Publisher Summary This chapter illustrates the development of a language and deductive system of epistemic logic for arithmetic. In addition to the usual connectives and quantifiers, the language contains an epistemic operator K. The epistemic system can essentially help illuminate the understanding and formalization of mathematical practice. Essentially, the language of intuitionistic arithmetic can be translated into the language, and thus, the system is capable of expressing formulas of both classical and intuitionistic arithmetic, as well as formulas of mixed constructivity. This also indicates that the system can contribute to an understanding of the difference between classical and constructive arithmetic and help understand the constructive and epistemic aspects of normal, non-intuitionistic mathematical practice. The chapter discusses possible understandings and interpretations of the operator K. Further, the chapter provides brief illustration on the possibility of applying Hintikka's semantics for ideal knowledge to the present language. The chapter concludes with a description of development of some further applications of language and deductive system to the formalization of mathematical practice.

Epistemic and Intuitionistic Arithmetic

This work describes a family of homomorphisms that contract natural deductions into typed ^-expressions, with the property that a convergence proof for an untyped program for function / is contracted to a typed program for /. The main novelties, compared to previous works on extracting algorithms from proofs, are the reading of deductions themselves as programs, and that instead of a constructive reading of 3, we use a Leibnitzian view of objects as sets of properties. The method is based on the observation that object-level components of natural deductions can be ignored computationally. It is applicable to every formalism in which there are no axioms or rules for objects in general, only for properties. Formalisms of this type include Peano's first order axiomatization of arithmetic (in its original fonn, with a primitive predicate identifier AO. second and higher order logic (in which data types, like the natural numbers, can be defined explicitly), and various variants of fixpoint extensions of first order logic. Among the technical offshoots of the method are very simple and transparent proofs of Guard's Theorem, that the provaWy recursive functions of second order arithmetic are all representaUe in the second order X-calculus, and of GOdel's "Dialectica" Theorem, that the provaMy recursive functions of first order arithmetic are all computable using primitive recursion in all finite types. This research was partially supported by ONR Grant N00014-844C-0415, and in part by the Defense Advanced Research Projects Agency (DOD), ARPA Order No. 4976, Amendment 20, under Contract Number F33615-87C-1499, monitored by the Avionics Laboratory, Air Force Wright Aeronautical Laboratories, Aeronautical Systems Division (AFCS), Wright-Patterson AFB, Ohio 45433-6543. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of ONR, DARPA, or the U.S. Government Introduct ion Most techniques for extracting the algorithmic contents of proofs fall into three methods. The interpretational method maps a formula ^ to a formula y that renders the constructive contents of <p. One shows that if <p is provable constructively, then y> is true. This is the method underlying Kleene's realizability interpretations, Godel's "Dialectica" interpretation, and Kreisel-Goodman's Theory of Constructions. The proof normalization method is based on the special form taken by normal proofs in semantic directed calculi for constructive theories, such as natural deductions for Intuitionistic (Heyting's) First Order Arithmetic. The key property used is existential instantiation: a normal proof of a formula 3x.<p must have, as premise of its last inference, y>[t/x] for some term r. The proof combinatorics method attempts to interpret a natural deduction proof as comprising an algorithm. The method is motivated by Curry and Howard's "formula as type" isomorphism [CF58,How80,Lau70,deB70], and underlies the programming language of Per Martin-L6f [Mar79], the family of PRL systems [Con86], and Coquand and Huet's Calculus of Constructions [CH88, Coq]. Our method is a variant, originating in [Lei83], of the proof combinatorics method. It is based on the observation that object-level components of natural deductions can be ignored computationally, leading to a homomorphism from proofs to typed programs, that extracts the computational contents of deductions. The underlying rationale goes back to Leibnitz: in a general setting where objects have a priori no special property, reasoning and computing are with respect to objects' properties, rather than over the objects themselves. The fact that an object x has property P will be reflected in the combinatorics of (potential) proofs of P{x). For instance, if N is a suitable rendition of the property of being a natural number, then a proof of N(x) will have a structure that reflects the construction of x. This approach is applicable to every formalism in which there are no axioms or rules for objects in general, only for properties. Formalisms of this type include Peano's first order axiomatization of arithmetic (in its original form, with a primitive predicate identifier N), second and higher order logic (in which data types, like the natural numbers, can be defined explicitly), and various variants of fixpoint extensions of first order logic. Our approach differs in important respects from previous works on proofs as programs, such as Martin-Lfif's Type Theory (MLTT). We disregard abstraction over individual objects, whereas object abstraction is at the core of the MLTT style. The constructive contents of proofs rests, for us, in the combinatorics of abstraction and application within proofs, in contrast to MLTT, where the constructive contents lies mainly in a constructive interpretation of existential quantification. Our approach is committed to a "semantic" view of typing (types as properties) as opposed to the "ontological" view of (most variants of) MLTT, where objects come equipped with their type. Finally, our method is particularly suited for reasoning about functions that are partial with respect to data types. Among the technical offshoots of the method are very simple and transparent proofs of Girard's Theorem, that the provably recursive functions of second order arithmetic are all representable in the second order A-calculus, and Godel's "Dialectica" Theorem, that the provably recursive functions of first order arithmetic are all computable using primitive recursion in all . finite types. We outline the use of our contraction homomorphisms for three types of calculi. In §§1-4 we present the method and some of its ramifications and applications for pure second order logic, which is mapped to Girard-Reynolds's pure second order A-calculus. We start with this instance of the method because the target formalism is a pure A calculus. In §5 we exhibit the method for a variant of Peano's Arithmetic and similar "generative axiomatizations" of inductively generated data types. §6 outlines applications to "inductive axiomatizations", based on first order inductive definitions. Finally, §7 touches on the contraction of second order proofs with restricted forms of comprehension, to second order typed A-expressions with restrictions on type arguments. Since we deal here with functions provable in second order logic, restricting comprehension leads to computational classes well below the provably recursive functions of first order arithmetic, such as the primitive recursive functions, the elementary, and the super-elementary functions [Lei89, Leia, Lei/3, Lei7]. A number of technical elaborations are factored out into appendices, to avoid distraction from the main development. The main results of §§1-3 were reported in [Lei83]. They were rediscovered by Krivine and Parigot [Kri86, KP87]. Closely related results are described in [Gir89] (see discussion following Theorem V below). The main results of §§4,5 are contained in §2 of [Lei84], and were reported in December 1983 at the Workshop on Logic in Computer Science at Brooklyn College of CUNY. Acknowledgements. I am grateful to Phokion Kolaitis, Georg Kreisel, Michel Parigot, Jonathan Seldin, and Paul Taylor for comments on a preliminary version of this work. Research partially supported by ONR grant N00014-84-K-0415 and by DARPA grant F33615-87-C-1499, ARPA Order 4976, Amendment 20.

