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Math Forum / Mathematics / Mathematical Logic / November 2008Godel cant tell us what makes a mathematical statement true
the australian philosopher colin leslie dean points out http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE thus his incompleteness theorems are meaningless rubbish mathematician have so much invested in godels incompleteness theorem much maths is reliant on it but at the time godel wrote his theorem he ha no idea of what truth was as peter smith the Cambridge expert on Gode admitts http://groups.google.com/group/sci.logic/browse_thread/thread/ebde70bc932fc0a7/d e566912ee69f0a8?lnk=gst&q=G%C3%B6del+didn%27t+rely+on+the+notion+PETER+smith#de5 66912ee69f0a8 Quote: Gödel didn't rely on the notion of truth but truth is central to his theorem as peter smith kindly tellls us http://assets.cambridge.org/97805218...40_excerpt.pdf Quote: Godel did is find a general method that enabled him to take any theory T strong enough to capture a modest amount of basic arithmetic and construct a corresponding arithmetical sentence GT which encodes the clai ‘The sentence GT itself is unprovable in theory T’. So G T is true i and only if T can’t prove it If we can locate GT , a Godel sentence for our favourite nicely ax- iomatized theory of arithmetic T, and can argue that G T is true-but-unprovable, and godels theorem is http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir... Quote: Gödel's first incompleteness theorem, perhaps the single most celebrate result in mathematical logic, states that: For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but no provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. you see godel referes to true statement but Gödel didn't rely on the notion of truth now because Gödel didn't rely on the notion of truth he cant tell us what true statements are thus his theorem is meaningless this puts mathematicians in deep sh.t because all the modern idea derived from godels theorem have no epistemological or mathematical worth for we dont know what true statement are - Message posted using http://www.talkaboutscience.com/group/sci.logic More information at http://www.talkaboutscience.com/faq.htm > the australian philosopher colin leslie dean points out > [quoted text clipped - 58 lines] > Message posted usinghttp://www.talkaboutscience.com/group/sci.logic/ > More information athttp://www.talkaboutscience.com/faq.html A statement is true just because it is true not because somebody 'relies on the notion of truth'. You probably confuse truth and provability. Of course, if you do so you will never be able to understand Godel's work. To think that something is true only because it has been proven is the wrongest idea you can conceive. Baudouin > A statement is true just because it is true not because somebody > 'relies on the notion of truth'. Could you think of any circumstance in which 0=1 is true? > To think that something is true only because > it has been proven is the wrongest idea you can conceive. Suppose for a given formal system T, we define true sentences as the following: - T(F) = true iff T |- F. - T(F) = false otherwise. What would you think as "wrong" with this definition? > Baudouin  Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") >> A statement is true just because it is true not because somebody >> 'relies on the notion of truth'. [quoted text clipped - 11 lines] > > What would you think as "wrong" with this definition? For clarity: - Truth(F) = true, iff T |- F. - Truth(F) = false, otherwise. Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > > A statement is true just because it is true not because somebody > > 'relies on the notion of truth'. > > Could you think of any circumstance in which 0=1 is true? Sure. Let "0" be my dad. Let "1" be me. Let "=" be the relation "father of". Not sure where you're going with this. > > To think that something is true only because > > it has been proven is the wrongest idea you can conceive. [quoted text clipped - 6 lines] > > What would you think as "wrong" with this definition? Well, if T |/- F and T |/- ~F then both F and ~F are false. Is that problematic for you? -- hz >>> A statement is true just because it is true not because somebody >>> 'relies on the notion of truth'. >> Could you think of any circumstance in which 0=1 is true? > > Sure. Let "0" be my dad. Let "1" be me. Let "=" be the > relation "father of". Can't you think a bit more mathematical, like: A1: AxEy[Sx=y] A2: Ax[0=x] Then (as a theorem) 0=S0, which is 0=1. Right? > Not sure where you're going with this. Now T df= {A1 + A2} is consistent, so 0=1 is *true* in T. So, unlike what Baudouin said, whether 0=1, or 0=/=1, is true does rely on some notion of truth! There's no such thing as a formula being true "just because it is true", and not relying on a notion of truth. >>> To think that something is true only because >>> it has been proven is the wrongest idea you can conceive. [quoted text clipped - 8 lines] > Well, if T |/- F and T |/- ~F then both F and ~F are false. > Is that problematic for you? Of course not. It's *not a perfect definition* but: a) If a formula is not defined to be true w.r.t. T, then there's nothing wrong to equate it with being false. In this way then: - A consistent system is one in which one of {F, ~F} is true and the other false. - An inconsistent system is one in which both F and ~F are true. - F is undecidable in T if both F and ~F are false (which only means F and ~F aren't provable in T!) Nothing would seem "wrong" at all! b) The standard definition of truth is *not perfect either*: there are some T in which if T is consistent, there would be some formula F which you can't tell whether or not F is true in any model of T! But the point here is a formula can't be just simply true. You have to *choose* *some* selected/defined notion of truth! Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > Now T df= {A1 + A2} is consistent, so 0=1 is *true* in T. Of course I meant "0=1 is *true* in any model of T." > >>> A statement is true just because it is true not because somebody > >>> 'relies on the notion of truth'. [quoted text clipped - 4 lines] > > Can't you think a bit more mathematical, Well, I do what I can. > like: > > A1: AxEy[Sx=y] > A2: Ax[0=x] > > Then (as a theorem) 0=S0, which is 0=1. Right? Right. > > Not sure where you're going with this. > > Now T df= {A1 + A2} is consistent, so 0=1 is *true* in [any > model of] T. Right. It's false in some structures that are not models of T. > So, unlike what Baudouin said, whether 0=1, or 0=/=1, > is true does rely on some notion of truth! What you've shown is that whether 0=1 (or 0=/=1) is true relies on the structure in which it is interpreted. This, in itself, relies on a model theoretic notion of truth. > There's no such thing > as a formula being true "just because it is true", and not relying > on a notion of truth. Right. Truth attaches to statements, not to signs such as formulae. A sign must be interpreted to express a statement. Whether an interpreted sign, a statement, expresses a truth or not will rely on what we take as the relationship between statements and truth, i.e. upon some notion of truth. The model theoretic notion of truth is that an interpreted formula is true in a structure when the formula is satisfied by that structure. This is essentialy a rigorised formulation of Aristotle's correspondence theory of truth, in which a statement is true when it corresponds to reality. A sympathetic reading of M. Le Charlier's assertion would be that the correspondence of a statement to reality, or the satisfaction of an interpreted formula in a structure, is not a matter of opinion, or of proof, or of our knowledge of such correspondence or satisfaction. Of course you are right: to assert of any statement that it "is true" or "is false" is to already have some notion of what it means for a statement to "be true" or to "be false". > >>> To think that something is true only because > >>> it has been proven is the wrongest idea you can conceive. [quoted text clipped - 20 lines] > - F is undecidable in T if both F and ~F are false (which only means > F and ~F aren't provable in T!) If both F and ~F are false (which only means F and ~F aren't provable in T) then F is undecidable in T (by your third clause) and T is not a consistent system (by your first clause), since it will not be the case that one of {F, ~F} is true (provable) and the other false (not provable). So we have that a system that is is not consistent is not necessarily the same as a system that is inconsistent (by your second clause): F and ~F are both true (provable). I don't find this terminology quite satisfactory. Also, I think you have to think about the difference between the concept of a formula F being true in a theory T (being true in every model of T) and the the anterior notion of a formula F being true in a given model (structure). > Nothing would seem "wrong" at all! > > b) The standard definition of truth is *not perfect either*: there > are some T in which if T is consistent, there would be some formula > F which you can't tell whether or not F is true in any model > of T! You seem to wish to assert that the truth of a formula F relies on our knowledge (whether we can "tell") of its truth in some model. Commonly (but not universally) it is taken that the truth or falsehood of a statement is not dependent on its epistemological status of being known to be true or not. This is part of the realist (Platonist) position that there is an objective reality that is not dependent on our minds, or what we happen to know. Whether or to what degree mathematical reality partakes of this objective status is, of course, a venerable debate. (You have provoked in me some thought about how far the model theoretic notion of truth is implicitly a realist position, or not.) > But the point here is a formula can't be just simply true. You have to > *choose* *some* selected/defined notion of truth! True. But once having chosen some notion of what it means for a statement to "be true" or to "be false", then it is no longer a matter of opinion as to whether a statement is true or false -- it is only a question of whether the statement is in accord with the notion adopted. Of course, some notions of what "truth" is will be sillier than others. You seem to have two issues here: that a formula must be interpreted to have any meaning at all (much less be true or false), and that a formula has to be known to be true (by some criteria of truth) in order for it to be true (by that criteria). -- hz >> So, unlike what Baudouin said, whether 0=1, or 0=/=1, >> is true does rely on some notion of truth! > > What you've shown is that whether 0=1 (or 0=/=1) is true relies > on the structure in which it is interpreted. This, in itself, > relies on a model theoretic notion of truth. That's right. >> There's no such thing >> as a formula being true "just because it is true", and not relying [quoted text clipped - 5 lines] > not a matter of opinion, or of proof, or of our knowledge > of such correspondence or satisfaction. I think I understand what was stated; it's just I disagree with the alleged *absolute* nature of it! > Of course you are right: to assert of any statement that > it "is true" or "is false" is to already have some notion [quoted text clipped - 27 lines] > be the case that one of {F, ~F} is true (provable) and the other > false (not provable). OK, so I typed it too quickly. What is meant here should have been typed: >> - A consistent system is one in which one of {F, ~F} is true and >> the other false, for an F. >> - An inconsistent system is one in which both F and ~F are true, >> for any F. >> - F is undecidable in T if both F and ~F are false (which only means >> F and ~F aren't provable in T!), for an F. > So we have that a system that is is not consistent is not necessarily > the same as a system that is inconsistent (by your second clause): > F and ~F are both true (provable). > > I don't find this terminology quite satisfactory. With the revised above, I hope you'd be convinced to change your mind. > Also, I think you have to think about the difference between the > concept of a formula F being true in a theory T (being true in [quoted text clipped - 10 lines] > You seem to wish to assert that the truth of a formula F relies > on our knowledge (whether we can "tell") of its truth in some model. That's what I'd like to convey. The truth of a formula F is relative to whatever definition of truth one is please. And in my case, I've "relativized" it to that of syntactical proof. > Commonly (but not universally) it is taken that the truth or > falsehood of a statement is not dependent on its epistemological > status of being known to be true or not. This is part of the > realist (Platonist) position that there is an objective reality > that is not dependent on our minds, or what we happen to know. But the moment one is able to demonstrate logically that the formula could assume the opposite truth value, this common would be on a shaky ground! > Whether or to what degree mathematical reality partakes of this > objective status is, of course, a venerable debate. > > (You have provoked in me some thought about how far the model > theoretic notion of truth is implicitly a realist position, > or not.) All I know is if mathematical reasoning is about (or dependent on) knowledge, then the reasoning can't go too far: either by means of provability - or truth! >> But the point here is a formula can't be just simply true. You have to >> *choose* *some* selected/defined notion of truth! [quoted text clipped - 9 lines] > You seem to have two issues here: that a formula must be interpreted > to have any meaning at all (much less be true or false), No! Formula's semantic and truth don't have to be identical. For example, GC has some meaning but whether or not it's true or false (by whatever the chosen underlying truth definition) is a different matter. > and that a formula has to be known to be true (by some criteria of truth) > in order for it to be true (by that criteria). That's I think is meant by something like "mathematical truth is relative", and is what I'd like to convey. But that in itself is not an "issue", imho. Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") >>> So, unlike what Baudouin said, whether 0=1, or 0=/=1, >>> is true does rely on some notion of truth! [quoted text clipped - 4 lines] > > That's right. Actually that's not quite right. What I showed is that at best (when T is consistent) the normal truth is just an alias of provability. Therefore, in technical sense, the notion of truth can be discarded! Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >>> So, unlike what Baudouin said, whether 0=1, or 0=/=1, > >>> is true does rely on some notion of truth! [quoted text clipped - 9 lines] > provability. Therefore, in technical sense, the notion of truth > can be discarded! So, is every even number < 2 the sum of two primes? Yes, no, maybe, indeterminate, "it depends", ... what? -- hz >>>>> So, unlike what Baudouin said, whether 0=1, or 0=/=1, >>>>> is true does rely on some notion of truth! [quoted text clipped - 10 lines] > > Yes, no, maybe, indeterminate, "it depends", ... what? "It depends": on what axioms governing "<", "2", and "primes" one would have in mind. Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >> So, unlike what Baudouin said, whether 0=1, or 0=/=1, > >> is true does rely on some notion of truth! [quoted text clipped - 4 lines] > > That's right. We're in simple agreement for once. :-) > >> There's no such thing > >> as a formula being true "just because it is true", and not relying [quoted text clipped - 8 lines] > I think I understand what was stated; it's just I disagree > with the alleged *absolute* nature of it! I didn't say that you didn't understand what was stated. > > Of course you are right: to assert of any statement that > > it "is true" or "is false" is to already have some notion [quoted text clipped - 47 lines] > > With the revised above, I hope you'd be convinced to change your mind. It's pretty standard as amended, other than the fact that you are identifying truth with provability. Still, we have the situation that if T |/- F and T |/- ~F then F and ~F are both false. This doesn't distinguish refuted formulae (the negations of proven formulae) from undecidable formulae. > > Also, I think you have to think about the difference between the > > concept of a formula F being true in a theory T (being true in [quoted text clipped - 7 lines] > >> F which you can't tell whether or not F is true in any model > >> of T! In the standard definition of truth, a formula F that is undecidable in a theory T is true in some model of T. If F is false in every model of T, then F is decidable in T: it is the negation of a formula provable in T (F is refuted in T). I don't see that as an imperfection. > > You seem to wish to assert that the truth of a formula F relies > > on our knowledge (whether we can "tell") of its truth in some model. > > That's what I'd like to convey. The truth of a formula F is relative > to whatever definition of truth one is please. And in my case, I've > "relativized" it to that of syntactical proof. What if a formula F has a proof but you don't know that it has a proof? Is F false until you discover the proof? Or was it true all along? > > Commonly (but not universally) it is taken that the truth or > > falsehood of a statement is not dependent on its epistemological [quoted text clipped - 5 lines] > could assume the opposite truth value, this common would be on a shaky > ground! I didn't say "formula", I said "statement". Of course a formula can assume different truth values depending on the structure in which you interpret it. This doesn't mean that you automatically know its truth value in that structure. > > Whether or to what degree mathematical reality partakes of this > > objective status is, of course, a venerable debate. [quoted text clipped - 6 lines] > knowledge, then the reasoning can't go too far: either by means > of provability - or truth! I don't know what this means. > >> But the point here is a formula can't be just simply true. You have to > >> *choose* *some* selected/defined notion of truth! [quoted text clipped - 11 lines] > > No! Formula's semantic