositions logically imply ~A? It seems rather that what they imply is that A cannot be asserted" — Leontiskos

What does "cannot be asserted" mean?

Anything utterable can be asserted. So do you mean "cannot truthfully be asserted"?

If both the propositions can be truthfully asserted then "A is not the case can be truthfully asserted".

((a→b)∧(a→¬b))↔¬a is valid — Lionino

That is true if "both props" is understood as (A → B) ^ (A → ¬B) and "imply ¬A" as the proposition being True means A is False — Lionino

((a→b)∧(a→¬b))↔¬a

The ↔ operator means that everytime the left side is 1 (True), the right side is also 1, and same for 0 (False). So (a→b)∧(a→¬b) does not mean that A is False, unless we say (a→b)∧(a→¬b) is True. If (a→b)∧(a→¬b) is False, ¬A is False, so A can be True or not¹. So (a→b)∧(a→¬b) may be taken as a proposition p which can take the values of 0 or 1 as well.

More confusions stemming from whether 'A' means 'A is true' or a variable which may take the values True or False. It would be better if there were a way to easily distinguish the two. Perhaps a doctor thesis for someone out there.

1 – We know (a→b)∧(a→¬b) is the same as a→(b∧¬b), and (b∧¬b) is a contradiction. So (a→b)∧(a→¬b) just means A implies a contradiction. If (a→b)∧(a→¬b) is True, A cannot be True, it has to be False. But let's say (a→b)∧(a→¬b) is False, does that mean A is true? That is what the logical tables would say:

The original question was, "Do (A implies B) and (A implies notB) contradict each other?"

On natural language they contradict each other. — Leontiskos

Woa, woa, easy on the draw there, pardner.

"On natural language they contradict each other" is pretty categorical.

You can't speak for all speakers of all natural languages. People have all kinds of notions of 'imply' and 'contradiction', ranging from thinking of implication as contexts as at least partially related to material implication and even in some contexts wholly related, through everyday senses of relevance, necessity, common sense in some unarticulated and not explicitly conceived way, to not even being able to make sense of the question. And what is 'everyday'? May it not include professions that concern circuits, models of weather and that kind of thing?

You don't have public surveys that would show how many people would regard those as contradictory, or nonsense, or to account for some who would be evidence against your universal claim.

Moreover, it's not just a question of language but of the kind of reasoning people recognize as correct or incorrect in everyday situations. The English language, for example, doesn't imply what people consider good reasoning. You can't look in a dictionary and grammar book to find out whether the propositions are to be regarded as contradictory, not even a description of colloquial use.

On the understanding of contradiction that I gave in the first post, they do not contradict each other, and their conjunction is not a contradiction. — Leontiskos

I answered that. I don't recall whether you addressed my answer.

1 – We know (a→b)∧(a→¬b) is the same as a→(b∧¬b), and (b∧¬b) is a contradiction. So (a→b)∧(a→¬b) just means A implies a contradiction. If (a→b)∧(a→¬b) is True, A cannot be True, it has to be False. But let's say (a→b)∧(a→¬b) is False, does that mean A is true? That is what the logical tables would say: — Lionino

But (a→b)∧(a→¬b) being False simply means that A does not imply a contradiction, it should not mean A is True automatically.

- Thanks Lionino. Good reminders and clarifying points.

Edit: I underestimated your post. Like your first post, this is far above and beyond anything else that is occurring in this thread.

I would want to say something along the lines of this, "A proposition containing (p∧¬p) is not well formed." — Leontiskos

I'd like to see what formation rules you come up with.

Similar to what I said earlier, "When we talk about contradiction there is a cleavage, insofar as it cannot strictly speaking be captured by logic. It is a violation of logic — Leontiskos

I asked what you mean by 'cleavage' and 'capture by logic'. I don't recall that you replied.

Whatever 'capture' means, the meaning of [P & ~P' is specified.

What does "violation of logic" mean that entails that 'P & ~P' is well formed. It is well understood that 'P & ~P' is a violation of the law of non-contradiction. But it is not a violation of syntax to write 'P & ~P' nor "P is true and P is false". Indeed, a statement of the law of non-contradiction would itself involve stating a contradiction in order to deny it.

