Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: Torkel Franzen on 3 Feb 2005 00:12 lrudolph(a)panix.com (Lee Rudolph) writes: > It would be kind of you > to remind me, so I'll have something more recent to forget. It wouldn't surprise me if all if I have picked this up from Kreisel, but I can't at the moment recall any comments of his.
From: examachine on 3 Feb 2005 12:37 tchow(a)lsa.umich.edu wrote: > In article <367nq8F4qcoivU1(a)individual.net>, > Jamie Andrews; real address @ bottom of message <me(a)privacy.net> wrote: > <In comp.theory tchow(a)lsa.umich.edu wrote: > <> (*) Formal sentences (in PA or ZFC for example) adequately express > <> their informal counterparts. > <> Any candidates for a catchy name for (*)? > < > <Is this not the central tenet of "logicism"? > > Not really. Logicism is the doctrine that mathematics reduces to logic. > Exactly what is involved in this "reduction" varies from one thinker to > another; for example, in one version, mathematical assertions such as > "1+1=2" that are prima facie about natural numbers are actually logical > assertions of the form "`1+1=2' follows from the axioms for arithmetic." > > The sentence (*), or improved versions of it elsewhere in this thread, > doesn't have much to do with reducing mathematics to logic. It just > says that mathematical assertions can be mirrored in a formal language. > The formal language might express non-logical propositions. Dear Tim, This was a very nice explanation I think. But I would like to point out that it looks incomplete without also being tied to a broader metaphysical point of view, say that of empiricism. So, our dear Torkel is a logicist and we knew that, but from where should he think the truth of an axiom, say axiom of infinity, follow? Regards, -- Eray Ozkural
From: Stephen Harris on 3 Feb 2005 12:32 "Stephen Harris" <cyberguard1048-usenet(a)yahoo.com> wrote in message news:y8tMd.196$ZZ.107(a)newssvr23.news.prodigy.net... > > "Jamie Andrews; real address @ bottom of message" <me(a)privacy.net> wrote > in message news:36a8vlF4v8123U1(a)individual.net... >>>> In comp.theory tchow(a)lsa.umich.edu wrote: > It seems to me clear that the cardinality quantifiers E_K > for ? uncountable belong to mathematics (specifically, set > theory) and not to logic; they are all excluded by the > homomorphism invariance condition, along with the E_K for > K countable. As just remarked, the finite ones are recovered > once one includes the identity I. The quantifier"there exist > infinitely many", for K = Aleph_0 is a borderline case to > which intuition and experience do not provide a clearcut > answer as to its status. It can, however, be assimilated to > logical notions under the homomorphism invariance criterion > simply by restricting one's consideration to those operations > which are invariant over infinite domains M_0, without > thereby including the E_K for K uncountable. The > "completeness" argument for logicality (suggested by Quine > in the case of =) here gives quite anomalous results, since > one has a complete logic for E_K for the case that ? = Aleph_1 > by the work of Keisler (1970) while, as is well known, there > is no such logic for the case that ? = Aleph_0. > The "?" stand for K Condensed from Sol Feferman: 'David Lewis makes a persuasive case that we do have an independent grasp of plural quantification that doesn't have to be explained in terms of second-order quantification, though there appears to be an asymmetry between existential plurals (natural) and universal plurals (not natural) in English.'