and truth don't have to be identical. That's not what I said. An uninterpreted formula has no meaning or truth value. Statements have truth values. > For example, > GC has some meaning but whether or not it's true or false (by whatever the > chosen underlying truth definition) is a different matter. Well maybe you have a point here: Assume the standard semantics for arithmetic, that is, assume, e.g., that Euclid's theorem on the infinitude of prime numbers means what we usually take it to mean. How do you vary its truth-value within the usual semantics? I can weaken PA to where it can't prove Euclid's theorem, but the statement of the theorem will still be true in the standard model of PA, no? > > and that a formula has to be known to be true (by some criteria of truth) > > in order for it to be true (by that criteria). > > That's I think is meant by something like "mathematical truth is relative", > and is what I'd like to convey. But that in itself is not an "issue", imho. You wish to say that mathematical truth of a statement is relative not only to the chosen criteria of truth, but also to whether we know the statement meets the criteria? IOW, a statement is false if it fails to meet the chosen truth definition, but also if we don't know whether it meets the truth definition or not? -- hz >> With the revised above, I hope you'd be convinced to change your mind. > > It's pretty standard as amended, other than the fact that you > are identifying truth with provability. Arguably, there are "forces" within the community who'd tend to oppose such a truth definition I've given. I think in their opinion that would reduce their sacred arithmetical truths to just syntactical provabilities within Q; and as such, Godel's results would be groundless. > Still, we have the situation that if T |/- F and T |/- ~F then > F and ~F are both false. I assume the "situation" you alluded to below. > This doesn't distinguish refuted > formulae (the negations of proven formulae) from undecidable > formulae. But it does. According to the definition, if F is a refuted formula then ~F is provable (criteria 1). On the other hand, if F is undecidable then both F *and* ~F aren't provable (i.e. false)! There's nothing to be confused about and nothing technically wrong here. In other words such a definition is *just an alias* for the syntactical definitions of provability, reputability, (un)decidability, (in)consistency! >>> Also, I think you have to think about the difference between the >>> concept of a formula F being true in a theory T (being true in [quoted text clipped - 13 lines] > formula provable in T (F is refuted in T). I don't see that > as an imperfection. The imperfection lies in that for complex truth systems, such as arithmetic truth, the canonical definition (which is intuitive) would be treated as *another entirely independent regime of reason*, independent from syntactical provability through inference rules. And that's not only "dangerous" to reasoning it's also wrong as a framework that we could *rely* on - to obtain *new knowledge*. >>> You seem to wish to assert that the truth of a formula F relies >>> on our knowledge (whether we can "tell") of its truth in some model. [quoted text clipped - 4 lines] > What if a formula F has a proof but you don't know that it has > a proof? What about it? Are you saying that one human being must know all 1st order proofs there are? > Is F false until you discover the proof? Or was it > true all along? According to the definition, F is true when T |- F, or false if it's not true. Period. Whether or not one discovers the proof or the un-proof is immaterial to its intrinsic truth value (again, per this definition). You sounded as if "F's being provable" and "one's not knowing its proof" must be contradictory! They're not, unfortunately. That's actually a legacy of FOL (a.k.a. the "Characterization Problem" for F)! > I didn't say "formula", I said "statement". In the context we're discussing, what's the significant difference between a formula and a statement? (We're not talking about meta statements here, are we?) To many "zero is equal to one" is a statement, but that could be converted to the formula "0=S0", given the context. No? > Of course a formula > can assume different truth values depending on the structure in > which you interpret it. This doesn't mean that you automatically > know its truth value in that structure. I didn't say anything about "automatically", did I? I'm not sure I follow your argument here. >>> Whether or to what degree mathematical reality partakes of this >>> objective status is, of course, a venerable debate. [quoted text clipped - 7 lines] > > I don't know what this means. It means basically that, e.g., if you have a problem of *not* knowing a syntactical proof for (F /\ ~F), say, in Q, then *assuming* there's a model of Q is just that: an *assumption*, *not* a fact! > >>>> But the point here is a formula can't be just simply true. You have to [quoted text clipped - 13 lines] > That's not what I said. An uninterpreted formula has no meaning > or truth value. Statements have truth values. There's no such thing as just a true formula: usually it's understood by that we mean an interpreted-as-true formula. As such an interpreted formula needs to have a meaning. But not the other way around. For example, if the formula F = (0=0) /\ (0=1) has a meaning, its meaning is *independent* from whether or not F has any truth value! >> For example, >> GC has some meaning but whether or not it's true or false (by whatever the [quoted text clipped - 10 lines] >>> and that a formula has to be known to be true (by some criteria of truth) >>> in order for it to be true (by that criteria). The above is what you stipulated as one of my issue. >> That's I think is meant by something like "mathematical truth is relative", >> and is what I'd like to convey. But that in itself is not an "issue", imho. A slight mis-communication here, imho. (Though I did say "like"). Had your statement above been "a formula has to be known to be true (by some criteria of truth) in order for it to be *asserted as* true (by that criteria)", then that wouldn't be an issue (mine or otherwise), and I'd agree, in saying "mathematical truth is relative". But I now realize that's not what you really said. So... > You wish to say that mathematical truth of a statement is > relative not only to the chosen criteria of truth, but > also to whether we know the statement meets the criteria? that's not exactly what I wish to convey: "not only" isn't part of what I intended to say! > IOW, a statement is false if it fails to meet the chosen > truth definition, but also if we don't know whether it meets > the truth definition or not? A formula is false when "it fails to meet the chosen truth definition", whether or not we know about that is another issue all together. Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > There's no such thing as just a true formula: usually it's understood > by that we mean an interpreted-as-true formula. As such an interpreted [quoted text clipped - 5 lines] > >> GC has some meaning but whether or not it's true or false (by whatever the > >> chosen underlying truth definition) is a different matter. Well this is a technical question of some interest. I was under the impression that interpreting a formula in a given structure entailed a truth-value assignment also. Perhaps I am mistaken on this point. Can you explain how a formula F can be assigned a meaning (interpreted) without thereby arriving a truth-value for F? I'm, admittedly, slightly fuzzy on this point. -- hz >> There's no such thing as just a true formula: usually it's understood >> by that we mean an interpreted-as-true formula. As such an interpreted [quoted text clipped - 8 lines] > the impression that interpreting a formula in a given structure > entailed a truth-value assignment also. That's correct. Interpreting a formula is basically assigning the formula to a binary value (e.g. T or F). The only thing here is by "a formula", we by default mean "a meaningful formula", because every formula is a wff and every wff would have at least one meaning: the combination of meanings of the individual meanings of logical and non-logical symbols. (The caveat here is that one could do away with meaning, hence with truth, all together in FOL!) For example, consider the formula: (1) Ax[~(Sx=0)]. One could view FOL reasoning as nothing more than just a game of symbols, a manipulation of symbols, and in such view "Ax","~","S","=","0" don't mean anything, and yet you could infer the following formula as a consequence of the rules of the manipulation (i.e. rules of inference), given (1): (2) Ex[~(Sx=0)] On the other hand, if we *choose* to assign all of FOL symbols to canonical meanings then (1) would "mean", roughly in English: (1') For all x's, it's not the case that (x,0) is in the binary relation S, which is also an unary function. But the meaning (1') is not yet associated with (assigned to) any particular binary value! > Perhaps I am mistaken on this point. Imho, by *convenience*, we tend to treat a true statement as meaningful, while a false one meaningless. For example, "The Sun rises in the East" is true and meaningful, while "The Sun rises in the West" is false *and* meaningless! But from FOL's point of view, if we care about "meaning" at all for a formula, then *all* of these: "President Kennedy was assassinated", "President Kennedy is alive", etc... are *meaningful*. Whether any of them is true, or false, will be a matter of interpretation, or assignment, w.r.t. to a binary (truth) value. But the statements: "President Kennedy was 'done-in'", "It's not true somebody 'did him in'",... are meaningless, and can't be assigned to a truth value, until we know exactly what the syntactical language/grammar roles would 'done-in' and 'did him in' play! Imho. > Can you explain how a formula F can be assigned > a meaning (interpreted) without thereby arriving a truth-value > for F? I'm, admittedly, slightly fuzzy on this point. I hope the above would clarify the situation. Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > Imho, by *convenience*, we tend to treat a true statement as meaningful, > while a false one meaningless. For example, "The Sun rises in the East" > is true and meaningful, while "The Sun rises in the West" is false *and* > meaningless! I don't consider false statements to be meaningless. And I hardly doubt there is a consensus (especially among philosophers and mathematicians) that false statements are meaningless. Indeed, certain philosophers and mathematicians are clear that false statements DO have meaning. > But from FOL's point of view, if we care about "meaning" > at all for a formula, then *all* of these: [quoted text clipped - 11 lines] > exactly what the syntactical language/grammar roles would 'done-in' and > 'did him in' play! Imho. I don't especially disagree with that. But I don't see reason to accept your opinion that generally people take falsehoods to be meaningless. MoeBlee > >> There's no such thing as just a true formula: usually it's understood > >> by that we mean an interpreted-as-true formula. As such an interpreted [quoted text clipped - 35 lines] > But the meaning (1') is not yet associated with (assigned to) any particular > binary value! OK -- Now if we specify the extension of the binary relation S such that for some domain of objects D which includes 0 as an element we have that the pair (d_i, 0) is not an element of S for any element d_i of D, can we now say that (1') is true w.r.t. D? That is to say, is the truth value of (1') settled when we specify a domain D and the extension of S? -- hz > > There's no such thing as just a true formula: usually it's understood > > by that we mean an interpreted-as-true formula. As such an interpreted [quoted text clipped - 12 lines] > a meaning (interpreted) without thereby arriving a truth-value > for F? I'm, admittedly, slightly fuzzy on this point. A structure gives the truth value of SENTENCES but not of formulas in general. A structure PLUS an assignment for the individual variables does provide for a formula to be satisfied or not satisfied by that structure with that assignment for the individual variables. And a structure alone does provide for whether a formula is satisfiable or not satisfiable per that structure. But for truth or falsehood in a structure onto itself, we have that only for sentences. I don't opine as to what herbzet has in mind by 'meaning', but, aside from intensional meaning, at least the extensional meaning of a formula per a structure might be taken to be the set of assignments for the individual variables under which the formula is satisfied by that structure. MoeBlee >>> There's no such thing as just a true formula: usually it's understood >>> by that we mean an interpreted-as-true formula. As such an interpreted [quoted text clipped - 26 lines] > for the individual variables under which the formula is satisfied by > that structure. Are you responding to herbzet, or Nam? > MoeBlee Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >>> There's no such thing as just a true formula: usually it's understood > >>> by that we mean an interpreted-as-true formula. As such an interpreted [quoted text clipped - 28 lines] > > Are you responding to herbzet, or Nam? To herbzet. It was my mistake in the second paragraph to refer to him in the third person. MoeBlee > > > There's no such thing as just a true formula: usually it's understood > > > by that we mean an interpreted-as-true formula. As such an interpreted [quoted text clipped - 15 lines] > A structure gives the truth value of SENTENCES but not of formulas in > general. Yes, I've been talking sloppy. I meant, basically, sentences. We're discussing truth, so we need sentences if we want to have bearers of truth-values. > A structure PLUS an assignment for the individual variables does > provide for a formula to be satisfied or not satisfied by that [quoted text clipped - 8 lines] > for the individual variables under which the formula is satisfied by > that structure. Yes, I meant extensional meaning for sentences. Don't mix me up talking about satisfaction of open formula (wff). Once we have assigned an extension to the predicate symbols of the language and the individual constants (if any), drawn from a non-empty domain D over which the quantifiers are presumed to range, is the truth value of the sentences of the language thereby settled? -- hz > >> With the revised above, I hope you'd be convinced to change your mind. > [quoted text clipped - 5 lines] > that would reduce their sacred arithmetical truths to just syntactical > provabilities within Q; and as such, Godel's results would be groundless. Suppose we define 'true' as 'provable' as you suggest. Then we can define 'troo' as we had previously defined 'true'. Now we're right back to the usual results in mathematical logic (including variations on the incompleteness theorem that mention 'true') except 'troo' occurs where 'true' used to appear. > > Still, we have the situation that if T |/- F and T |/- ~F then > > F and ~F are both false. [quoted text clipped - 11 lines] > words such a definition is *just an alias* for the syntactical definitions > of provability, reputability, (un)decidability, (in)consistency! But why would we WANT an alias? We have clear definitions of those those things. Why can't we have a different definition of 'true', I mean 'troo'? And doesn't "Both F and ~F are both false" seem at least a bit odd to you as far as an ordinary non-technical meaning of 'false'? And I think what herbzet meant is that, with your approach, both a refuted formula and an undecidable formula both come out as false, which does seem at least counterintuitive. > >>> Also, I think you have to think about the difference between the > >>> concept of a formula F being true in a theory T (being true in [quoted text clipped - 20 lines] > "dangerous" to reasoning it's also wrong as a framework that we could > *rely* on - to obtain *new knowledge*. But we can formalize the mathematical defintion of 'truth' so that we can formally prove that certain sentences are true or false in certain models. Of course, we don't have an algorithm to determine such questions; but we don't have an algorithm to determine questions of plain first order provability either. MoeBlee >>>> With the revised above, I hope you'd be convinced to change your mind. >>> It's pretty standard as amended, other than the fact that you [quoted text clipped - 6 lines] > Suppose we define 'true' as 'provable' as you suggest. Then we can > define 'troo' as we had previously defined 'true'. What do you mean by "Then" here? Anybody can define 'true' in any which way and create an alias ('troo', 'truthful', ...) *at will*, so "Then" isn't necessary. The point I made here is "provable" is a well established syntactical notion in FOL; so if we use it to define 'true', as opposed to the canonical definition of 'true' which is based on intuition, then we have to accept certain consequences. And I've mentioned these consequences in the above sentence "I think ... Godel's results would be groundless." > Now we're right > back to the usual results in mathematical logic (including variations > on the incompleteness theorem that mention 'true') except 'troo' > occurs where 'true' used to appear. No! Not if Incompleteness results we're talking about. You misunderstood the issue. Here, 'troo' is an alias for the old notion of 'true', while my 'true' is an alias for 'provable', but the old 'true' and the new 'true' *aren't* of the same sense. And whether we'd arrive at the same Godel's results would depend *which definition* of 'true'/'false' we're using. >>> Still, we have the situation that if T |/- F and T |/- ~F then >>> F and ~F are both false. [quoted text clipped - 12 lines] > But why would we WANT an alias? We have clear definitions of those > those things. The aliases are here only to demonstrate to the readers that, for *better* reasoning, we should base mathematical assertions on syntactical provability rather than the old intuitive notion of "truth". For example, to prove F is undecidable, with 100% certainty, we should prove so using *only* syntactical rules of inferences and axioms! > Why can't we have a different definition of 'true', I mean 'troo'? We'd