My idea would be that (p∧¬p) is outside the domain of the logic at hand/quote]

(1) Which logic at hand?

(2) Even in everyday contexts, people say things like "You said Jack is to be trusted and you said he's not to be trusted. Which is it?" "You said to turn right at Elm and you said not to turn right at Elm. Would you please give me directions that don't contradict?'

and to try to use the logic at hand to manipulate it results in paradoxes. — Leontiskos

Did you mean "try not to"?

Anyway, I don't see the paradoxes arising from undo use of negation. Would you give an example?

I'm sure others have said this better than I — Leontiskos

I would be interested to know who.

There are systems of negationless logic, but that is not what you have mind since you don't want to toss even negation. There might be systems that have negation but do not allow writing 'P & ~P', but I don't know of any.

And is your idea only for everyday reasoning, or do you want wipe out being able to state a contradiction in formal ogic, informal academic logic and rhetoric, mathematics, science, philosophy and other acdademic fields?

Moreover, even though formal logic and everyday reasoning often converge, wouldn't you want to allow formal reasoning to be expressed in everyday terms?

If I write a formal argument deriving a contraction from P, shouldn't I be taken as making sense when I instantiate it with natural language?

And what about an inconsistent statement that is not in the form of 'P & ~P'? I certain sentential logic and monadic predicate logic, it is checkable whether a formula is inconsistent, but not in dyadic or greater predicate logic. But, for excellent reasons, formation rules for formal language are checkable.

principle of explosion is in fact relevant here insofar as it too relies on the incorporation of a contradiction into the interior logical flow of arguments. — Leontiskos

Does 'interior logic flow of arguments' just mean 'proof steps'?

It is formal logic pretending to say something. — Leontiskos

It is not "pretending" anything. It has a precise meaning.

"The premise that Tom reneged on his library fines leads to a contradiction, therefore, Tom did not renege on his library fines".

I'd rather have a lawyer who understands contradiction than one who follows only Lenontiskos logic.

"The car is green" and "The car is red" is not a contradiction. But if we add the premise: "If the car is red then the car is not green," then the three statements together are inconsistent. That's for classical logic and for symbolic rendering for classical logic too. — TonesInDeepFreeze

Yep. Worth noting that parsing this correctly shows that the original was incomplete - implied nothing.

More generally, parsing natural languages in formal languages, while not definitive, does occasionally provide such clarification. That's kinda why we do it.

Also worth noting that (A → B) ^ (A → ¬B), while not a contradiction, does imply one, given A:

(A → B) ∧ (A → ¬B)→(A→(B∧¬B))

So in answer to the OP

Do (A implies B) and (A implies notB) contradict each other? — flannel jesus

Taking "implies" as material implication, they are not contradictory but show that A implies a contradiction.

I'd like to see what formation rules you come up with. — TonesInDeepFreeze

I had the same thought when I read that. It's wellformed. It is also invalid: A∧¬A

This thread is bringing out some rather odd attitudes towards the relation between logic and natural languages.

If (a→b)∧(a→¬b) is False, ¬A is False, so A can be True or not¹. — Lionino

But (a→b)∧(a→¬b) being False simply means that A does not imply a contradiction, it should not mean A is True automatically. — Lionino

Isn't this a fairly big problem given that (¬¬A↔A)? I take it that this is the same thing I have pointed out coming out in a different way? Namely the quasi-equivocation on falsity?

...Or is it simpler: ¬(a→(b∧¬b)) → a

Back to material implication:

As noted in my original post, your interpretation will involve Sue in the implausible claims that attend the material logic of ~(A → B), such as the claim that A is true and B is false. Sue is obviously not claiming that (e.g. that lizards are purple). The negation (and contradictory) of Bob's assertion is not ~(A → B), it is, "Supposing A, B would not follow." — Leontiskos

Given material implication there is no way to deny a conditional without affirming the antecedent, just as there is no way to deny the antecedent without affirming the conditional.

quotes an article:

"Dogs have four legs, and Lassie has four legs, therefore Lassie is a dog" is not a valid argument. The conclusion ("Lassie is a dog") may be true, but it has not been proved by this argument. It does not "follow" from the premises.

