• What can we say about logical formulas/propositions?
    ↪Tarskian
    And constructivism uses the law of identity, so it is not the case that the only one of those three laws allowed by constructivism is non-contradiction.
    1h     — TonesInDeepFreeze

    I didn't write '1h'.
  • What can we say about logical formulas/propositions?
    Using a word to mean something other than what it does is exactly a violation of grammar. — Lionino

    "What it does" meaning its syntactical role, yes.

    "What it means", no.

    If I think 'red' means 'loud' and I say "The trombone is red", then still "The trombone is red" is grammatical even though it is false and false due to the speaker's mistake in the meaning of the word 'red'.
  • What can we say about logical formulas/propositions?
    The only foundational law that seems to withstand foundational scrutiny by constructive mathematics, is the law of non-contradiction: — Tarskian

    The law of identity is allowed by constructivism. It "withstands foundational scrutiny" by constructivism. No strawman.
  • Infinity
    ↪Lionino


    Inference rules may be rigorously defined as relations on the power set of the finite set of formulas cross the set of formulas. So, if sets are mathematical objects, then, as rules themselves are sets, rules also are mathematical objects.

    Let S be the set of formulas. Let T be the set of finite subsets of S. Let PT be the power set of T. Let x be the Cartesian product. Then:

    An inference rule is a subset of PT x S.

    Every rule is a set of ordered pairs, such that for each pair <G P>, G is a finite set of formulas (the premises) and P is a formula (the conclusion).

    For example, with that definition, the rule of modus ponens is:

    {<G P> | P is a formula, and there is a formula Q such that G = {P -> Q, P}}

    All the rules of natural deduction can be written in that manner.

    And then 'proof' may be defined as a sequence of formulas such that latter entries are conclusions from previous premises per the rules. So, proofs also are mathematical objects.

    In general, languages, syntaxes, axiom sets, inference rules, systems, theories, and interpretations are also formalizable as mathematical objects.
  • Can we reset at this point?
    .9999... = x
    9.9999... = 10x
    10x-x = 9.999... - .999...
    9x = 9
    x = 1     — flannel jesus

    That's not a proof. It's handwaving by using an undefined operation of subtraction involving infinite sequences. Actual proofs are available though.
  • Can we reset at this point?
    I would say that whether 0.999...=1 is crucially dependent upon which number line is presupposed. — alan1000

    No, it depends on what is meant by '...'. In ordinary mathematics, '...' in that context refers to the limit of a certain sequence, and we prove that that limit is 1.

    in the (classical) real number line, 0.999... is the largest real number which is less than 1 — alan1000

    Wrong. In classical mathematics, '.9...' is notation for the limit of a certain sequence, and that limit is proven to be 1.

    Cantor's Diagonal Argument certainly seems to support this interpretation — alan1000

    Cantor's argument has nothing to do with it. They are different matters.

    Abraham Robinson — alan1000

    Robinson came up with non-standard analysis. That is a different context.
  • The Liar Paradox - Is it even a valid statement?
    You can perfectly know the construction logic of a system but that does still not allow you to know its complete truth. So, even if we manage to figure out the perfect theory of the physical universe, we will still not be able to predict most of its facts. — Tarskian

    That is ridiculously overbroad and vague.
  • The Liar Paradox - Is it even a valid statement?
    ↪Gregory


    My point was that the incompleteness theorem is not a conjecture.

    /

    I take it that 'postulates' means axioms.

    For every sentence, there is a system of which the sentence is an axiom.

    A sentence is provable or not relative to a given system.

    Every axiom is trivially provable in a system in which it is an axiom (by the trivial proof of putting the axiom itself as the only line in a proof).

    For every sentence, trivially, there is a system in which the sentence is provable (by making the sentence an axiom).

    So, the incompleteness theorem is not about what is provable simpliciter, but what is provable in certain kinds of systems.

