Comments

  • Do (A implies B) and (A implies notB) contradict each other?
    In fact it says it isn't valid tout court up above. — Count Timothy von Icarus

    No, not tout court. There's an 'however'. They correctly say it is invalid but they also incorrectly say that, however, symbolic logic disagrees.

    It's text in broad daylight.
  • Do (A implies B) and (A implies notB) contradict each other?
    I somehow find it more plausible that they were trying to highlight the incongruity between the fact "Lassie has four legs" does not imply Lassie is a dog in symbolic logic in the argument:

    All dogs have four legs
    Lassie has four legs
    Therefore Lassie is a dog

    And the fact that "Lassie has four legs" does imply Lassie is a dog if "Lassie is a dog" is true.     — Count Timothy von Icarus

    Whatever you think was meant to be highlighted, or plausible, or your interpretation, the plain fact is that the paper says that symbolic logic regards the argument as valid.

    That's enough right there. It's incontrovertible.

    But, going on:

    The reason they make that claim is to make a strawman against symbolic logic. It's a strawman because symbolic logic does not say the argument is valid. Then they incorrectly argue that the reason symbolic logic says the argument is valid is because "Lassie is a dog" is true and symbolic logic says that any true sentence is proven by any sentences.

    the straightforward purpose given the context — Count Timothy von Icarus

    Whatever you think the purpose is, it is utterly straightforward that the paper claims that symbolic regards the argument as valid. They incorrectly base that claim on the claim that symbolic logic regards the argument as valid because "Lassie is a dog" is true and symbolic logic says that any true sentence is proven by any sentences.

    the text is not particularly hostile towards symbolic logic aside from arguing that it isn't particularly helpful for most people's use cases. — Count Timothy von Icarus

    The article uses the Lassie example to argue that symbolic logic departs from common sense logic. But that is a specious argument, since symbolic logic is right with common sense regarding the Lassie argument.

    This, rather than assuming they are trying to imply an falsehood to cast shade on symbolic logic in an extremely roundabout way using an example obfuscates their point (if that was the point they were making)—doing all this to try to suggest something that is easily verifiable as false for ... what purpose? — Count Timothy von Icarus

    (1) You are obfuscating. Your argument is that they couldn't have meant what they exactly wrote because of something else. It is in broad daylight that they claimed that symbolic logic disagrees that the argument is invalid. And what they say right after is premised on that. Again: They make the claim, then argue that the reason symbolic logic regards the argument as valid is that symbolic logic says "true proven by anything" (by the way, that is not what symbolic logic says, so another false claim in the paper).

    (2) The example and argument is not roundabout. And it does not obfuscate their point. It makes their point blazingly clear as they themselves state it right after the example. They try to make symbolic look ridiculous for saying the argument is valid, then incorrectly say what is at root in symbolic logic that allows symbolic logic to say the argument is valid.

    (3) The purpose is to present an argument about symbolic logic. I don't why they resorted to egregious sophistry to do that. Why does anyone resort to sophistry? Possibilities include: (a) They think they can get away with it, (b) They got caught up in themselves thinking they had a clever argument, (c) They don't understand symbolic logic ...

    IDK, maybe I am letting the principle of charity run amok. — Count Timothy von Icarus

    What is that supposed to mean?

    Anyway, you continue to evade exactly what they wrote.
  • Do (A implies B) and (A implies notB) contradict each other?
    In fact it says it isn't valid tout court up above. — Count Timothy von Icarus

    It correctly says it is invalid, but incorrectly says that symbolic logic "disagrees". It's right there in the paper. It is amazing that you ignore that plain fact.

    If that was the point, it could have been stated much clearer — Count Timothy von Icarus

    It was stated in perfect clarity that symbolic logic "disagrees" that the premises don't prove the conclusion.

    If they wanted to make the point you ascribe to them why wouldn't they use an example like:

    All monkeys have tails.
    Garfield the cat has a tail.
    Therefore Garfield is a monkey.     — Count Timothy von Icarus

    Because their point in that paragraph was that "Lassie is a dog" is true but not provable from the premises but that symbolic logic disagrees that is not provable from the premises.