Now in Aristotelian logic, a true conclusion logically follows from, or is proved by, or is "implied" by, or is validly inferred from, only some premises and not others. The above argument about Lassie is not a valid argument according to Aristotelian logic. Its premises do not prove its conclusion. And common sense, or our innate logical sense, agrees. However, modern symbolic logic disagrees. One of its principles is that "if a statement is true, then that statement is implied by any statement whatever."

Symbolic logic definitely does not hold that that Lassie argument is valid.

That claim in the article is either sneaky sophistry or egregious ignorance. It is a ludicrous claim. It goes dramatically against what obtains in symbolic logic.

Let:

'Fx' stand for 'x has 4 legs'

's' stand for 'Lassie'

'Dx' stand for 'x is a dog'

The argument is:

Ax(Dx -> Fx)
Fs
Therefore Dx

[EDIT: There's a typo there. It should be:

Ax(Dx -> Fx)
Fs
Therefore Ds]

Symbolic not only does not say that that is valid, and not only does symbolic logic say it is invalid, but symbolic logic proves it is invalid.

Here is where the authors try to pull a fast one:

Correct: A valid formula is implied by any set of formulas.

Correct: If P is true, then, for any formula Q, we have that Q -> P is true.

Incorrect: If P is true, then Q -> P is valid.

Look what the authors did:

By saying "'Lassie is a dog; is true", they are adopting Dx [edit typo: should be Ds] as a premise. So, of course,

Ax(Dx -> Fx)
Fs
Dx
Therefore Dx

is valid.

[EDIT: There's a typo there. It should be:

Ax(Dx -> Fx)
Fs
Ds
Therefore Ds]

Or I invite the authors to show any symbolic logic system for ordinary predicate logic that provides a derivation of:

Ax(Dx -> Fx)
Fs
Therefore Dx

[EDIT: There's a typo there. It should be:

Ax(Dx -> Fx)
Fs
Therefore Ds]

Moreover, we prove that classical logic provides that its proof method ensures that the the premises indeed entail the conclusion. That is, if the conclusion is not entailed by the premises, then the conclusion is not proved by the premises. And that goes for both true and false conclusions. If the truth that Lassie is a dog is not entailed by the premises, then 'Lassie is a dog' is not provable from the premises.

That's a disgustingly specious and disinformational start of an article. And unfortunate that that speciousness and disinformation is propagated by another poster quoting it here.

But this is good:

Logician: So, class, you see, if you begin with a false premise, anything follows.
Student: I just can't understand that.
Logician: Are you sure you don't understand that?
Student: If I understand that, I'm a monkey's uncle.
Logician: My point exactly. (Snickers.)
Student: What's so funny?
Logician: You just can't understand that.[/quoye]

Quite so.

Logicians have an answer to the above charge, and the answer is perfectly tight and logically consistent. That is part of the problem! Consistency is not enough.[/quote]

Indeed, that is why we prove both consistency and soundness.

So a proper answer is that the logic is consistent, sound and extaordinarily useful for many contexts. And it is useful even in everyday context where vacuousness doesn't even come up.

My point is that it is a vacuous instance of validity — Leontiskos

((a→b)∧(a→¬b))↔¬a is valid
— Lionino

My point is that it is a vacuous instance of validity — Leontiskos

The proof does not use vacuousness.

As I claimed above, there is no actual use case for such a proposition — Leontiskos

(1) I'd like to see you forrmulate a grammar, formal or informal, that restricts to only locutions that "have use".

(2) Even though the form mentioned is not found in everyday discourse, it is equivalent with very basic everyday reasoning

and I want to say that propositions which contain (b∧¬b) are not well formed. — Leontiskos

Not well formed in everyday language? Not well formed in formal languages?

And I would like to see your formation rules for either.

They lead to an exaggerated form of the problems that ↪Count Timothy von Icarus has referenced. We can argue about material implication, but it has its uses. I don't think propositions which contain contradictions have their uses. — Leontiskos

Being able to write a contradiction is useful, as I explained.