    The incompleteness theorem is: If a theory is formal, sufficient for a certain amount of arithmetic and consistent, then the theory is incomplete. That is highly informative: It tells us that there is no axiomatization of arithmetic such that every sentence of arithmetic is a theorem or its negation is a theorem. It tells us that there is no axiomatization that proves all the true sentences of arithmetic. It tells us that there is no algorithm to determine whether any given sentence of arithmetic is true. And the methods of the proof lead to profoundly informative results such as the unsolvability of the halting theorem and that there is no algorithm to determine whether a given Diophantine equation is solvable.

    You seem to not be distinguishing between (1) For any given system, the axioms are not proven from previous theorems and (2) Given any consistent set of axioms sufficient for a certain amount of arithmetic, there are sentences of arithmetic such that neither the sentence not its negation is provable from the axioms, thus there are true sentences of arithmetic not provable from the axioms.

    (1) is a trivial given. (2) is a remarkable result.
  • What can we say about logical formulas/propositions?
    "the sentence or its negation is a theorem" ignores the existence of true but unprovable sentences. So, it should rather be "the sentence or its negation is true". They don't need to be provable theorems. — Tarskian

    I'm just telling you what the definition is. It doesn't matter what you think "should" be or what "needs" to be.

    I do not see the difference between "the sentence or its negation is true" and "P v ~P". — Tarskian

    The definition of 'decidable' is not "the sentence or its negation is true".

    I was referring to the identity of indiscernibles — Tarskian

    And that is not the law of identity. And it doesn't bear on the law of identity the way you claimed it does.

    You think that the only law that constructivism allows is non-contradiction? You've gone through all other laws and found that they are not constructivisitically acceptable?
    — TonesInDeepFreeze

    I was referring to Boole's laws of thought:

    - the law of identity (ID)
    - the law of contradiction (or non-contradiction; NC)
    - the law of excluded middle (EM)     — Tarskian

    And constructivism uses the law of identity, so it is not the case that the only one of those three laws allowed by constructivism is non-contradiction.
  • The Liar Paradox - Is it even a valid statement?
    ↪Gregory


    There are two different things:

    (1) The incompleteness theorem. It's not a conjecture. It is proven. It is a theorem about certain kinds of object theories.

    (2) The Godel sentence. In proving the incompleteness theorem, we prove that the Godel sentence is a sentence in the language of the object theory. And we prove that, if the object theory is consistent, then the Godel sentence is not provable in the object theory.
  • The Liar Paradox - Is it even a valid statement?
    The differences between the Godel sentence and "this is a statement and this is not provable"?

    The Godel sentence is a sentence in the language of arithmetic. Given the standard interpretation of the language of arithmetic, the Godel sentence says something about natural numbers. But also, the Godel sentence is true in the standard interpretation if and only if it is not provable in the system (whichever system the incompleteness theorem is being proven about). In that sense, the Godel sentence says "I am not provable", but keep in mind that "I" is only our informal description; the language doesn't have such pronouns. And, the Godel sentence does not say "I am a sentence" nor mention "sentence". Rather, it is in the meta-theory that we show that the Godel sentence is indeed a sentence in the language of arithmetic and that, if the system is consistent, then the Godel sentence is not provable in the system, and the negation of the Godel sentence is not provable in the object theory, and that the Godel sentence is true in the standard model for the language. (Note that Godel did not specify formal models, as formal models were not explicated until later, and he proved for a different kind of system. Instead, he simply worked in ordinary mathematics regarding natural numbers without putting a fine point on that in terms of models and a standard interpretation.)
  • Mathematical truth is not orderly but highly chaotic
    very humble of you. — fishfry

