Abstract
We can associate with each consistent formula F of first-order logic a computing device as its representation. This computing device is one which will calculate the Skolem functions of F (for a denumerable domain). When two such devices are operating in parallel, the resulting architecture does not necessarily represent any ordinary first-order formula, but it will represent a formula in independence-friendly (IF) logic, which hence can be considered as a true logic of parallel processing. In order to preserve representability by a digital automaton (Turing machine), a nonstandard (constructivistic) interpretation of the logic in question has to be adopted. It is obtained by restricting the Skolem functions available to verify a formula F to recursive ones, as in the Gödel’s Dialectica interpretation.
Written jointly with Gabriel Sandu
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Notes
For a survey of results in this area, see e.g. the article of J. L. McLelland, D. E. Rumelhart and G. E. Hinton, The appeal of parallel distributed processing, in David E. Rumelhart et al., Parallel Distributing Processing, vol. 1, Foundations, MIT Press, Cambridge, 1986.
See, e.g. Warren D. Goldfarb, Logic in the twenties: The nature of a quantifier, J. Symbolic Logic 44 (1979) 351–68.
This result goes back to A. Mostowski, “On a system of axioms which has no recursively enumerable model”, Fundamenta Mathematicae 40 (1953) 56–61, and “A formula with no recursively enumerable model”, ibid. 42 (1955) 125–140, and to G. Kreisel, “A note on arithemetic models for constant formulae of the predicate calculus”, in Proc. XIth Int. Congress of Philosophy, vol. 14, Amsterdam and Louvain, 1953, pp. 39–49.
See Kurt Gödel, “On a hitherto unexploited extension of the finitary standpoint”, J. Philosophical Logic 9, 133–142.
It will nevertheless turn out that negation behaves in certain respects in a nonclassical way in the languages we will end up embracing.
The reasons why we can (and why Gödel cannot) avoid the climb towards higher orders are spelled out in Jaakko Hintikka, “Gödel’s functional interpretation in a wider perspective”, forthcoming in the Proc. 1991 Meeting of the Int. Côdel Society The simpler functional interpretation proposed here is also mentioned there and commented on.
See e.g. R. L. Vaught, “Sentences true in all constructible models”, in J. Symbolic Logic 25 (1960), 39–53.
This follows from the observation, first made by Stanley Tennenbaum in “Nonarchimedean models for arithmetic”, Notices of the American Mathematical Society 6 (1959) 270, that the only model of Peano arithmetic where the interpretations of sum and product are recursive is the standard one. For the entire range of questions arising here, see Richard Kaye, Models of Peano Arithmetic, Clarendon Press, Oxford, 1991, especially 11.3.
Op. cit., p. 10.
For partially ordered quantifier structures, including branching quantifiers, see, e.g. Leon Henkin, “Some remarks on infinitely long formulas”, in lnfsnitistic Methods,Pergamon Press, Oxford, 1959, pp. 167–83 (the first paper on the subject) or Jon Barwise, “On branching quantifiers in English”, J. Philosophical Logic 8 (1979) 47–80 (with further reference).
Cf. Jaakko Hintikka, What is elementary logic, forthcoming; Jaakko Hintikka and Gabriel Sandu, “Informational independence as a semantical phenomenon”, in Logic Methodology and Philosophy of Science VIII, Jens Erik Fenstad et al., Elsevier Science Publishers, Amsterdam, 1989, pp. 571–589.
The proof can be found in The Game of Language,Jaakko Hintikka and Jack Kulas, D. Reidel, Dordrecht, 1983, and in Hodges, Wilfrid, 1989, “Elementary predicate logic”, in F. Gabbay and F. Guenther, eds., Handbook of Philosophical Logic,D. Reidel, Dordrecht, Reidel.
These games have been introduced in Jaakko Hintikka, and Gabriel Sandu, 1989, “Information independence as a semantical phenomenon”, in J. E. Fenstad et al., eds., Logic Methodology and Philosophy of Science VIII,Elsevier Science Publishers, pp. 571–589.
