Comments

  • Do (A implies B) and (A implies notB) contradict each other?
    A reductio requires special background conditions. In this case it would require the background condition that (1) is more plausible than (2). — Leontiskos

    That is a bad misconception. It's not how it works.

    Proof is from a set of premises. A natural deduction proof lists premises. A reductio premise however is discharged at the end of the proof. It is not an ordinary premise.

    Let G be a set of premises and a sentence P is not a member of G. And we want to show that G proves ~P. Then we may use any of the members of G in our argument. But, along with members of G, we also may suppose P to derive a contradiction, thus to show that G proves ~P.

    "plausibility" is not invoked.
  • Do (A implies B) and (A implies notB) contradict each other?
    They will be required to examine the logic machine itself instead of just assuming that it is working. — Leontiskos

    Logicians and philosophers of mathematics examine logics with intense scrutiny and may be interested in all kinds of formal and philosophical alternatives. And critiques of classical logic should be welcomed; indeed they have been vital in the history of logic and mathematics. But when critiques are ill-premised and describe the logic in inaccurate ways, then, of course, how can that be dealt with other than indeed explaining how the "machine" does operate?

    But in what sense is the "machine" working?

    Well, it does not work to express much of everyday discourse. Is there anyone who has denied that? And it does not accord with certain philosophies and alternative formal systems. And it does not provide answers to all the philosophical questions. But it does work as an axiomatization for mathematics for the sciences (indeed, the P vs NP problem is commonly described as the most important mathematical problems for practical implications for purposes of business, with a one million dollar prize for its solution). It does work as the basis for the computers we're using when we post. It does work in the sense that it gives rigorous definitions and unambiguous formulations of mathematics and for other fields of study. It does work in the sense that, in principle, we can most objectively (that is algorithmically) check that any given purported proof is indeed a proof in the system. It does work in the sense that the soundness and completeness of the logic is rigorously proven. And it does work in the sense that it provides beauty to its students and experts, and such that it induces great creativity and insight and questions that themselves lead to not only flourishing of mathematical inquiry but discoveries applicable in other fields. And it works in the sense that it provides the starting point and the basic rubrics for the invention of alternatives to it.
  • Do (A implies B) and (A implies notB) contradict each other?
    His ready-made approach doesn't answer the questions that are being asked — Leontiskos

    It would be rare that any one poster can answer all questions. I have given corrections, explanations about the logic that is being critiqued, answers to specific questions, and perspective on the subject. And describing my approach as "ready made" is empty polemics.

    A person may have all kinds of independent, self-fashioned, not ready-made ideas. And one may critique "ready-made" classical logic, but when a critique misconstrues or misrepesents what it is critiquing, then it is quite appropriate to point out the errors and even explain why they are errors. Moreover, it is appropriate to point it out when what is critiqued is subjected to criteria not pertinent. Of course, classical logic doesn't accord with much of everyday discourse. And a pencil doesn't accord with the colors in the world. That doesn't entail that, for certain purposes, a pencil is not preferable to a box of crayons. Black and white photography doesn't accord with the colors in the world. That doesn't entail that black and white photography is wrong. An x-ray doesn't accord with the way we see things in everyday life. That doesn't make an x-ray wrong. And it is apt to explain how x-rays work, what they represent, and their purpose, just as it is apt to explain how classical logic works, what it represents and its purpose, especially in context of ill-premised remarks about it.
  • Do (A implies B) and (A implies notB) contradict each other?
    If A is false, then A implies anything. — flannel jesus

    We need to be careful to recognize that "A is false" is not simpliciter. Rather, "A is false in model M" (with sentential logic, a model may be represented as a row in a truth table).

    You can check the truth-table on implication: A -> B is always true if A is false. — flannel jesus

    But I would add (and this was my point to @Leonino) it is not the case that

    "Winston Churchill was French" implies every sentence, simpliciter.

    Rather, it is only in interpretations in which "Winston Churchill is French" is false is it the case that for every sentence "P", ""Winston Churchill is French" implies P" is true.
  • Do (A implies B) and (A implies notB) contradict each other?
    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

    Tones replied that that is not true for all contradictions but for some interpretations.     — Lionino

    That's not what I said.

    If I recall correctly, you said that "A -> (B & ~B)"* may be translated as "A implies a contradiction". (*Or it might have been a related formula; not crucial since my point pertains to all such examples.)

    That is not the case as follows:

    (1) The sentence has a sub-sentence that is a contradiction, but the sentence itself does not mention the notion of 'contradiction'.

    (2) To say "a contradiction" is to implicitly quantify: "There exists a contradiction such that A implies it". And that quantifies over sentences. If we unpack, we get: "There exists a sentence Q such that Q is a contradiction and A implies it".

    A translation of "A -> (B & ~B)" is:

    If A, then both B and it is not the case that B.

    and not

    "A implies a contradiction".

    (3) "B & ~B" is a particular contradiction, not just "a contradiction". Even though all contradictions are equivalent, a translation should not throw away the particular sentences that happened to be mentioned.

    (4) If we have that A implies B & ~B, then of course, we correctly say "A implies a contradiction". But that is a statement about A, not part of a translation.

    "If A implies B & ~B, then A implies a contradiction" is true, but it is a statement about the sentences, not a translation of them.
  • 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?
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