    Read the paragraph slowly, line by line, and you will see:

    "Its premises do not prove its conclusion."

    "modern symbolic logic disagrees."
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪Count Timothy von Icarus


    Again, look at the exact words in the paper:

    "Its premises do not prove its conclusion."

    'it' refers to the Lassie argument.

    "modern symbolic logic disagrees."

    That is, modern symbolic disagrees that the Lassie argument's premises do not prove the conclusion.

    That is plain as day. All you need to do is read the exact words. It is amazing that you won't recognize it.

    Discussing other aspects of the paper does not change that the paper claims that symbolic logic regards the Lassie argument is valid. Indeed, the paper goes on to give its incorrect explanation of why symbolic logic regards the Lassie argument to be valid.
  • Do (A implies B) and (A implies notB) contradict each other?
    https://thephilosophyforum.com/discussion/comment/916812

    Since there were so many typos in my reply, here it is corrected:

    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 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 Ds as a premise. So, of course,

    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 Ds

    Moreover, we prove that classical logic provides that its proof method ensures that 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.
  • Do (A implies B) and (A implies notB) contradict each other?
    There's a paper that says the premises prove the conclusion of this argument? — flannel jesus

    No, the paper says the argument is invalid, but that symbolic logic says it's valid.

    The paper is a polemic against symbolic logic, and it argues (egregiously incorrectly) that symbolic logic regards the argument as valid because symbolic logic says that any true statement (such as "Lassie is a dog") is implied by any statements. In my first post about it, I explained exactly where the sophistry occurs. (There are some typos in my post, but I made edit notes to correct them.)

    It's quoted in a post earlier in this thread.
  • Infinity
    Is Hilbertian formalism incompatible with platonism? I'd like to see an argument that it is.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪flannel jesus


    That is what the paper says. The paper is incorrect.

    Not just incorrect, but incorrect due to egregious sophistry, ignorance or blatant lack of reasoning skills.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪Count Timothy von Icarus


    It's not ambiguous. It's as plain as day, with a plain reading:

    "Its premises do not prove its conclusion."

    'it' refers to the Lassie argument.

    "modern symbolic logic disagrees."

    That is, modern symbolic disagrees that the Lassie argument's premises do not prove the conclusion.
  • Do (A implies B) and (A implies notB) contradict each other?
    "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 [correct -TIDF] according to Aristotelian logic.Its premises do not prove its conclusion. And common sense, or our innate logical sense, agrees. [correct -TIDF] However, modern symbolic logic disagrees. [incorrect - TIDF] One of its principles is that "if a statement is true, then that statement is implied by any statement whatever."
  • Do (A implies B) and (A implies notB) contradict each other?
    No, your reading of it is incorrect because you seem to think it is saying:

    All dogs have four legs
    Lassie has four legs
    Lassie is a dog

    ...is valid in symbolic logic. It doesn't say that. It says the exact opposite, that this is not valid.     — Count Timothy von Icarus

    Read it again. The paper says it is invalid, but that symbolic logic "disagrees".

    Again, the paper is correct that it is invalid, but the paper is incorrect that symbolic logic disagrees.

    If in a given interpretation, P is true, then per that interpretation, for any statement Q, Q->P is true. — TonesInDeepFreeze

    That's perfectly clear.
  • Mathematical truth is not orderly but highly chaotic
    model theory will map the symbols. However, the actual purpose of doing that is to achieve what is described in the correspondence theory of truth. — Tarskian

    That seems okay as a broad synopsis.

    simplify — Tarskian

    Simplifications are okay if they don't mislead by omitting crucial conditions and distinctions.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪Lionino


    I gave a direct translation, symbol by symbol to word by word.

    The formula has a subformula that is a contradiction, but the formula doesn't itself say that it is a contradiction. The word 'contradiction' is not in the formula, and the formula doesn't quantify over contradictions. Again:

    The formula does not say:

    For all contradictions, if A does not imply the contradiction then A is true.

    "Winston Churchill was French" does not imply a contradiction. But that does not imply that "Winston Churchill was French" is true.

