Comments

  • Do (A implies B) and (A implies notB) contradict each other?
    1. A→(B∧¬B) assumption
    2. A assumption
    3. B∧¬B 1,2, conditional proof
    4. ~A 2, 3 reductio
    — Banno

    As I noted earlier in response to Tones' reductio, a reductio is an indirect proof which is not valid in the same way that direct proofs are.     — Leontiskos

    This caveat will cover my posts in general: People may have different contexts with different meanings of terms, different fundamental conceptions, different philosophical positions, and especially different formal systems that may diverge from classical logic, contract it, or extend it. If a context is not made explicit, then it is reasonable to respond in the context of classical logic, which I will do. And doing so is especially apt when responding to criticisms or examination of classical logic.

    Reductio ad absurdum is proven valid as it is proven that inferences using it are truth preserving, just as with the other rules of classical sentential logic. An argument is valid if and only if there are no models in which the premises are all true and the conclusion is false. Reductio ad absurdum is proven to provide only valid arguments.

    You can see this by examining your conclusion. In your conclusion you rejected assumption (2) instead of assumption (1). Why did you do that? In fact it was mere whim on your part, and that is the weakness of a reductio. — Leontiskos

    The argument may be better written:

    (1) A -> (B & ~B) ... premise

    (2) A ... assumption toward a contradiction

    (3) B & ~B ... 1, 2 modus ponens

    (4) ~ A ... 2, 3 RAA with 2 discharged

    There we see that the premise A -> (B & ~B) proves ~A, while the temporary assumption A was discharged.

    And it is valid, as every model in which "A -> (B & ~B)" is true is a model in which "A" is true.

    I am attributing the modus tollens to you because you are the one arguing for ¬A. If you are not using modus tollens to draw ¬A then how are you doing it? By reductio? — Leontiskos

    Modus tollens and reductio are two sides of the same coin.

    A Hilbert system may have modus tollens (or to have classical, the non-intuitionistic version) in axiom form, and a natural deduction system may use RAA (and, to have classical, the non-intuitionistic version) in rule form.

    To prove a negation, there have to be axioms or rules to do it. Modus tollens and RAA are in a sense versions of each other and they both do the job.
  • Do (A implies B) and (A implies notB) contradict each other?
    "the presence of water implies the presences of oxygen"

    is not an "if then" statement, since 'the presence of water' and 'the presence of oxygen' are noun phrases, not propositions.
    — TonesInDeepFreeze

    An alternative way of putting it would be 'if water then oxygen'. 'If water then no oxygen' contradicts 'if water then oxygen' according to the logic of everyday parlance.     — Janus

    I don't think that way, except possibly as elliptic. Especially if the context demands analysis of the logical connections, I would says something like this instead:

    'If water is present then oxygen is not present' contradicts 'if water is present then oxygen is present'.

    As to to whether the above is true, of course, in many everyday contexts, it is regarded as true. In other contexts it is not regarded as true.

    My point earlier with taking an alternative interpretation, that is with the 'notB' not being interpreted as 'not oxygen' but rather as signifying something other than oxygen, say hydrogen, then the two statements would not contradict one another. — Janus

    If I understand you correctly, I agree, and I touched on a similar example a while back.
  • Do (A implies B) and (A implies notB) contradict each other?
    one can’t pretend to represent a contradiction in the form of a proposition and then apply the LEM — Leontiskos

    Does "apply" there refer to proofs of "(A -> (B & ~B)) -> ~A" or "~(B & ~B)"? LEM is not needed to prove those.
  • Do (A implies B) and (A implies notB) contradict each other?
    I understand the proviso "in same time in all respects". But that proviso may be given more generally, upfront about all the statements under consideration:

    (1) Caveat: We are considering only statements that are definite enough that they are unambiguous as to such things as time, aspects, etc. So we're covered in that regard.

    Then we have:

    (2) Law: For all statements A, it is not the case that both A and not-A.

    Would (1) and (2) suffice for you as the law of non-contradiction?
    — TonesInDeepFreeze

     — javra
    How does your newly provided caveat (1) added to your previously made statement (2) not fully equate semantically to what I initially explicitly defined the law of noncontradiction to be in full? — javra

    I asked because I wanted to know whether you think they are equivalent, and if not, then knowing in what ways they are different would shed light on the differences in how the law of non-contradiction is taken. And, if they are equivalent then I could use my formulation in also your context, as I prefer my formation with which we don't have to give the caveat each time we talk about the kind of statements we're studying, including the law of non-contradiction and others.