This example comes from A. Blass and Y. Gurevich, “Henkin quantifiers and complete problems”, in Annals Pure App. Logic 32 (1986) 1–16.
What we mean by the use of independent quantifiers in Ramsey Theory can be seen from an example. As such an example, let us consider one of the central results of Ramsey Theory, the Hales—Jewett theorem. (See Ronald L. Graham et al., Ramsey Theory,John Wiley, New York, 1980.) It can be formulated in the conventional terminology as follows:
For all r, t there exists N’ = HJ(r, t) so that for N x N’,the following holds: If the vertices of C“ are r-colored there exists a monochromatic line.
One does not even have to know the precise definitions of the concepts used here to realize what is going on. (As an aid to visualization, Cis’ is roughly speaking an N-dimensional cube with edges of t units). It is easily seen that the Hales—Jewett theorem can be reformulated as follows:
For all t, the following holds: for each R there exists N’, and for each Cr together with an r-coloring of its vertices, there exists a line l in C14 such that, if N > N’and R>r, then 1 is monochromatic.
Here “existing for” means being functionally dependent on. The point is that N’ is a function of only t and R while 1 is a function of only t, N and the r-coloring. For instance, for l we can take the line in GI that has the largest number of elements of the same color. This brings it to the open that there is a Henkin quantifier structure as a part of the logical form of the Hales—Jewett theorem.
This follows from the results of W. Walkoe, Jr., Finite partially ordered quantification, J. Symbolic Logic 35 (1970) 535–555.
The contrast we have in mind here is that between ordinary first-order logic and higher-order logic (or such extensions in the direction of higher-order logic as IF logics). The usual modal logics, epistemic logic, etc., are in the sense intended here in the same boat as ordinary first-order logic.
With this provisio in mind, one can see that the majority of papers in such AI oriented volumes as Joseph Y. Halpern, ed., Theoretical Aspects of Reasoning About Knowledge: Proc. of the 1986 Conf,Morgan Kaufmann, Los Altos, CA, 1986, or its successor volume, Moshe Y. Vardi, ibid: Proc. Second Conf,Morgan Kaufmann, Los Altos, CA, 1988, use an essentially first-order logic.
For Henkin quantifiers, see Note 10 above, and also M. Krynicki, “Hierarchies of partially ordered connectives and quantifiers”, in Mathematical Logic Quarterley 39 (1993) 287–294.
See Jaakko Hintikka, “What is elementary logic? Independence-friendly logic as the true core area of logic”, forthcoming.
A proof is sketched in Jaakko Hintikka, Quantifiers versus quantification theory, Linguistic Inquiry 5 (1974) 153–77. Another proof may be found in M. Krynicki and A. Lachlan, “On the semantics of the Henkin quantifier”, J. Symbolic Logic 44 (1979) 184–200.
See the article of Blass and Gurevich in Note 14 above.
See David Marr, Vision, W H. Freeman, San Francisco, 1982, especially p. 25.
See Jaakko Hintikka, “Paradigms for language theory” Acta Philosophica Fennica (1990) 181–209.
For the relation between Henkin quantifiers and NP-problems, see Blass and Gurevich, Note 21 above.
For the metatheory of IF first-order logic, see Gabriel Sandu, On the logic of informational independence and its applications, J. Philosophical Logic, forthcoming; Jaakko Hintikka, “What is elementary logic? Independence-friendly logic as the true core area of logic”, forthcoming, and Jaakko Hintikka, “Defining truth, the whole truth and nothing but the truth”, forthcoming (preprint available as Reports from the Department of Philosophy, University of Helsinki, no. 2, 1991 ).
For this purpose, we have to use interpolation formulas which differ somewhat from the familiar Craigean ones. See Antti Koura and Jaakko Hintikka., “On the difficulty of deductions” forthcoming.
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Hintikka, J. (1998). What is the Logic of Parallel Processing?. In: Language, Truth and Logic in Mathematics. Jaakko Hintikka Selected Papers, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2045-8_10
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