    Half is humble, since my knowledge of modern logic is not extensive relative to people who study it a lot more intensely (though vastly greater than cranks and jokers - such as in this forum - who know don't know jack about it). And I've forgotten a lot of what I knew and am rusty on many details and more advanced topics. Also, in the last couple of weeks, very atypically, I made not just one or two reasoning errors but a series of them, though I exercised intellectual honesty to correct them. The other half is not humble since I do have a well developed perspective on jazz - technically, historically, discographically - and a well developed taste in it and an intense emotional and spiritual connection with it, though there are people who know a tremendous amount more than me.
  • The Liar Paradox - Is it even a valid statement?
    'This statement is not provable' means:

    1) this is a statement
    2) this is not provable     — Devans99

    That is not the Godel sentence.
  • The Liar Paradox - Is it even a valid statement?
    If we remove this class of malformed, contradictory statements then these limitations do not apply any more. — Devans99

    The incompleteness theorem does not rely on any sentences that can't be formed in the language of arithmetic.
  • The Liar Paradox - Is it even a valid statement?
    they are a consequence of the postulates — leo

    Regarding the liar sentence, what postulates?
  • The Liar Paradox - Is it even a valid statement?
    These self-denying statements are acceptable according to formal logic and they lead to Godel's Incompleteness Theorems. — Devans99

    The incompleteness theorem does not rely on the liar sentence.
  • The Liar Paradox - Is it even a valid statement?
    Not all statements in a given language can be given a truth value — leo

    For a formal language, per a given interpretation, every sentence has a truth value.
  • The Liar Paradox - Is it even a valid statement?
    So in the case of the Godel statement, ‘this statement is not provable’… means 'it is not provable that this is a statement'. — Devans99

    That is not the Godel sentence.

    If you can’t prove its a statement then you can even start to prove it. — Devans99

    We do prove it is a sentence in the language of the theory at hand. And we don't the sentence in the theory.
  • The Liar Paradox - Is it even a valid statement?
    1. this is a statement
    2. and it is false

    So 2 says 1 is false. IE it is not a statement.     — Devans99

    We can formulate the liar paradox without saying "is a statement".
  • The Liar Paradox - Is it even a valid statement?
    a statement declares a fact; it does not in addition instantiate that fact to a given truth value. — Devans99

    Some statements do mention truth values.

    I believe Godel's objections would go away though I need to look at that conjecture further — Devans99

    The incompleteness theorem is not at all disqualified by the liar sentence. And it's not a conjecture.
  • The Liar Paradox - Is it even a valid statement?
    The so-called "Liar's paradox" seems quite silly — Leontiskos

    Knowing something about logic and the context helps to understand why the liar paradox is of interest.

    I agree it's not much use to spend much time pondering about them
    — leo

    Me too.     — Leontiskos

    Good then that no one is forcing you to spend time on it. But meanwhile it is worth time to people who study logic.
  • Infinity
    ↪Lionino


    Do you disagree with the point that inference rules may themselves be a mathematical object?

    A→B being defined (convention) exactly by what it gives in a truth table according to each value of A and B, and A&B, etc. — Lionino

    The symbol '->' may be a primitive or defined from primitive symbols.

    The truth or falsehood, in a model M, of a sentence of the form 'P -> Q' is determined by the definition of 'S is true in model M'.

    Meanwhile, 'P -> Q' is a formula (if 'P' and 'Q' are formulas) or it stands for a set of formulas (if 'P' and 'Q' are meta-variables ranging over formulas). It is not something that is "defined". Rather, it is shown to be a formula from the defintion of 'is a formula'.
  • What can we say about logical formulas/propositions?
    .
  • What can we say about logical formulas/propositions?
    If — then — is only used in math/logic because it is clearer to look at than If —, —. — Lionino

    It's not used only in logic and mathematics. In everyday discourse, people write "If ___, then" commonly. The source you cited mentioned mentions "If ___, ___" only but I would not take that to preclude also "If ___, then". Are there grammarians who explicitly disallow it? Are there not grammarians who do allow it? Perhaps there are grammarians explicitly disallow "If ___, then ___", but that would be pedantic, especially in this context, in face of the fact that "If ___, then ___" is not only used in everyday discourse, but in all kinds of writing. Moreover, since it is taken as grammatical in logic and mathematics, then that's good enough here, since logic is the subject. I don't know what point you are making about logic when you rule out "If ___, then ___".