From: Stephen Harris on 3 Feb 2005 12:25 "Jamie Andrews; real address @ bottom of message" <me(a)privacy.net> wrote in message news:36a8vlF4v8123U1(a)individual.net... >>> In comp.theory tchow(a)lsa.umich.edu wrote: >>>> (*) Formal sentences (in PA or ZFC for example) adequately express >>>> their informal counterparts. >>>> Any candidates for a catchy name for (*)? > >> "Jamie Andrews; real address @ bottom of message" <me(a)privacy.net> wrote >> in >> message news:367nq8F4qcoivU1(a)individual.net... >>> Is this not the central tenet of "logicism"? > > In comp.theory Stephen Harris <cyberguard1048-usenet(a)yahoo.com> wrote: >> Logicism is the theory that mathematics is an extension of >> logic and therefore all mathematics is reducible to logic. >> Kurt Gýdel's incompleteness theorem ultimately undermined >> the purpose of the project. The attempted resurrection of >> this theory is styled neo-logicism. > > Please cite Wikipedia when you copy from it. > > I think the Wikipedia entry might not be entirely accurate. Point well taken. Some sources more likely to be accurate: http://www.helsinki.fi/collegium/eng/Raatikainen/godelfinal.pdf "There has been some dispute on the issue as to whether Godel's theorems conclusively refute logicism, that is, the claim that mathematics can be reduced to logic, as endorsed, for instance, by Frege and Russell. Obviously this issue dependsheavily on how one understands the essence of logicism. Clearly Gýdel's theorems show that all arithmetical truths are not reducible to the standard first-order logic, or indeed, to any recursively axiomatizable system. On the other hand, one may restrict the logicist thesis to some class of mathematical truths (such as known truths, or humanly knowable ones), and/or extend the scope of logic. There is, though, the threat that the issue becomes trivial or wholly verbal. ... Sternfeld (1976) and Rodrýguez-Consuegra (1993), on the other hand, argue that it is possible to defend logicism even after Godel's theorems. Sternfeld and Rodrýguez-Consuegra appeal to the fact that Godel's theorems do not provide an absolutely undecidable statement, but only a relative one. This is certainly true. Yet this defense apparently collapses logicism into the view that every mathematical truth is derivable in some formal system. This, however, makes the thesis completely trivial. Furthermore, would this not imply that not only mathematics but also all empirical facts are "logically true"? Geoffrey Hellman (Hellman 1981, see also Reinhard 1985) has analyzed the bearing of Gýdel's theorems on logicism in more detail. Hellman focuses only on the thesis that knowable mathematical truth can be identified with derivability in some formal system. Logicism so understood cannot be directly refuted by Gýdel's first theorem. Hellman subsequently gives a considerably more complicated argument which leans on Gýdel's second theorem, and breaks the argument down into two cases. First, he concludes that no finitely axiomatizable logicist system exists. Second, he considers non-finitely axiomatizable systems, and here the claim is weaker: such logicist systems may exist, but Godel's second theorem prohibits our being able to know of any particular system that it is one of them.5 Hellman's argument has the advantage of not depending on any particular restrictive way of drawing the controversial line between logic and non-logic." ---------------------------------------------------------- http://math.stanford.edu/~feferman/papers/logiclogicism.pdf (page 12) "One should be aware that the notion of absoluteness is itself relative, and is sensitive to a background set theory, hence again to the question of what entities exist. For examples of absolute operations which are patently set-theoretical yet come out as logical on the Tarski-Sher thesis, see the just mentioned reference. ... It is undeniable that the relation of identity has a "universal", accepted and stable logic (at least in the presence of totally defined predicates and functions, as is usual in the PC with =), and that argues for giving it a distinguished role in logic even if it should not turn out to be logical on its own under some cross-domain invariance criterion, such as under homomorphisms. Of course, even if a form of the latter is accepted as a criterion for logicality, one is still free to consider the operations which are defined from I by those provided in Theorem 6. That of course buys one the quantifiers E_K for ? finite, but not those for ? infinite, whose loss is discussed separately, next." (pages 21 & 22) It seems to me clear that the cardinality quantifiers E_? for ? uncountable belong to mathematics (specifically, set theory) and not to logic; they are all excluded by the homomorphism invariance condition, along with the E_? for ? countable. As just remarked, the finite ones are recovered once one includes the identity I. The quantifier"there exist infinitely many", for ? = Aleph_0 is a borderline case to which intuition and experience do not provide a clearcut answer as to its status. It can, however, be assimilated to logical notions under the homomorphism invariance criterion simply by restricting one's consideration to those operations which are invariant over infinite domains M_0, without thereby including the E_? for ? uncountable. The "completeness" argument for logicality (suggested by Quine in the case of =) here gives quite anomalous results, since one has a complete logic for E_? for the case that ? = Aleph_1 by the work of Keisler (1970) while, as is well known, there is no such logic for the case that ? = Aleph_0. I also agree with Quine (1986, pp. 64 ff) that second-order and higher order quantification go beyond the bounds of logic. He takes these (famously) to be "set theory in sheep's clothing", and it is certainly true that the understood meaning of such quantifiers depends on what sets exist, or alternatively - if such quantifiers are regarded as binding predicate variables- of what predicates exist. 