want to have a different and new definition of 'true' because the old one isn't adequate. But your "Why can't we have a different definition of 'troo'?" wouldn't quite make sense, because there isn't such a thing as the old definition of 'troo'! > And doesn't "Both F and ~F are both false" seem at least a bit odd to > you as far as an ordinary non-technical meaning of 'false'? No. We're supposed to live in a binary (logic) world aren't we? If an F isn't provable then got to be non-provable. So, in this sense, if F isn't true then it got to be false. Nothing is odd, it seems to me! > And I think what herbzet meant is that, with your approach, both a > refuted formula and an undecidable formula both come out as false, [quoted text clipped - 24 lines] > can formally prove that certain sentences are true or false in certain > models. Any of us can formalize a particular of 'truth' and use it to "prove" something else. That's not the problem! The problem is if such truth definition has anything to do with a particular formal system T, then how well would such definition would ryhme with the consistency of T? I believe my definition does very well but the canonical doesn't! > Of course, we don't have an algorithm to determine such > questions; but we don't have an algorithm to determine questions of > plain first order provability either. Correct. But that's why if we assume the natural is a model of Q then we have to admit that's just an assumption and *not a proof*. Consequently, Godel's results are hypothetical results, not assertions! For them to be assertions, the required additional hypothesis would be: "If Q is syntactically consistent" Godel didn't stipulate that one hypothesis. And neither have we, after 70+ years! Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >>>> With the revised above, I hope you'd be convinced to change your mind. > >>> It's pretty standard as amended, other than the fact that you [quoted text clipped - 49 lines] > certainty, we should prove so using *only* syntactical rules > of inferences and axioms! If a theory T proves a sentence is undecidable in T, then T is inconsistent. (Not actually sure this is correct ("true") in complete generality.) > > Why can't we have a different definition of 'true', I mean 'troo'? > [quoted text clipped - 64 lines] > we must therefore examine the methods of the mathematician." > (Shoenfield, "Mathematical Logic") >> The aliases are here only to demonstrate to the readers that, >> for *better* reasoning, we should base mathematical assertions [quoted text clipped - 6 lines] > inconsistent. (Not actually sure this is correct ("true") in > complete generality.) Note that I said "we should prove" but didn't say "we can prove". The point being is *in general* if you can't syntactically prove F is undecidable, then no matter what else you might say using the canonical definition of truth it wouldn't be sufficient. (And I can demonstrate this, but perhaps in different post, if you'd like to). But I've never said or believed we can always syntactically prove an undecidability. Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >> The aliases are here only to demonstrate to the readers that, > >> for *better* reasoning, we should base mathematical assertions [quoted text clipped - 15 lines] > But I've never said or believed we can always syntactically prove > an undecidability. If one can prove in T' that there is a proof in T of the formula F, then naturally the proof in T' should be completely formalizable. In fact, we wouldn't accept it _as_ proof if we didn't think it was reducible to a completely formal proof. No argument there. That holds for any proof, come to think of it. -- hz >>>> The aliases are here only to demonstrate to the readers that, >>>> for *better* reasoning, we should base mathematical assertions [quoted text clipped - 16 lines] > If one can prove in T' that there is a proof in T of the formula F, > then naturally the proof in T' should be completely formalizable. Right. The proof in T' is just a normal proof, like countless other 1st order proof! > In fact, we wouldn't accept it _as_ proof if we didn't think > it was reducible to a completely formal proof. No argument > there. What does "it" refer to, here? Nonetheless, I'd agree: a proof is a proof, and no argument from me either! > That holds for any proof, come to think of it. Sure: any proof is a proof! What's the point of debate here though? Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >>>> The aliases are here only to demonstrate to the readers that, > >>>> for *better* reasoning, we should base mathematical assertions [quoted text clipped - 25 lines] > > What does "it" refer to, here? The proof in T'. > Nonetheless, I'd agree: a proof is a proof, > and no argument from me either! [quoted text clipped - 4 lines] > > What's the point of debate here though? You are arguing against the practice of giving intuitive, unformalizable arguments as proof, a practice which no one is advocating. -- hz > You are arguing against the practice of giving intuitive, > unformalizable arguments as proof, a practice which no one > is advocating. Your statement could be interpreted in 2 ways. First way: The "practice of giving intuitive unformalizable arguments as proof" is a practice "which no one "is advocating". And I'm arguing against such a dismissed practice. So I must be doing something right, correct? Second way: My "arguing against the practice..." is something "no one is advocating". So I'm doing something not popular here. But why should "intuitive, unformalizable arguments" be accepted as proof, while *proof* has a very rigor definition that everyone already accepts? Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > > You are arguing against the practice of giving intuitive, > > unformalizable arguments as proof, a practice which no one [quoted text clipped - 7 lines] > dismissed practice. So I must be doing something > right, correct? The above is the intended interpretation. > Second way: My "arguing against the practice..." is something > "no one is advocating". So I'm doing something not > popular here. But why should "intuitive, unformalizable > arguments" be accepted as proof, while *proof* has a > very rigor definition that everyone already accepts? -- hz >>> You are arguing against the practice of giving intuitive, >>> unformalizable arguments as proof, a practice which no one [quoted text clipped - 8 lines] > > The above is the intended interpretation. Unfortunately, that's how we've done reasoning after Hilbert's Program. I'm not defending Hilbert's one-size-catch-all formal system, only his syntactical provability. What we've done since Godel is to replace that formal system with one-encoding-size-catch-all arithmetic (interpretation) truth system. Talking about avoid one "devil", just to meet another one! >> Second way: My "arguing against the practice..." is something >> "no one is advocating". So I'm doing something not [quoted text clipped - 4 lines] > -- > hz Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >>> You are arguing against the practice of giving intuitive, > >>> unformalizable arguments as proof, a practice which no one [quoted text clipped - 14 lines] > Godel is to replace that formal system with one-encoding-size-catch-all > arithmetic (interpretation) truth system. I'm not sure what you're talking about with this last sentence. Could you expand a bit on what you mean by "What we've done since Godel is to replace that formal system with one-encoding- size-catch-all arithmetic (interpretation) truth system"? -- hz > Talking about avoid one "devil", just to meet another one! > [quoted text clipped - 11 lines] > we must therefore examine the methods of the mathematician." > (Shoenfield, "Mathematical Logic") >> Unfortunately, that's how we've done reasoning after Hilbert's >> Program. I'm not defending Hilbert's one-size-catch-all formal [quoted text clipped - 7 lines] > since Godel is to replace that formal system with one-encoding- > size-catch-all arithmetic (interpretation) truth system"? Let's just say the formal system in question, for undecidability, is ZF and the suspected undecidable formula in ZF is G(ZF). Suppose now we adhere to Hilbert's syntactical-ism where we don't have the concept of arithmetical truth and where formulas are devoid of meanings; and proving theorems is exactly nothing more than just a manipulation of strings of symbols, based on some syntactical rules of inference. Then G(ZF)'s undecidability means ZF proves neither G(ZF) nor ~G(ZF). But how would we demonstrate a system T not prove a particular F in general? After all, a formal system is designed to prove formulas, not to un-prove them! The long and short of it is, we've learned through Godel's work, there's some "faint" hope if we somehow encode G(ZF) and ~G(ZF), using an "external system", say, named Ar (for Arithmetic), and using some of Ar's properties, we'd perhaps have some satisfactory (mental) mapping between these properties and the suspected unprovability of both G(ZF) and ~G(ZF). But what's exactly the nature of Ar, as an "external system"? Well, if we stay with Hilbert's syntactical-ism