Contracting proofs to programs

(1998). The Surprise Examination or Unexpected Hanging Paradox. The American Mathematical Monthly: Vol. 105, No. 1, pp. 41-51.

https://www.jstor.org/stable/pdf/2589525.pdf

The Surprise Examination or Unexpected Hanging Paradox

Peter Wegner’s definition of computability differs markedly from the classical term as established by Church, Kleene, Markov, Post, Turing et al. Wegner identifies interaction as the main feature of today’s systems which is lacking in the classical treatment of computability. We compare the different approaches and argue whether or not Wegner’s criticism is appropriate. Taking into account the major arguments from the literature, we show that Church’s thesis still holds.

Why Church's Thesis Still Holds. Some Notes on Peter Wegner's Tracts on Interaction and Computability

Traditional combinatory logic uses combinators S and K to represent all Turing-computable functions on natural numbers, but there are Turing-computable functions on the combinators themselves that cannot be so represented, because they access internal structure in ways that S and K cannot. Much of this expressive power is captured by adding a factorisation combinator F. The resulting SF-calculus is structure complete, in that it supports all pattern-matching functions whose patterns are in normal form, including a function that decides structural equality of arbitrary normal forms. A general characterisation of the structure complete, confluent combinatory calculi is given along with some examples. These are able to represent all their Turing-computable functions whose domain is limited to normal forms. The combinator F can be typed using an existential type to represent internal type information.