6 Tarski and Boolos on logicism. In his "What are logical notions?" lecture that was the starting point for this paper, Tarski concluded with a discussion of its relevance to the logicist program, as follows: "The question is often asked whether mathematics is a part of logic. Here we are interested in only one aspect of this problem, whether mathematical notions are logical notions, and not, for example, in whether mathematical truths are logical truths, which is outside our domain of discussion. (Tarski 1986, p. 151)" SF: His answer is, curiously: "As you wish"! The argument is that since "the whole of mathematics can be constructed within set theory, or the theory of classes", and since "all usual set-theoretical notions" can be defined in terms of the relation of membership, the determination comes down to whether membership is a logical notion. *11 But -Tarski goes on- two methods have been provided for the foundations of set theory following the discovery of paradoxes in that subject, namely the theory of types as exemplified in Principia Mathematica (which he takes implicitly in unramified form), and axiomatic set theory as formulated by Zermelo, et al. If one follows the method of the theory of types then membership is a part of logic, since it is invariant under the extension to higher types of any permutation of the domain of individuals. On the other hand, if axiomatic set theory is followed, there is "only one universe of discourse and the membership relation between its individuals is an undefined relation, a primitive notion." On that account, membership is not a logical notion, since as Tarski had shown earlier, there are only four permutation-invariant relations between individuals, the universal relation, the empty relation, the identity relation and its complement. {*11. It is also curious that Tarski ignores the fact due to his fundamental result on the nondefinability of truth-in-L within a language L, that the mathematical notion of truth of sentences of the language of set theory cannot be defined within set theory (and similarly for type theory).} Tarski winds up these considerations as follows: "This conclusion ["As you wish!"] is interesting, it seems to me, because the two possible answers correspond to two different types of mind. Am onistic conception of logic, set theory, and mathematics, where the whole of mathematics would be a part of logic, appeals, I think, to a fundamental tendency of modern philosophers. Mathematicians, on the other hand, would be disappointed to hear that mathematics, which they consider the highest discipline in the world, is a part of something so trivial as logic; and they therefore prefer a development of set theory in which set-theoretical notions are not logical notions. The suggestion which I have made does not, by itself, imply any answer to the question of whether mathematical notions are logical."(Tarski 1986,p.153) SF: Though Tarski's consideration only of the question "whether mathematical notions are logical notions" and not of "whether mathematical truths are logical truths" appears at first sight to be a reasonable one, it is not clear that the two can be separated so neatly. For, any argument one way or the other about the first question must necessarily invoke assumptions about various properties of the notions involved, and those lead one into the second question. By contrast, Boolos shows that FA, which does not use extensions, is consistent. Then, following the lead of Wright (1983), he shows that "(o)nce Hume's principle is proved, Frege makes no further use of extensions." (Boolos 1998, p. 191). In his discussion of the significance of this work, Boolos comes to the following provocative conclusions (op. cit., p. 200): "(1) Numbers is no logical truth; and therefore (2) Frege did not demonstrate the truth of logicism in the Foundations of Arithmetic. (3) Logic is synthetic if mathematics is, because (4) there are many interesting, logically true conditionals with antecedent Numbers whose mathematical content is not appreciably less than that of their consequents." And he adds to these: "(5) Since we have no understanding of the role of logic or mathematics in cognition, the failure of logicism is at present quite without significance for our understanding of mentality." In view of my working identification of logic with the first-order predicate calculus PC, I am in agreement with (1) and (2). I am more or less in disagreement with (3), though I don't have strong feelings about what being synthetic amounts to. I don't see (4) since all results of mathematics can be represented as logical consequences of mathematical hypotheses. As to (5), I agree with the conclusion, but not the premise; it seems to me that we do have some understanding of the role of logic, and to some extent of mathematics, in cognition, though we surely have much farther to go in both respects. To reiterate my introductory remarks, I think that the theoretical study of what a logical operation is, and hence of what the scope of logic is, must be connected with the more empirical study of the role of logic in the exercise of human rationality. I am optimistic that a better understanding of either will inform the other."