then it must be just another formal system. But in this case, we'd come the a full circle: how could we in Ar demonstrate encoded(G(ZF)) and ~encoded(G(ZF)) aren't provable? The long and short of it is, if we could, Ar would be that "one-size-catch-all" formal system, because *all* the similarly constructed encoded(G(T))'s and ~encoded(G(T))'s [with all the Ts satisfying some conditions of course] would be proven in Ar; (and I think PM would be the natural choice for this external encoding Ar, at the time). In a nutshell, Godel couldn't have used PM to encode G(ZF), ~G(ZF), or what had he: because of the circularity. What he did, probably through some flash of intuition, is to use some key strategies, instead: (1) Having alternative definition of "undecidability" using the concept of model, which is based on intuition and not on Hilbert's syntactical-ism: G(T) would be undecidable if it's true in a model of T, but false in another one. (This in itself isn't new: we did that with the the 5th postulate). (2) Mapping the model truths and falsehoods in T to those in a "fictitious formal" system Ar. (3) Mentally (or rather in a meta level), ignore/remove the complete/full formality of Ar, and retain only its models: at least one of its models we'd deem as "the standard model of arithmetic". In a nutshell, that's my summary of how we end-up replacing a would-be- "one-size-catch-all" formal by aother "one-encoding-size-catch-all" "the-standard-model-of-arithmetic": which is really just the natural numbers, collectively! Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > In a nutshell, Godel couldn't have used PM to encode G(ZF), ~G(ZF), > or what had he: because of the circularity. What he did, probably > through some flash of intuition, is to use some key strategies, instead: Please replace "couldn't have used PM to encode G(ZF), ~G(ZF)" with "couldn't have used PM to prove the encoded G(ZF), ~G(ZF)". Thanks. >>> Unfortunately, that's how we've done reasoning after Hilbert's >>> Program. I'm not defending Hilbert's one-size-catch-all formal [quoted text clipped - 57 lines] > "the-standard-model-of-arithmetic": which is really just the natural > numbers, collectively! [The above is with some typo-fixes.] Of course, I meant there's a problem in this kind of replacement: fundamentally, if we can't know through provability, we can't simply just replace that un-knowledge with a mere hand-waving "assumed" intuition-knowledge. At least for a complex provability-issue such as the encoded(G(T)) or its negation. But let me try to explain that in another post. Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >> Unfortunately, that's how we've done reasoning after Hilbert's > >> Program. I'm not defending Hilbert's one-size-catch-all formal [quoted text clipped - 43 lines] > of model, which is based on intuition and not on Hilbert's > syntactical-ism: Why do you say that using the concept of "model" is based on intuition? > G(T) would be undecidable if it's true in a model > of T, but false in another one. (This in itself isn't new: we did [quoted text clipped - 11 lines] > "the-standard-model-of-arithmetic": which is really just the natural numbers, > collectively! -- hz >>>> Unfortunately, that's how we've done reasoning after Hilbert's >>>> Program. I'm not defending Hilbert's one-size-catch-all formal [quoted text clipped - 43 lines] > > Why do you say that using the concept of "model" is based on intuition? Because model is based on "interpretation" which is subjective: subjective to what an individual would intuit. For example, you're looking for a model of the T1 = {a < b}, while I'm looking for one of T2 = {~(a < b)}. Now, say, we both come across the a particular ordered-pair (c,d) (which is, say, a ZF set). Now to you, your intuition might map (c,d) to the formula (a < b), but to me my intuition would map the very same set (c,d) to ~(a < b). So for the same set (c,d), we now have 2 different models, simply because we have *2 opposite but valid intuitions*! In brief, model is based on truth, and truth can legitimately vary per (subjective) intuitions. On the other hand, e.g., from T1 = {a < b}, we can prove the theorem Ex(x < b). But each of us either knows or doesn't know the proof, but we could never conclude *2 opposite theorems*. In brief, unlike model truth, provability is based on syntactical rules and formulas. Even if each of us has different intuitions about a theorem, in the end it isn't anyone's intuition that has the final say if indeed a formula is a theorem. >> G(T) would be undecidable if it's true in a model >> of T, but false in another one. (This in itself isn't new: we did [quoted text clipped - 14 lines] > -- > hz Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >>>> Unfortunately, that's how we've done reasoning after Hilbert's > >>>> Program. I'm not defending Hilbert's one-size-catch-all formal [quoted text clipped - 54 lines] > model is based on truth, and truth can legitimately vary per (subjective) > intuitions. Such mappings are just mathematical functions. It is subjective as to which functions one may wish to study, but it is not subjective as to what functions do exist. In that sense there is nothing subjective about models that isn't subjective about functions generally in mathematics. Again, a structure for a language is just a certain kind of function (or something similar depending on the author). That different people are interested in different structures makes such mappings no more subjective than are functions generally in mathematics subjective merely because different people prefer to study different functions and different kinds of functions too. > On the other hand, e.g., from T1 = {a < b}, we can prove the theorem > Ex(x < b). But each of us either knows or doesn't know the proof, but > we could never conclude *2 opposite theorems*. In brief, unlike model > truth, provability is based on syntactical rules and formulas. No, "Sentence S is true in model M" is itself a mathematical statement that is subject to proof and refutation. There is no subjectivity as to whether a given sentence is true in a given model. There is subjectivity as to what models one may wish to study, but that is no more a subjectivity than as to what functions in mathematics one wishes to study. > Even if > each of us has different intuitions about a theorem, in the end it isn't > anyone's intuition that has the final say if indeed a formula is a > theorem. And it is never any person's intuition that is the final say as to whether a given sentence is true in a given model. MoeBlee > > Why do you say that using the concept of "model" is based on intuition? > [quoted text clipped - 5 lines] > (a < b), but to me my intuition would map the very same set (c,d) to > ~(a < b). If 'a' is mapped to c, and 'b' is mapped to d, and '<' is mapped to a set of ordered pairs that includes (c,d), then the formula (a < b) is true per the mapping. If 'a' is mapped to c and 'b' is mapped to d, and '<' is mapped to a set of ordered pairs that does *not* include (c,d), then ~(a < b) is true per this different mapping. We could map 'a' to d and 'b' to c, and if '<' is mapped to a set of ordered pairs that does *not* include (d,c) (but maybe includes (c,d)) then ~(a < b) will be true per _this_ mapping. Etc. It's not clear what is meant by mapping either formula to (c,d). We map constants to elements of some domain D and predicates to ordered sets of elements of D. > So for the same set (c,d), we now have 2 different models, You're saying that the set (c,d) is a model of two different theories. > simply because we have *2 opposite but valid intuitions*! Don't know what intuitions have to do with it. > In brief, > model is based on truth, and truth can legitimately vary per (subjective) > intuitions. The truth of a sentence will vary in different models, yes. > On the other hand, e.g., from T1 = {a < b}, we can prove the theorem > Ex(x < b). But each of us either knows or doesn't know the proof, but [quoted text clipped - 3 lines] > anyone's intuition that has the final say if indeed a formula is a > theorem. MoeBlee already gave a good reply here. Truth in a structure is a rigorously defined concept, not a matter of intuition. -- hz Correction: > We map constants to elements of some domain D and predicates > to [sets of] ordered sets of elements of D. Also, we map n-ary function symbols to sets of ordered sets of (n+1) elements of D, with the provision that blah blah blah. -- hz >>> Why do you say that using the concept of "model" is based on intuition? >> Because model is based on "interpretation" which is subjective: subjective [quoted text clipped - 21 lines] > We map constants to elements of some domain D and predicates > to ordered sets of elements of D. Let me clarify the mapping a bit further. But for a reason that would be clear later, let's use the 2 language constants 'e', 'm' instead of 'a', 'b'. Also, let T1 = {e < m}, T2 = {~(e < m)}. Now let's consider these 4 mappings: (1) ((e < m), R={(c,d}}) -> True [This basically says R is a model of T1] (2) ((e < m), R={(c,d}}) -> False [This basically says R isn't a model of T1] (3) (~(e < m),R={(c,d}}) -> True [This basically says R is a model of T2] (4) (~(e < m),R={(c,d}}) -> False [This basically says R isn't a model of T2] >> So for the same set (c,d), we now have 2 different models, > > You're saying that the set (c,d) is a model of two different > theories. What I says is (1) and (2). What you said is (1) and (3). The 2 situations aren't quite the same, in general. >> simply because we have *2 opposite but valid intuitions*! > > Don't know what intuitions have to do with it. Think of (1) and (2): you're free to choose R as model of T1 - *or not*! Does FOL reasoning framework *force you to choose one*, at the expense of the other? Of course not. Your intuition is free to choose whichever between (1) and (2) that it would behold! Think of these mappings in the high level as the following: (1') The Moon is above the Earth is true (2') The Moon is above the Earth is false (3') Not the Moon is above the Earth is true (4') Not the Moon is above the Earth is false Your intuition is free to accept any of the 4 above, on 2 fundamental grounds: - You could take the perspective of being on Earth, or on Moon. - You could interchange the semantics of Earth or Moon: By 'Earth' you mean the Moon, and vice versa! But either ground is based on subjective intuition that one would have, or choose to have. >> In brief, >> model is based on truth, and truth can legitimately vary per (subjective) >> intuitions. > > The truth of a sentence will vary in different models, yes. So you and MoeBlee would agree with me then, without subjective intuition, truth can *not vary*? > >> On the other hand, e.g., from T1 = {a < b}, we can prove the theorem [quoted text clipped - 7 lines] > MoeBlee already gave a good reply here. Truth in a structure is > a rigorously defined concept, not a matter of intuition. "Rigorously defined concept" still doesn't help that between any 2 thinkers, one could choose (1) while the other (2), which of course is a *matter of intuition*. In brief, Moeblee was wrong here. What he seems to fail to realize is that the very definition of true sentences would depend on the *intuition* of the natural numbers, hence can not be a "rigorously defined concept"! If arithmetic of the natural numbers were rigorously well defined, then we'd as well consider the word "arithmetic" is nothing more than just an alias of the formal system Q (a la Shoenfield's book), which is of course not the case. Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > >>> Why do you say that using the concept of "model" is based on intuition? > >> [quoted text clipped - 29 lines] > > (1) ((e < m), R={(c,d}}) -> True [This basically says R is a model of T1] It says nothing of the sort. Apparently you are unclear on what a model is, despite my examples given *immediately above*. For a simple discussion (in the context of defining the concept of logical consequence) please see http://www.iep.utm.edu/l/logcon-m.htm . > (2) ((e < m), R={(c,d}}) -> False [This basically says R isn't a model of T1] > (3) (~(e < m),R={(c,d}}) -> True [This basically says R is a model of T2] [quoted text clipped - 7 lines] > What I says is (1) and (2). What you said is (1) and (3). The 2 situations > aren't quite the same, in general. What you are saying is in your own private language. I gave three different interpretations of the formulae. > >> simply because we have *2 opposite but valid intuitions*! > > > > Don't know what intuitions have to do with it. > > Think of (1) and (2): you're free to choose R as model of T1 - *or not*! We are free to interpret the predicate symbol '<' so that R is a model of '<' or not. We will need to also interpret the constants 'e' and 'm' in order to determine the truth of T1. It is incomplete and just meaningless to say that R is or is not a model of T1. It is therefore meaningless to say that we're free to choose R as a model of T1, since it is meaningless to say that R is or is not a model of T1. A model of T1 requires more than an interpretation of the predicate symbol. Aside from all that technical business, you are again using language in an idiosyncratic way when you assert that, because there are a variety of ways to interpret a language, that interpreting a language is a matter of intuition. An interpretation of a formal language is a perfectly rigorous and well-defined object. Also, you use far too many exclamation points. > Does FOL reasoning framework *force you to choose one*, at the expense > of the other? Of course not. Your intuition is free to choose whichever > between (1) and (2) that it would behold! Yes, we have a choice of interpretations for a given sentence or theory. So what? > Think of these mappings in the high level as the following: > [quoted text clipped - 9 lines] > - You could interchange the semantics of Earth or Moon: By 'Earth' you > mean the Moon, and vice versa! Your arguments are one long relentless insistence on the equivocal nature of words. Wonderful point -- words, and formulae, can be interpreted to mean different things. Can we move on now? > But either ground is based on subjective intuition that one would have, > or choose to have. So is "You are a giant a.s." > >> In brief, > >> model is based on truth, and truth can legitimately vary per (subjective) [quoted text clipped - 4 lines] > So you and MoeBlee would agree with me then, without subjective intuition, > truth can *not vary*? I can't speak for MoeBlee, but you appear to be making the entirely trivial point that the truth of "The Empire State building is in New York" is a matter of "subjective intuition" because the sentence could be interpreted as meaning "The Grand Canyon is in France". You wish to assert that Ex[x < m] is true in theory T1 = (e < g) because there is a deduction from the latter to the former. But this is precisely to assert the relativity of the truth of the formula to a theory (rather than a model) since in a different theory the formula will not be deducible. It is no more an absolute notion of truth than that of truth-in-a-model. > >> On the other hand, e.g., from T1 = {a < b}, we can prove the theorem > >> Ex(x < b). But each of us either knows or doesn't know the proof, but [quoted text clipped - 10 lines] > one could choose (1) while the other (2), which of course is a *matter > of intuition*. I choose to interpret your words as saying "Nam is a giant a.s." > In brief, Moeblee was wrong here. What he seems to fail to realize is that > the very definition of true sentences would depend on the *intuition* of the [quoted text clipped - 4 lines] > of the formal system Q (a la Shoenfield's book), which is of course not the > case. I think I'll forgo asking what is the case, since what you might mean would be entirely a matter of "intuition". -- hz > I choose to interpret your words as saying "Nam is a giant a.s." It's really pointless to discuss technical issues at foundation of mathematical reasoning with you, herbzet , given your "insight" above. Bye. Signature"To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician." (Shoenfield, "Mathematical Logic") > > I choose to interpret your words as saying "Nam is a giant a.s." > [quoted text clipped - 3 lines] > > Bye. What's the problem? Just interpret my words as you like. -- hz > > MoeBlee already gave a good reply here. Truth in a structure is > > a rigorously defined concept, not a matter of intuition. > > "Rigorously defined concept" still doesn't help that between any 2 thinkers, > one could choose (1) while the other (2), which of course is a *matter An intrepretation for a language is a mathematical function. Two thinkers could diverge as to which mathematical function is more of interest to them. They might have subjective, intuitive, or philosophical reasons for preferring to study one interpretation as opposed to another; they might even consider a particular interpretation to be a reasonable, fruitful, or even a correct one. But the actual intrepretations themselves are mathematical objects; they're functions. And the definition of 'the sentence S is true in the model M' is a rigorous mathematical definition. > In brief, Moeblee was wrong here. What he seems to fail to realize is that > the very definition of true sentences would depend on the *intuition* of the > natural numbers, hence can not be a "rigorously defined concept"! No, because I NEVER MENTIONED a definition of 'true sentence'. What I have talked about is 'true sentence PER a given model'. And the definition of THAT is rigorous. Of course, the definition may be motivated by an intuition or sense or even conviction as to what it means for a sentence to be true per a given state of affairs. But the definition itself is entirely rigorous mathematics. MoeBlee > > > MoeBlee already gave a good reply here. Truth in a structure is > > > a rigorously defined concept, not a matter of intuition. [quoted text clipped - 22 lines] > means for a sentence to be true per a given state of affairs. But the > definition itself is entirely rigorous mathematics. The lack of rigor in your assertions above actually sticks out like a sore thumb. You have not said what it means to be "given" a model. Can you tell me what is a "given" model? This is where all