/pdf/a-combinatory-account-of-internal-structure-1goksq32jz.pdf

A combinatory account of internal structure

?1. Four values. In this paper I will develop a semantic account for modal logic by considering only the values of sentences (and formulas). This account makes no use of possible worlds. To develop such an account, we must recognize four values. These are obtained by subdividing (plain) truth into necessary truth (T) and contingent truth (t); and by subdividing falsity into contingent falsity (f) and necessary falsity (impossibility: F). The semantic account results from reflecting on these concepts and on the meanings of the logical operators. To begin with, we shall consider the propositional language Lo. The language Lo has (1) infinitely many atomic sentences, (2) the two truth-functional connectives %, V, and the modal operator Cl. (Square brackets are used for punctuation.) The other logical expressions are defined as follows: Dl [A&B] =(def)-[ -A V -B], D2 [A v B] (def)[-A V B], D3 0 A =(def)FIp-A. I shall use matrices to give partial characterizations of the significance of logical expressions in Lo. For negation, this matrix is wholly adequate:

Modal Semantics without Possible Worlds

In this paper, I will develop a logical system intended to capture (and explain) the use of conditional sentences to make conditional assertions, as well as their use to make conditional denials and suppositions. There is a large literature dealing with natural-language conditionals, but there is no entirely satisfactory treatment of conditionals. I think that illocutionary logic provides the resources needed for a successful account of an important class of these speech acts. Daniel Vanderveken and John Searle have pioneered the study of illocutionary logic in Vanderveken and Searle (1985) and Vanderveken (1990). However, their approach is quite different from mine. Theirs might be characterized as a ‘‘top down’’ approach, while mine is ‘‘bottom up.’’ They view illocutionary logic as a supplement, or appendix, to standard logic, and they focus on very general principles / laws which characterize illocutionary acts of all kinds. In contrast, I understand illocutionary logic to be a very comprehensive subject matter that includes standard logic as a proper part. I seek to develop systems which deal with specific kinds of illocutionary acts, and favor a multiplicity of different systems for ‘‘capturing’’ the different kinds of illocutionary acts. The present treatment of conditional assertions is developed within the framework of a general system of logic that hasn’t been developed simply to deal with conditionals. This treatment is intended to accommodate or account for what are ordinarily understood to be indicative conditionals as opposed to counterfactual or subjunctive conditionals. An (ordinary) assertion is here taken to be an act of accepting a statement (as true) or of reaffirming one’s acceptance; when so understood, such an act may or may not have an addressee. Conditional assertions differ from ordinary assertions in not being true or false—they are not acts of accepting true

Conditional assertion, denial, and supposition as illocutionary acts

In this paper, I present a modified and extended version of combinatory logic. Schonfinkel originated the study of combinatory logic (in [2]), but its development is primarily due to H. B. Curry. In the present paper, I will make use of both the symbolism (with some modification) and the results of Curry, as found in [1]. What is novel about my version of combinatory logic is a kind of combinators which I call discriminators . These combinators discriminate between different symbols, and yield values which are determined by the symbols which are their arguments. If the normal combinators investigated by Curry are considered as functions taking (combinations of) symbols as arguments and yielding symbols as values, these combinators can rearrange and cancel their arguments or introduce new symbols.

Combinatory Logic with Discriminators

This paper explores Church‘s Thesis and related claims made by Turing. Church‘s Thesis concerns computable numerical functions, while Turing‘s claims concern both procedures for manipulating uninterpreted marks and machines that generate the results that these procedures would yield. It is argued that Turing‘s claims are true, and that they support (the truth of) Church‘s Thesis. It is further argued that the truth of Turing‘s and Church‘s Theses has no interesting consequences for human cognition or cognitive abilities. The Theses don‘t even mean that computers can do as much as people can when it comes to carrying out effective procedures. For carrying out a procedure is a purposive, intentional activity. No actual machine does, or can do, as much.

Thinking Machines: Some Fundamental Confusions

This paper further develops the system of illocutionary logic presented in ‘Propositional logic of supposition and assertion’ (Notre Dame Journal of Formal Logic 1997, 38, 325-349) to accommodate an ‘I believe that’ operator and resolve Moore's Paradox. This resolution is accomplished by providing both a truth-conditional and a commitment-based semantics. An important feature of the logical system is that the correctness of some arguments depends on who it is that makes the argument. The paper then shows that the logical system can be expanded to resolve the surprise execution paradox puzzle. The prisoner's argument showing that he can't be executed by surprise is correct but his beliefs are incoherent. The judge's beliefs (and our beliefs) about this situation are not incoherent.

John T. Kearns

Papers

Modal Semantics without Possible Worlds

Conditional assertion, denial, and supposition as illocutionary acts

Combinatory Logic with Discriminators

Thinking Machines: Some Fundamental Confusions

An Illocutionary Logical Explanation of the Surprise Execution