    If (2) and the now explicitly stated (1) do fully equate semantically to what I initially stated explicitly, then you have your answer. “Yes.”

    A and notA do not occur — javra

    Is A a statement?
    — TonesInDeepFreeze

    obviously not when taken in proper given context. ("if a statement both does and does not occur [...]" ???)     — javra

    Then my formulation does not accord with your notion, since my formulation takes the law of non-contradiction to regard statements.

    if not [a statement], then what is A
    — TonesInDeepFreeze

    Anything whatsoever that can be the object of one’s awareness. For example, be this object of awareness mental (such as the concept of “rock”), physical (such as a rock), or otherwise conceived as a universal (were such to be real) that is neither specific to one’s mind or to physical reality (such as the quantities specified by “1” and “0”, as these can for example describe the number of rocks present or else addressed).

    and what does it mean for it to occur?
    — TonesInDeepFreeze

    In all cases, it minimally means for it to be that logical identity, A=A, which one is at least momentarily aware of. Ranging from anything one might specify when saying, "it occurred to me that [...]" to anything that occurs physically which one is in any way aware of.     — javra

    I get you. A broad sense.

    get the sense you might now ask further trivial questions devoid of any context regarding why they might be asked. — javra

    They're not trivial. And I asked them to better understand your view.

    Thank you for suggesting the SEP article. I am familiar with it even if I have not carefully studied all of it.
  • Do (A implies B) and (A implies notB) contradict each other?
    What is the definition 'analogical equivocity'?
    — TonesInDeepFreeze

    It is the kind of equivocity present in analogical predication, where a middle term is not univocal (i.e. it is strictly speaking equivocal) but there is an analogical relation between the different senses. This is the basis for the most straightforward kind of metabasis eis allo genos. The two different senses of falsity alluded to above are an example of two senses with an analogical relation.     — Leontiskos

    I am still looking at references to get a grasp of those terminologies. I think I have at least a picture of the notion of analogical equivocal and analogical univocal, mostly as I find the notions in certain philosophy of religion, but I guess found more generally in philosophy also. If I am not mistaken, the main idea, in greatest generality (not specifically regarding theological concerns) is that we have different kinds of subject to apply predicates to. When the predicate applied to the subjects means the same among the subjects, then that is univocal. But when the predicate means differently among the subjects then that is equivocal. That's the best come up with so far. But there is more terminology* and I don't follow what it means as applied to logic in this thread. *Especially it would help to know the translation of "metabasis eis allo genos".
  • Do (A implies B) and (A implies notB) contradict each other?
    Tones gave a translation of the latter as:
    "It is not the case that if A then B & ~B
    implies
    A"
    I still can't make sense of it.     — Lionino

    That might be because, for ease of clarity, the sentence needs parentheses.

    "¬(A→(B∧¬B)) entails A"

    In order not to conflate with 'entails' to stand for semantic entailment, I'll simplify:

    ~(A -> (B & ~B)) -> A

    I merely used the ordinary interrelations of the symbols.

    (it is not the case that (A implies (B and it is not the case that B))) implies A

    or

    (not (A implies (B and not B))) implies A

    or

    if (not (A implies (B and not B))) then A

    But you still have not told me what your point is in asking me this!
  • Do (A implies B) and (A implies notB) contradict each other?
    I read his responses to Lionino, but many of those posts are just completely blank. He deletes what he wrote. — Leontiskos

    What in the world?! The poster takes issue with the fact that it happens sometimes that one needs to delete! I deleted some posts that were only started, because those posting boxes were out of sequence when I came back to finish them. Then l finished the posts in posting boxes that were in better sequence. Wow, how petty he is!
  • Do (A implies B) and (A implies notB) contradict each other?
    [Tones] is a pill and iinundates me with an absurd number of replies (15 in just the last 24 hours). Presumably he is the only one you believe has "explained this at length"? — Leontiskos

    The hypocrisy there is astounding. The poster has written a whole lot of posts in this thread. Possibly a lot more characters I have written in this thread. Possibly more than anyone in this thread. I was away for a period while the poster had entered a lot posts, most of which are not short. When I got back, I replied to them. Only an arse would think that is not fair. And the poster, in a petty way, counts my posts, while some of them are individual for convenience. Moreover my posts have a lot of space in them due to my formatting. The poster writes stuff and I respond, often in detail. No one is a "pill' for that. In contrast with the poster's petty counting of posts, I don't begrudge the poster for filling up a lot of posting boxes. I think people should post as much as they want, explain as much as they want.