This is perhaps a difference over what logic is. Is it the art of reasoning and an aid to thought, or just the manipulation of symbols? — Leontiskos

And the study of formal logic is not just a study of manipulating symbols.

ones I have in mind are good at manipulating symbols, but they have no way of knowing when their logic machine is working and when it is not. They take it on faith that it is always working and they outsource their thinking to it without remainder. — Leontiskos

I take it that the machine Leontiskos was using when he wrote the paragraph above is working. That's a machine whose invention and development is steeped in formal logic, steeped in the use of sentential logic, steeped in syntax that has material implication, disjunction, conjunction and negation, steeped in truth functionality, steeped in 2-value Boolean algebra.

the only person on these forums who has shown a real interest in what I would call 'meta-logic' is — Leontiskos

Whatever you call 'meta-logic, the subject of meta-logic is discussed plenty on this forum, by me and others.

A parallel equivocation occurs here on 'false' and 'absurd' or 'contradictory'. Usually when we say 'false' we mean, "It could be true but it's not." In this case it could never be true. It is the opposite of a tautology—an absurdity or a contradiction. — Leontiskos

The method of models formalizes the idea that a statement "is true", "could be true, but it's not" or "is false" or "could be false, but it's not".

People use terminology differently, but we have quite unequivocal terminology:

a sentence S is false in model M if and only if [fill in the the rigorous definition].

a sentence S is a contradiction if and only if S is of the form 'P & ~P'

a sentence S is inconsistent if and only if S proves a contradiction.

a set of sentences G is inconsistent if and only if G implies a contradiction.

"absurd" doesn't usually get a formal definition, but its use is usually along the lines such that:

"S is absurd" is equivalent with "S implies a contradiction".

Usually when we say 'false' we mean, "It could be true but it's not." — Leontiskos

That is addressed also:

S is logically true if and only if S is true in all models.

S is logically false if and only if S is false in all models.

S is contingently true if and only if S is true in some models and false in other models.

Every time we make an inference on the basis of a contradiction a metabasis eis allo genos occurs (i.e. the sphere of discourse shifts in such a way that the demonstrative validity of the inference is precluded). Usually inferences made on the basis of a contradiction are not made on the basis of a contradiction “contained within the interior logical flow” of an argument. Or in other words, the metabasis is usually acknowledged to be a metabasis. As an example, when we posit some claim and then show that a contradiction would follow, we treat that contradiction as an outer bound on the logical system. — Leontiskos

I don't know what any of that means or a bunch of other stuff in a similar vein.

We do not incorporate it into the inferential structure and continue arguing. — Leontiskos

could also put this a different way and say that while the propositions ((A→(B∧¬B)) and (B∧¬B) have truth tables, they have no meaning. They are not logically coherent in a way that goes beyond mere symbol manipulation. — Leontiskos

I understand them just fine, and not as mere symbols.

((A→(B∧¬B))
∴ ¬A

Viz.:

Any consequent which is false proves the antecedent
(B∧¬B) is a consequent which is false
∴ (B∧¬B) proves the antecedent — Leontiskos

You have it quite wrong.

No, the conditional with the negation of the consequent prove the negation of the antecedent.

A ... antecedent

B & ~B ... consequent

~(B & ~B) ... negation of the consequent

~A ... negation of the antecedent

In this case the middle term is not univocal. It is analogical (i.e. it posses analogical equivocity). Therefore a metabasis is occurring. — Leontiskos

If that and bunch more in your post is not egregious doubletalk then I stand having it explained that it's not.

There's so much more in your post. Time is not enough.

A = There are vampires.
B = Vampires are dead.
Not-B = Vampires are living.

As you can clearly judge, this truth table works with Ts straight across the top, since vampires are members of the "living dead." Fools who think logic forces them to affirm ~A — Count Timothy von Icarus

That's just a matter of defining the words. If 'dead' and 'living' are defined so that they are not mutually exclusive, then of course we don't make the inference. It's silly to claim that sentential logic is impugned with the example.

I think Kreeft is involved in word games here — Leontiskos

Worse, his argument about Lassie and symbolic logic is specious, dishonest or stupid, and ludicrous.