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Sense and denotation as algorithm and value
 Lecture Notes in Logic
, 1994
"... In his classic 1892 paper On sense and denotation [12], Frege first contends that in addition to their denotation (reference, Bedeutung), proper names also have a sense (Sinn) “wherein the mode of presentation [of the denotation] is contained. ” Here proper names include common nouns like “the earth ..."
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In his classic 1892 paper On sense and denotation [12], Frege first contends that in addition to their denotation (reference, Bedeutung), proper names also have a sense (Sinn) “wherein the mode of presentation [of the denotation] is contained. ” Here proper names include common nouns like “the earth ” or “Odysseus ” and descriptive phrases like “the point of intersection of
What Is an Algorithm?
, 2000
"... Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (o ..."
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Cited by 23 (3 self)
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Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (or, more generally, on strings from a finite alphabet) is computable in principle exactly when it can be computed by a Turing Machine. The ChurchTuring Thesis grounds proofs of undecidability and it is essential for the most important applications of logic. On the other hand, it cannot be argued seriously that Turing machines model faithfully all algorithms on the natural numbers. If, for example, we code the input n in binary (rather than unary) notation, then the time needed for the computation of f(n) can sometimes be considerably shortened; and if we let the machine use two tapes rather than one, then (in some cases) we may gain a quadratic speedup of the computation, see [8]. This mea...
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
1 What Is an Algorithm?
"... When algorithms are defined rigorously in Computer Science literature machines, mathematical models of computers, sometimes idealized by allowing access to “unbounded memory”. 1 My aims here are to argue ..."
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When algorithms are defined rigorously in Computer Science literature machines, mathematical models of computers, sometimes idealized by allowing access to “unbounded memory”. 1 My aims here are to argue
DETAILED PROOF OF THEOREM 4.1 IN SENSE AND DENOTATION AS ALGORITHM AND VALUE
"... §4. Sense identity and indirect reference. Van Heijenoort [30] quotes an extensive passage from a 1906 letter from Frege to Husserl which begins with the following sentence: “It seems to me that we must have an objective criterion for recognizing a thought as the same thought, since without such a c ..."
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§4. Sense identity and indirect reference. Van Heijenoort [30] quotes an extensive passage from a 1906 letter from Frege to Husserl which begins with the following sentence: “It seems to me that we must have an objective criterion for recognizing a thought as the same thought, since without such a criterion a logical analysis is not possible.” This could be read as asserting the existence of a decision procedure for sense identity, but unfortunately, the letter goes on to suggest that logically equivalent sentences have the same sense, a position which is contrary to the whole spirit of [12]. It is apparently not clear what Frege thought of this question or if he seriously considered it at all. Kreisel and Takeuti [17] raise explicitly the question of synonymity of sentences which may be the same as that of identity of sense. If we identify sense with referential intension, the matter is happily settled by a theorem. Theorem 4.1. For each recursor structure A = (U1,..., Uk, f1,..., fn) of finite signature, the relation ∼A of intensional identity on the terms of FLR interpreted on A is decidable. For each structure A and arbitrary integers n, m, let SA(n, m) ⇐ ⇒ n, m are Gödel numbers of sentences or terms θn, θm of FLR (58) and θn ∼A θm. The rigorous meaning of 4.1 is that this relation SA is decidable, i.e., computable by a Turing machine. By the usual coding methods then, we get immediately: Corollary 4.2. The relation SA of intensional identity on Gödel numbers of expressions of FLR is elementary (definable in LPC), over each acceptable structure A. The Corollary is useful because it makes it possible to talk indirectly about FLR intensions within FLR. In general, we cannot do this directly because the intensions of a structure A are higher type objects over A which are not ordinarily 15 members of any basic set of the universe of A. One reason we might want to discuss FLR intensions within FLR is to express indirect reference, where Frege’s treatment deviates from his general doctrine of separate compositionality This manuscript is posted on