    More precisely, "Winston Churchill was French" does not imply a contradiction. But that does not imply that "Winston Churchill was French" is true in all interpretations, but it does imply that "Winston Churchill was French" is true in at least one interpretation.

    With symbols:

    If P does not imply a contradiction, then it is not implied that P is true in all interpretations, but it is implied that P is true in at least on interpretation.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪Lionino


    I then edited it with better demarcations. Said another way:

    "It is not the case that if A then both B and not-B" implies "A".
  • Do (A implies B) and (A implies notB) contradict each other?
    ~(A→(B &~B)) implies A

    Translation:

    It is not the case that if A then B & ~B
    implies
    A.

    We can't say:

    For all contradictions, if A does not imply the contradiction then A is true.

    (1) True in what interpretations?

    (2) "Winston Churchill was French" does not imply a contradiction. But that does not imply that "Winston Churchill was French" is true.

    More precisely, "Winston Churchill was French" does not imply a contradiction. But that does not imply that "Winston Churchill was French" is true in all interpretations, but it does imply that "Winston Churchill was French" is true in at least one interpretation.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪Count Timothy von Icarus


    The example is incorrect, no matter what its purpose is.

    "if a statement is true, then that statement is implied by any statement whatever." — Count Timothy von Icarus

    I would not say that.

    P may be true in some interpretations and not in others. If in a given interpretation, P is true, then per that interpretation, for any statement Q, Q->P is true.
  • Do (A implies B) and (A implies notB) contradict each other?
    (1) A -> (B & ~B) implies ~A

    (2) ~(A→(B &~B)) implies A

    "when it is not true that A implies a contradiction, we know A is true?"

    No that is not a correct translation of (2). The translation universally quantifies over contradictions, which (2) does not.
  • Do (A implies B) and (A implies notB) contradict each other?
    Let a proposition P be A→(B∧¬B)
    Whenever P is 1, A is 0.
    In natural language, we might say: when it is true that A implies a contradiction, we know A is false.

    Now a proposition Q: ¬(A→(B∧¬B))
    Whenever Q is 1, A is 1.
    Do you think it is correct to translate this as: when it is not true that A implies a contradiction, we know A is true?     — Lionino

    I'm recomposing.
  • Do (A implies B) and (A implies notB) contradict each other?
    Deleted by me. Recomposing below.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪Count Timothy von Icarus


    No, that was also a typo. I corrected it all in the post as it stands now.
  • Do (A implies B) and (A implies notB) contradict each other?
    I made a typo. It should be this:

    Ax(Dx -> Fx)
    Fs
    Therefore, Ds

    is not valid. And the paper is correct in saying it is not valid. But the paper is incorrect in saying that symbolic logic says it is valid.

    In other words:

    All dogs have four legs.
    Lassie has four legs.
    Therefore, Lassie is a dog.

    is not valid. And the paper is correct is saying it is not valid. But the paper is incorrect in saying that symbolic logic says it is valid.

    The paper says that symbolic logic says it is valid on the basis that "Lassie is a dog" is true.

    That is not correct, since if we are using "Lassie is a dog", then it has to be a premise in the argument, so the argument that the paper puts on symbolic logic would actually be:

    Ax(Dx -> Fx)
    Fs
    Ds
    Therefore Ds

    which is valid.

    /

    If you think the paper is correct in saying that symbolic logic regards this as valid

    Ax(Dx -> Fx)
    Fs
    Therefore, Ds

    In other words:

    All dogs have four legs.
    Lassie has four legs
    Therefore, Lassie is a dog.

    then you may prove that symbolic logic says it's valid. That is, show a derivation in symbolic logic that starts with the premises

    Ax(Dx -> Fx)
    Fs

    and ends with the conclusion

    Ds

    Hint: You won't be able to do it. Morever, using the method of counter-example, symbolic logic proves it is not valid.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪Count Timothy von Icarus


    The intent in that part of the paper was to make symbolic logic look like it says that this argument is valid:

    Dogs have four legs, and Lassie has four legs, therefore Lassie is a dog.