From: Stephen Harris on 3 Feb 2005 13:45
<tchow(a)lsa.umich.edu> wrote in message news:41fbeb5c$0$580$b45e6eb0(a)senator-bedfellow.mit.edu... > The Church-Turing thesis is familiar to many people, largely because it > has been widely discussed both in textbooks and in popular science > >writing. > Having a name helps, too. > > There is an analogous thesis that is relevant to logic and the foundations > of mathematics: > > (*) Formal sentences (in PA or ZFC for example) adequately express > their informal counterparts. > Since I think "quantification theory" is a synonym for (FOPL) First Order Predicate Logic, then there appear to be anomalies residing in translating from natural language, which I take to be the vehicle of informal logical statements about mathematics, and mapping all such translations onto formal sentences which are identical in meaning to the informal sentence which is to be represented formally, or vice-versa. Perhaps I am confusing your statement with Logicism. Or with Feferman's position: SF: "Though Tarski's consideration only of the question "whether mathematical notions are logical notions" and not of "whether mathematical truths are logical truths" appears at first sight to be a reasonable one, it is not clear that the two can be separated so neatly. For, any argument one way or the other about the first question must necessarily invoke assumptions about various properties of the notions involved, and those lead one into the second question." SH: Quantum theory has one standard underlying mathematical formalism. Yet, there are eight major interpretations about what that formalism describes about reality; interpretations which I would think would be classified as informal and which are not unique and seemingly contradictory in some cases, so the formal statement is not enough to distinguish between them. It is my impression that a "thesis" has no counter-examples, so that what you are describing is more accurately a guideline? http://www.fordham.edu/gsas/phil/klima/NLN.htm Gyula Klima: Approaching Natural Language via Mediaeval Logic I. ANOMALIES OF A PARADIGM Mismatch of syntax "As is well-known, natural language sentences of evidently the same syntactic structure are represented by formulae of quantification theory of entirely different structure, while the same formula may have different "readings", expressible by natural language sentences of widely different syntax. Regarding these discrepancies, of course, one might say that there is no justifiable need of a strict correspondence between the syntactic structure of natural language sentences and the formulae representing them. After all, a logical semantics, which is to be a general semantics for all kinds of human languages, should precisely disregard accidental grammatical features of particular natural language expressions, and hence also the delusive grammatical structure of natural language sentences in general. All that is required for correspondence is that the formula should state correctly the truth conditions of the sentence which it represents, since it is only these truth conditions that determine the logical relations of sentences among each other. Along these lines, mismatch of syntax may be made to appear entirely harmless, by making a distinction between logical form on the one hand, and grammatical form on the other, placing much confidence in the capability of quantification theory to express the former, and thereby justifiably ignoring the latter. Unrepresentable sentences There is, however, a further set of anomalies, which comes as a fatal blow to this interpretation of the relationship between quantification theory and natural languages. For, as it turned out, some apparently simple quantified sentences of natural languages are demonstrably unrepresentable in first order quantification theory in the sense that no first order formula is able to give their correct truth conditions.[3] As is well-known, examples of such sentences are those containing the determiners 'most' or 'more than half of', and so on. But if there are no formulae giving the correct truth conditions of such sentences, then quantification theory is simply unable to supply their logical form, and so the above-mentioned rationale for drawing the distinction between logical and grammatical form breaks down with these sentences. ... Intensional and intentional contexts To be sure, the above-mentioned "anomalies" may be considered as such only because they pose problems to quantification theory that everyone feels it should handle but cannot. It was clear from the beginning that there are large portions of natural language reasonings that simply fall outside the authority of quantification theory, namely those involving intensional contexts. Nevertheless, Frege's relegation of modal notions to the sphere of psychology notwithstanding, logicians have been working on expanding formal logic even to these contexts. Possible worlds semantics produced interesting results concerning modal notions and still seems to have some resources concerning tensed modal contexts. However, in virtue of the coarse-grained character of intensions available in possible worlds semantics, several intentional contexts, namely those created by attitude verbs, seem to defy analysis in terms of these intensions."[5] Looking forward to another learning opportunity, Stephen Condensed from Sol Feferman: 'David Lewis (IHO) makes a persuasive case that we do have an independent grasp of plural quantification that doesn't have to be explained in terms of second-order quantification, though there appears to be an asymmetry between existential plurals (natural) and universal plurals (not natural) in English.' |