    Presumably he is the only one you [Banno] believe has "explained this at length"? — Leontiskos

    A mere presumption indeed. And what is the point of the remark?
  • Mathematical truth is not orderly but highly chaotic
    It is (ZF\I)+~I that is bi-interpretable with PA. — TonesInDeepFreeze

    And actually we don't need the 'F'.

    (Z\I)+~I is bi-interpretable with PA.

    [EDIT CORRECTION: I think it is incorrect that (Z\I)+~I is bi-interpretable with PA. This is correct: If every set is finite, then the axiom schema of replacement obtains and (Z\I)+~I = (ZF\I)+~I. But I don't think that works; I was thinking that the negation of the axiom infinity implies that every set is finite. But I think that itself requires the axiom schema of replacement.]
  • Mathematical truth is not orderly but highly chaotic
    The quote is extreme. — Tarskian

    I said the quote is incorrect. You agreed. So I asked why you posted it.

    I don't think, however, that it is incorrect. — Tarskian

    Now you've reversed yourself.

    Platonism is not wrong either. It is just another way of looking at things. — Tarskian

    I don't offer this as a philosophy, not something I advocate others adopt, so not something I would need to defend, but my personal perspective is that different philosophies are not necessarily right or wrong but rather are framework options for organizing one's thoughts on subject matters.

    [PA and ZF\I] turn out to be perfectly bi-interpretable. — Tarskian

    ZF\I is not (ZF\I)+~I. The former is ZF without the axiom of infinity. The latter is ZF with the axiom of infinity replaced by the negation of he axiom of infinity.

    It is (ZF\I)+~I that is bi-interpretable with PA.

    [EDIT CORRECTION: I'm told that 'ZF-I' is a notation for '(ZF\I)+~I'.]
  • Mathematical truth is not orderly but highly chaotic
    Whatever the relative merits, do you see my point that the quote is incorrect, since there are approaches to formalism that don't view mathematics as being about nothing?
    — TonesInDeepFreeze

    Yes, of course.     — Tarskian

    So why do you quote something that is seriously incorrect?
  • Mathematical truth is not orderly but highly chaotic
    I find mathematics to be the opposite of boring.
  • Do (A implies B) and (A implies notB) contradict each other?
    given LEM — Lionino

    Usually we have to have LEM to have truth tables. For example, intutionistic sentential logic cannot be evaluated with truth tables with any finite number of truth values.
  • Do (A implies B) and (A implies notB) contradict each other?
    "if a statement is true, then that statement is implied by any statement whatever," which is straightforwardly counter intuitive.
    — Count Timothy von Icarus

    That's true of classical logic     — sime

    I explained why "if a statement is true, then that statement is implied by any statement whatever" is a misleading characterization.
  • Do (A implies B) and (A implies notB) contradict each other?
    Modern Symbolic Logic", it doesn't have a well-defined meaning since it refers to a plurality of logics — sime

    I think the context of the paper is classical logic, or a logic that has material implication.
  • Do (A implies B) and (A implies notB) contradict each other?
    "It is not the case that both water can be green and water can be not-green" is an instance of the law of non-contradiction.
    — TonesInDeepFreeze

    That looks accurate to me.     — Lionino

    To be clear, that is not my own claim.

    My question is, as we can see from the truth table I posted, (a → (b ∧ ¬b)) is False only when A is True. When we try to convert that to natural language, the result can be something that is evidently untrue — Lionino

    Not untrue to me.

    just because something does not imply a contradiction, it doesn't mean it is true). — Lionino

    You skipped what I said about that.

    j is not the case that if A then B & ~B
    implies
    A.
    — TonesInDeepFreeze

    does not enlighten me.     — Lionino

    I didn't intend to enlighten you. You asked me for a translation, so I gave you the most direct translation.

    You talked about interpretations/models, and my truth table shows all of them — given LEM. — Lionino

    I talked about interpretations so that we are clear about what we claim about these things.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪javra


    I understand the proviso "in same time in all respects". But that proviso may be given more generally, upfront about all the statements under consideration:

    (1) Caveat: We are considering only statements that are definite enough that they are unambiguous as to such things as time, aspects, etc. So we're covered in that regard.