    That is why I said "I am literally dying now" instead of "I am dying now". It is an incorrect usage of the word 'literally' if you are not really dying, therefore grammatically incorrect. — Lionino

    As far as I can tell, it is grammatical. 'literally' is an adjective to the noun 'dying'. But the sentence is false. "I am hopelessly dying", "I am unhappily dying", "I am literally dying". Grammatical as far as I know.

    their usage of the word is often just grammatically incorrect. — Lionino

    What rule of grammar is violated. I wouldn't take using a word with an incorrect meaning is not a violation of grammar. If someone thought 'choleric' means 'melancholic', then "Jack is choleric" is still grammatical even though Jack is not choleric.

    not lying or confused about their health — Lionino

    Yes, they are not lying or confused about their health. They simply mispoke while still grammatical.

    "I am literally dying now" may be true or it may be false. But in either case, it is grammatical.

    Dialetheism and the denial of LNC — Lionino

    I would need to re-read that article, but, as I recall, dialetheism is a philosophy not a system. Though, as you mention, there are paraconsistent systems. Yes, that is an example. But, for any for any law of thought there may be a system that denies the law, so any law of thought could be denied.

    If your point is that one is free to choose any system one wants to use, then, of course, one could not dispute that. But also one is free to choose whatever ways of thinking one wants to choose.

    The laws of thought are facts of the matter. Whatever they are, without them human rationality is not possible — otherwise they wouldn't be laws. — Lionino

    That something is necessary for rationality (under a given definition of 'rationality') doesn't entail that people may not break "laws of thought".

    Can you conceive something as other than what it is? — Lionino

    Whether or not I can conceive it doesn't entail that others cannot. It is not precluded that, for example, people in mystic states do experience suspension of non-contradiction. And it does not dialetheism permit conceiving such things?

    You said, "Some laws of logic may express those laws of thought. But that is just a semantic contention."

    Now:

    Leontiskos said laws of logic can't be broken. I said that it is the laws of thought that can't be broken instead. Despite the disagreement in choice of words, I still understand the content of his post. — Lionino

    I guess 'that' referred to the difference in the way you two stated the idea. Okay.

    /

    I asked, "Do you mean there are cases in which no law applies? Or do you mean that, for any law, there are cases in which that law does not apply?"

    I surmise you mean the latter.
  • Infinity
    and logical terms can be taken as primitives defined from their truth tables
    — Lionino
    Those are set up by convention.     — Lionino

    It's not clear to me what you're claiming. Example?
  • Infinity
    ↪Lionino


    My point is that there may be many views as to what mathematical objects are or are not, including realism, fictionalism, nominalism... But that, in that inquiry, not just things like sets, numbers, algebras may be considered, but also rules.
  • Infinity
    ↪Lionino


    Rules themselves may be mathematical objects. Languages, axioms, rules, systems, theories, and proofs can be defined and named in set theory. Even informally, when, for example, we say "by the rule of modus ponens", the rule of modus ponens is a thing named by 'the rule of modus ponens'. — TonesInDeepFreeze

    I should add that the above does not opine that those things are platonic things. Moreover, there is not a particular sense in which I am saying they are things. Moreover, I'm not opining that saying "things" or "objects" requires anything more than an "operational" sense: we use 'thing' or 'object' in order to talk about mathematics, as those notions are inherent in communication; it would be extraordinarily unwieldy to talk about, say, numbers without speaking, at least, as if they are things of some sort. But, it is not inappropriate to discuss the ways such things as rules are or are not mathematical things of some kind.
  • What can we say about logical formulas/propositions?
    Some laws of logic may express those laws of thought. But that is just a semantic contention.
    — Lionino

    What "semantic contention"?     — TonesInDeepFreeze
  • What can we say about logical formulas/propositions?
    the law of the excluded middle (LEM), which implicitly assumes that the question at hand is decidable. — Tarskian

    In context of modern logic, 'decidable' means either (1) the sentence or its negation is a theorem, or (2) There is an algorithm to decide whether the sentence is a member of a given set, such as the set of sentences that are valid, or the set of sentences that are true in a given model.