the action is. I maintain that you can be "given" a model if and only if *all* the truths in that model are defined, by YOU. It is the truths in the model that actually define the model. But then it is really silly to talk of a "true sentence PER a given model". Because the truth of that sentence, if it does not follow via provability from the axioms of the theory (whose model you are considering), is something that YOU need to specify before you can be "given" a model. If you accept this, then you have accepted the fundamental premise of NAFL outlined below. Unless, like all classical logicians/thinkers, you are tacitly assuming that models are already "out there" and you happen to be "discovering" the truths in that model, which you have somehow been "given". Which is, at the very least, non-rgorous and at worst, nonsensical. The position that I have outlined above is the NAFL position, of course. Models are not "pre-existing" objects, but actually constructed via axiomatic assertions made in the human mind. So if you consider a model M of a NAFL theory T, this model is generated by the "interpretation" T* of T. Here T* is also a NAFL theory which must prove all the axioms of T and could decide sentences that are undecidable in T. Sentences that are undecidable in T* are assigned a truth value "neither true nor false" in the model M. Thus you can see that the truths in the model M of a theory T are completely defined by this procedure. Here it is taken for granted that every sentence in the language of T* (or T) is either provable or refutable or undecidable in T*, even if at the moment we do not know which is the case. But the fact remains that the T* is decided by the free will of the human mind and *that* is what it means to be "given" a model in NAFL. We have *specified* the model by specifying T*. Now where can I find a logician/philosopher honest enough to openly acknowledge and discuss NAFL? Regards, RS >> > > MoeBlee already gave a good reply here. Truth in a structure is >> > > a rigorously defined concept, not a matter of intuition. [quoted text clipped - 28 lines] > sore thumb. You have not said what it means to be "given" a model. Can > you tell me what is a "given" model? This is where all the action is. Good grief. He simply means "any arbitrary model". > I maintain that you can be "given" a model if and only if *all* the > truths in that model are defined, by YOU. It is the truths in the > model that actually define the model. No, what defines the model is its domain and its assignment of semantic values to the names and predicates of the language it is interpreting. You appear not to understand classical model theory. > But then it is really silly to talk of a "true sentence PER a given > model". To the contrary, it is perfectly coherent in the context of classical model theory. > Because the truth of that sentence, if it does not follow via > provability from the axioms of the theory (whose model you are > considering), is something that YOU need to specify before you can be > "given" a model. If you accept this, then you have accepted the > fundamental premise of NAFL outlined below. You are simply using the word "model" in a completely different sense than the way it is used in classical model theory. How about we call what *you* are talking about "schmodels". Then it will be clear that you simply want to talk about a different subject. That's fine. But all you do is muddy very clear waters when you fail to clarify that you are using words differently than the people you are talking to, and suggest that perfectly simple theorems of classical model theory are false or lack rigor. > Unless, like all classical logicians/thinkers, you are tacitly > assuming that models are already "out there" and you happen to be > "discovering" the truths in that model, which you have somehow been > "given". Which is, at the very least, non-rgorous and at worst, > nonsensical. Model theory is done in some implicit set theory like ZF. Everything Moeblee has said follows rigorously from ZF and relevant definitions, and is completely independent of any philosophical stance toward the objective truth of ZF and the objective existence of models. It is in fact *your* views that are deeply biased in a very specific philosophical direction. > The position that I have outlined above is the NAFL position, of > course. Models are not "pre-existing" objects, but actually > constructed via axiomatic assertions made in the human mind. Well, fine and dandy. So you have a *different* view -- though, unlike classical model theory, one that appears to be intrinsically non-rigorous, if not nonsensical, because, unlike the classical model theorist, you refuse to provide a mathematically rigorous formalization of your theory. > Now where can I find a logician/philosopher honest enough to openly > acknowledge and discuss NAFL? I'm sure lots (well, a few, at least) of them would be happy to if only you were willing to state the axioms of your metatheory and provide rigorous definitions of such notions as "truth" and "model". Short of that there just isn't anything to discuss with sufficient rigor. A lack of honesty just isn't the problem here. > > I NEVER MENTIONED a definition of 'true sentence'. What I > > have talked about is 'true sentence PER a given model'. And the [quoted text clipped - 6 lines] > sore thumb. You have not said what it means to be "given" a model. Can > you tell me what is a "given" model? This is where all the action is. Oh for godsakes, you are truly ridiculous. The word 'a given' is ordinary English, in such contexts, for 'a particular'. By saying 'given' I'm only emphasizing that sentences are evaluated as true or false only per a model and not just true or false simply. You want rigor, then it's as simple as this: S is true in M <-> (S is a sentence & M is a model for the langauge of S & S is true in M) where each of the rubrics 'sentence', 'model for the language', and 'true in' are given as ordinariy in textbooks in mathematical logic. And my point to Nam is that I have never proferred a definition for plain S is true. Got it now? (Though, I do have a notion of plain 'true' for certain kinds of finitistic statements. But that is merely an informal explanatory notion I have, and, for me personally, not even a doctrine or thesis.) > I maintain that you can be "given" a model if and only if *all* the > truths in that model are defined, by YOU. It is the truths in the > model that actually define the model. Then you are utterly confused about this basic matter of mathematical logic. A model is a certain kind of function (or a function is part of the model, depending on the author's exact definition). For example, using Enderton's defintion: A model maps the universal quantifier to a non-empty set, and each n-place predicate symbol to an n-place relation on that set, and each n-ary function symbol to an n-ary function on the set. (I'll use that formulation from now on in this discussion. Other formulations can be accomodated for just by simple adjustments to my remarks.) THEN whether a sentence is true or not in the model is determined by the recursive definition (a la Tarski) of 'true in the model'. It is NOT the case that we define a model by what sentences are true in that model. That would make no sense. > But then it is really silly to > talk of a "true sentence PER a given model". Because the truth of that > sentence, if it does not follow via provability from the axioms of the > theory (whose model you are considering), No, you're completely mixed up. Truth in a model is not defined with regard to a theory. We do not have to consider any theory in the langauge at all just to define truth in the model. Rather, the model is a function from a certain set of SYMBOLS of a language. Then truth or falsehood of sentences of the language per the model is determined by the recursive definition. > is something that YOU need > to specify before you can be "given" a model. If you accept this, then > you have accepted the fundamental premise of NAFL outlined below. Well, lordy be, OBVIOUSLY my remarks are not based on your NAFL. > Unless, like all classical logicians/thinkers, you are tacitly > assuming that models are already "out there" and you happen to be > "discovering" the truths in that model, which you have somehow been > "given". Which is, at the very least, non-rgorous and at worst, > nonsensical. It is quite rigorous and quite sensical. Meanwhile, YOU still won't tell me what an NAFL theory IS. > The position that I have outlined above is the NAFL position, of > course. Models are not "pre-existing" objects, but actually > constructed via axiomatic assertions made in the human mind. So if you > consider a model M of a NAFL theory T, And an NAFL theory is what? > this model is generated by the > "interpretation" T* of T. Here T* is also a NAFL theory which must [quoted text clipped - 11 lines] > Now where can I find a logician/philosopher honest enough to openly > acknowledge and discuss NAFL? Where can we find a proponent of NAFL communicative enough to tell us what an NAFL theory IS? MoeBlee |
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