    But symbolic logic does not say that argument is valid.
  • Do (A implies B) and (A implies notB) contradict each other?
    Hopefully I'll have time at some point to address the logic question. I want to look at some materials and think through my reply. But first I need to clean up more of your misrepresentations and mischaracterizations:

    I know you want a strict definition, but the wonderful irony is that someone like yourself who requires the sort of precision reminiscent of truth-functional logic can't understand analogical equivocity or the subtle problems that attend your argument for ¬A. As we have seen in the thread, those who require such "precision" tend to have a distaste for natural language itself. — Leontiskos

    A rigorous definition is best, for obvious reasons. And giving one is intellectual considerateness. But, again you put words in my mouth, thus a strawman. I said a certain form would be helpful. I did not say that I "require" it or any specific degree of rigor let alone "precision".

    And I don't have "distaste" for language. You're just making that up. I love language and revel in its expressiveness, freedom, complexities, nuances and even its vagaries. You don't know jack when you presume to mischaracterize my sensibilities.

    Your method is to characterize me in a certain way, to wedge in a prejudice that my arguments are tainted by a point of view, even though I don't have that point of view. You are arguing by characterization of an interlocutor, actually characterization.

    You lied that I don't distinguish between formal language and natural language. I had said at least a few times that they don't agree in certain respects. Depending on the conversation, they don't agree on "if then".

    You would do well to knock it off with trying to impugn me with lies about what I've said a sloppy mischaracterizations of my point of view and sensibilities.
  • Do (A implies B) and (A implies notB) contradict each other?
    You seem to be missing the point of the example — Count Timothy von Icarus

    I am exactly on point. The paper says that symbolic logic permits a certain inference. But symbolic logic does not permit that inference. The paper's argument is specious exactly as I showed.

    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.

    ...when it comes to vamps it's deadly serious. :death: :death: :death:     — Count Timothy von Icarus

    Death is serious. The example is silly, for the reason you quoted.
  • Mathematical truth is not orderly but highly chaotic
    It just says that true(n) is not a legitimate predicate. — Tarskian

    It says that for certain formal interpreted languages, there is no predicate in the language that defines the set of sentences true in the interpretation.

    soundness theorem: provable ==> true — Tarskian

    That's not the soundness theorem.

    The soundness theorem stated in two equivalent ways:

    If a set of sentences G proves a sentence S, then every model of G is a model of S.

    If a set of sentences G proves a sentence S, then for all models M, if every member of G is true in M then S is true in M.

    So, the correspondentist mapping of truth occurs between theory and "model" (or "universe"). — Tarskian

    No, the mapping is from the symbols of the language:

    each individual constant map to a member of the universe

    each n-place predicate symbol maps to an n-ary relation on the universe

    each n-place operation symbol maps to an n-place function on the unviverse

    And ""model" (or "universe")" is wrong since a model is not just a universe. Rather, for every model there is a universe for that model.
  • Do (A implies B) and (A implies notB) contradict each other?
    Compare:

    ((A→(B∧¬B))
    ∴ ¬A

    With:

    ((A→(B∧¬B))
    ¬(B∧¬B)
    ∴ ¬A

    With:

    ((A→(B∧¬B))
    ¬(B∧¬B)
    ¬(B∧¬B) = "True"
    ∴ A does not follow

    This demonstrates the analogical equivocity     — Leontiskos

    What is the definition 'analogical equivocity'? Most helpfully it would be something like:

    An thing has analogical equivoicity if and only if [fill in definiens here, and such that the definiens doesn't use terminology that hasn't itself been defined or common anyway]
  • Do (A implies B) and (A implies notB) contradict each other?
    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)?     — Leontiskos

    A formal problem, philosophical problem, or problem in not adhering to everyday discourse.

    It's not a formal problem. I don't know what philosophical problem you suggest. And I've discussed symbolic logic and everyday discourse.
  • Do (A implies B) and (A implies notB) contradict each other?
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪Count Timothy von Icarus
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    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'?
  • Do (A implies B) and (A implies notB) contradict each other?
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    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.
  • Do (A implies B) and (A implies notB) contradict each other?
    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".
  • Do (A implies B) and (A implies notB) contradict each other?
    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."
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