    LEM is not that. LEM syntactically is the theorem: P v ~P, and LEM semantically is the theorem that for a given model M, either P is true in M or P is false in M (so, either P is true in M or ~P is true in M)

    The law of identity may also be problematic because of the existence of indiscernible numbers. — Tarskian

    The law of identity, the indiscernibility of identicals, and the identity of indiscernibles are different. What specific problem with the law of identity are you referring to?

    The only foundational law that seems to withstand foundational scrutiny by constructive mathematics, is the law of non-contradiction: — Tarskian

    You think that the only law that constructivism allows is non-contradiction? You've gone through all other laws and found that they are not constructivisitically acceptable?
  • What can we say about logical formulas/propositions?
    sentences of the kind "If --, then --" are not grammatically correct. — Lionino

    They are grammatically correct in English. Why would you claim otherwise?

    Every time someone says "If ___ then ___" they are incorrect?
    — TonesInDeepFreeze

    Yes, just like when someone says "I am literally dying right now" but they are alive and well.     — Lionino

    "If ____, then ___" is ordinary grammatical English.

    "I am dying now" said when not dying is ordinary grammatical English, but is a false sentence.

    "The laws of physics don't apply here", the meaning is clear. You yourself use the word without any apparent confusion:

    for any law, there are cases in which that law does not apply
    — TonesInDeepFreeze     — Lionino

    (1) I know the ordinary general sense of 'apply'. But this is a particular subject, and I'm wondering whether you have an explication of your use or whether 'apply' should just be taken as undefined by you. (2) I was asking you about your use of 'apply'; I didn't assert my own use of it. I didn't assert what you quoted of me; it was part of a question to you.

    And the question still stands:

    And do you mean there are cases in which no law applies? Or do you mean that, for any law, there are cases in which that law does not apply? — TonesInDeepFreeze

    But you do say:

    for any law, there are cases in which that law does not apply
    — TonesInDeepFreeze

    This, but one can make up scenarios and/or systems where that law does not apply. That was one of the answers at least to the liar paradox: making a completely different system.     — Lionino

    What law and system are you referring to?

    What are some of those laws of thought that can't be broken but are not laws of logic?
    — TonesInDeepFreeze

    I don't think there any, as soon as we can express our thoughts in language we can also express the rules our thoughts follow in language (this language being logic sometimes).     — Lionino

    You said that there are laws of thought that can't be broken. And you said laws of logic can be broken. What are some laws of thought that can't be broken but are not laws of logic?

    What are the obvious reasons they can't be broken?
    — TonesInDeepFreeze

    For example, I can't conceive of anything as being other than it is, because as soon as I conceive it, it is what it is, and not something else. I cannot imagine something as being otherwise.     — Lionino

    You can't conceive it. But that doesn't entail that others cannot conceive it. Also, conceiving that a contradiction holds does not entail that the contradiction holds.

    Yes, the periods are "missing". — Lionino

    If we put a period at the end of "If ___ then ___" , then it is a punctuated English sentence. Just as with that sentence itself.
  • Mathematical truth is not orderly but highly chaotic
    Theologician, redeem thyself.
  • What can we say about logical formulas/propositions?
    I don't think there are laws of logic that cannot be broken — Lionino

    What do you mean by not being able to "break"?
    — TonesInDeepFreeze

    There being cases in which a law does not apply.     — Lionino

    What do you mean by "apply"?

    And do you mean there are cases in which no law applies? Or do you mean that, for any law, there are cases in which that law does not apply?