    Then we have:

    (2) Law: For all statements A, it is not the case that both A and not-A.

    Would (1) and (2) suffice for you as the law of non-contradiction?

    if A and notA do not occur — javra

    Is A a statement? If so, what do you mean for a statement to occur? If not, then what is A and what does it mean for it to occur?
  • Infinity
    ↪Lionino


    Yes. Well put.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪javra


    So when you say you claim the opposite, do you mean you claim the denial of:

    "It is not the case that both water can be green and water can be not-green" is an instance of the law of non-contradiction.

    If so, I don't understand. I would think that you are claiming:

    "It is not the case that both water can be green and water can be not-green" is an instance of the law of non-contradiction.

    Or maybe you're not seeing that the scope of 'it is not the case that' is only 'both water can be green and water can be not-green'.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪Lionino


    At least for my sake, you don't need to link me to a generator for such simple matters.

    In the case that ~(A -> (B & ~B)) is true, A is true.

    I asked what is your point in asking me these questions.
  • Do (A implies B) and (A implies notB) contradict each other?

    I meant 'non-contradiction', not 'contradiction'. I meant:

    Do you take

    "It is not the case that both water can be green and water can be not-green."

    as an instance of the law of non-contradiction?     — javra
  • Mathematical truth is not orderly but highly chaotic
    Whatever the relative merits, do you see my point that the quote is incorrect, since there are approaches to formalism that don't view mathematics as being about nothing?
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪javra


    If I understand, you take

    It is not the case that both water can be green and water can be not-green.

    as an instance of the law of contradiction. (?)

    Do you also take

    It is not the case that both water can be green and water cannot be green

    as an instance of the law of contradiction?
  • Mathematical truth is not orderly but highly chaotic
    ↪Tarskian


    You may hold that the view has merits. I'm only pointing out that formalism is not confined to that view.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪javra


    Would allow simplifying that to:

    For any statement A, it is not the case that both A and not-A.
  • Mathematical truth is not orderly but highly chaotic
    In principle, mathematics proper is about nothing at all — Tarskian

    That is one extreme view.

    According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.

    That is extreme formalism. It does not speak for all formalists.
  • Do (A implies B) and (A implies notB) contradict each other?
    ↪javra


    What is your statement of the principle?
  • Do (A implies B) and (A implies notB) contradict each other?
    the proposition that A entails both B and notB will be logically contradictory — javra

    I haven't followed your posts, so there may be context I need. That said, (1) What is your context? Classical logic? Some other formal logic? Notions in everyday reasoning? (2) By 'logically contradictory' do you mean the proposition implies a proposition of the form P and ~P? Or something else?
  • Do (A implies B) and (A implies notB) contradict each other?
    "the presence of water implies the presences of oxygen" and (A implies notB) — Janus

    I haven't followed your posts, so there may be context I need. But at least at face value:

    "the presence of water implies the presences of oxygen"

    is not an "if then" statement, since 'the presence of water' and 'the presence of oxygen' are noun phrases, not propositions.
  • Infinity
    Yes, I can see a distinction between metaphysics/ontology and foundations. Perhaps though the distinctions can be quite less than sharp.
  • Infinity
    Couldn't a logicist not be a platonist? — Lionino

    Why not? Maybe if 'logical truth' was regarded as a property of formal semantics? I mean, can't we regard 'logical axiom' as merely a logical notion without ontological commitment?
  • Infinity
    ↪Lionino


    Right.

    I wonder about the categories. The schools could be something like:

    realist

    logicist

    formalist

    structuralist

    constructivist
  • Infinity
    ↪Lionino


    Right, I caught that a moment later, and edited mine.
  • Infinity
    ↪Lionino


    So you are asking "couldn't a formalist not be a nominalist?"

    I'll try to check it out.
  • Do (A implies B) and (A implies notB) contradict each other?
    An interpretation, aka 'a model'.

    For sentential logic, an interpretation assigns to each sentence letter a value True of value False.

    Most commonly this is represented as columns in a truth table.

    This is ordinary semantics for sentential logic.

    /

    I hope my explanation regarding my answer to your question was satisfactory to you. What was your purpose in asking the question?
  • 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.
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