    What are some laws and cases you have in mind?

    there are laws of thought that can't be broken (for obvious reasons). — Lionino

    What are some of those laws of thought that can't be broken but are not laws of logic? How do you state the difference between laws of logic and laws of thought? What are the obvious reasons they can't be broken?

    Some laws of logic may express those laws of thought. But that is just a semantic contention. — Lionino

    What "semantic contention"?
  • What can we say about logical formulas/propositions?
    "If X, then Y" is incorrect.
    "If X, Y" or "X, therefore Y", not both.     — Lionino

    "If you go, then I will go" is not okay grammatically. — Lionino

    "If X then Y" is incorrect because you think "If you go, then I will go" is not grammatical?

    Why would an ordinary sentence form be incorrect? Every time someone says "If ___ then ___" they are incorrect?

    And "If you go, then I will go" is missing a period. But otherwise it seems fine to me.

    "If X, Y" or "X, therefore Y", not both. — Lionino

    I don't know what you mean to say there.
  • What can we say about logical formulas/propositions?
    That is actually the main difference between classical logic and mathematical logic. — Tarskian

    Usually, mathematical logic is studied by means of classical logic. Indeed, mathematical logic is formulated by classical set theory. The theorems of mathematical logic, if formalized, are themselves theorems of set theory.
  • What can we say about logical formulas/propositions?
    After the semantic contention, a syntactic contention:

    "If X, then Y" is incorrect.
    "If X, Y" or "X, therefore Y", not both.     — Lionino

    What does that mean?
  • What can we say about logical formulas/propositions?
    If relevance is required between the antecedent and consequent for meaningfulness, then we don't know whether a given conditional is meaningful until we've settled whether there is relevance between the antecedent and the consequent. So if the question of relevance is unsettled, we have to wait before taking the conditional to be meaningful or not. "If Jackie has blue hair then London is noiser this year than last year". We don't know whether the antecedent is relevant to the consequent without knowing more. Maybe Jackie having blue hair causes a big fashion trend in which people go to London to be seen having blue hair or many other possibilities. For that matter, when would we ever be certain that there is no relevance between two sentences? A butterfly flapping its wings in Tierra del Fuego, so to speak.
  • What can we say about logical formulas/propositions?
    ↪Lionino


    What do you mean by "cannot be broken"? Do you mean "cannot break without being in error"?
  • What can we say about logical formulas/propositions?
    "In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q."

    I speak English, and I don't take "if P then Q" (whether in the sense of material implication or in everyday senses, including necessity or relevance) to be about presences. Indeed, where 'P' and 'Q' are sentences, I would take "the sentence P is present", etc., to be nonsense unless it meant that the sentence P was being displayed in some way, such as on a page or screen. Indeed, I've never heard an English speaker in everyday conversation say something like "The sentence P is present". Moreover, let P stand for a sentence such as "The world is big", then I've never heard any English speaker say anything like ""The world is big" is present". Indeed, if an English said "If the world is big, then the sun is huge" then I don't know any English speaker who would say, "Yes, the presence of "The world is big" indicates the presence of "The sun is huge". Not only is that dialogue not idiomatic, but it registers as nonsense.
  • What can we say about logical formulas/propositions?
    ↪Lionino


    Ah, I see the problem, and I carelessly extended it.

    I'm dumping this:

    "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".

    That is wrong.

    As I mentioned before, there are two different notions:

    (1) "If A then B"

    and

    (2) "A entails B"


    (1) in the sense of material implication means "(A is true and B is true) or (A is false and B is true) or (A is false and B is false)". And that reduces to "A is false or B is true".

    (2) means "Every instance in which A is true is an instance in which B is true".


    (1) is symbolized as 'A -> B'

    (2) is symbolized as A |= B


    Indeed, they are not equivalent.
  • Mathematical truth is not orderly but highly chaotic
    ↪fishfry


    Anytime you want jazz album recommendations, just ring.

    Jazz is one thing I know a lot about, unlike logic.
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