R. Smullyan, Diagonalization and self-reference, Oxford University Press, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and with the advent of computer technology the programming languages in which we represent, model, simulate, and act upon both physical and human reality. Reflection has been vigorously pursued in logic by many researchers since the fundamental work of Godel and Tarski(see the survey [17]) In computer science it has been present from the beginning in the form of universal Turing machines. Many researchers have recognized its great importance and usefulness in programming languages, in theorem proving, in concurrent and distributed computation, and in many other areas such as ....

R. M. Smullyan. Diagonalization and Self-Reference. Oxford University Press, 1994.

....logic for the notion of statement. It is shown in sections 5 and 7 of this paper that LP enjoys the soundness completeness property: LP F , F is true under any interpretation . Any system of proofs with a proof checker operation capable of internalizing its own proofs as terms (cf.[66]) may be within the scope of LP. In particular, any proof system for first order Peano Arithmetic PA (cf. 12] 14] 51] 68] provides a model for LP with Godel numbers of proofs being an instrument for internalizing proofs as terms. The soundness ( does not necessarily refer to arithmetical ....

R. Smullyan, Diagonalization and Self-Reference, Oxford University Press, 1994 44

....logic for the notion of statement. It is shown in Sections 2 and 4 of this paper that LP enjoys the soundness completeness property: LP F , F is true under any interpretation . Any system of proofs with a proof checker operation capable of internalizing its own proofs as terms (cf. [46]) may be in the scope of LP . In particular, any proof system for the first order Peano Arithmetic PA (cf. 9] 10] 34] 47] provides a model for LP with Godel numbers of proofs being a instrument of internalizing proofs as terms. The soundness ( does not necessarily refer to the ....

R. Smullyan, Diagonalization and Self-Reference, Oxford University Press, 1994

....The meaning of P may be regarded as a function from strings over an alphabet V into strings over the same alphabet, i.e. a function f : V # # V # . Such a program P on input x . may eventually stop, in which case it prints a string, or . it may run forever, in which case we say, following [134], that the program has been silenced by x. We shall prove that every universal program can be silenced by some input. Here is the argument. List all valid programs, P 1 ,P 2 , P n , 18 Now suppose, by absurdity, that we have a universal program that cannot be silenced, so, by ....

R. M. Smullyan, Diagonalization and Self--Reference, Clarendon Press, Oxford, 1994.

....an agent does not have, in general, an appropriate sentence for representing the required necessary and sufficient conditions involved in a definition. What we need is a sentence whose denotation depends on its context, like the denotation of the word here depends on the place it is uttered (Smullyan, 1994). A well known device for naming a context is to treat it as a designator, i.e. a parameter which acquires its meaning through a suitable substitution (see (Smullyan, 1957) for a discussion on the role of substitution in these cases) For example, to express that a given sentence ff is consistent ....

Smullyan, R. (1994). Diagonalization and Self-Reference. Oxford.

....in the car or somewhere else. We just need the minimal amount of information. Analogously we do not need the context in the water . The only necessary and sufficient context is heat on . This self referential view of contexts is not new, both Smullyan and Perlis in their papers on self reference [26] and [21] refer to contexts. Now, 3) make sense if their definitional form is, indeed, a fixed point equation, and thus they are logical truths in some logical system. The only way to ensure this is to show that, for each context, there is at least a fixed point equation that satisfies it. That ....

R.M. Smullyan. Diagonalization and Self-Reference. Oxford, 1994. 12

....of LP and ILP The Logic of Proofs is meant to play for a notion of proof a role similar to that played by the boolean propositional logic for the notion of statement. In principle LP models any system of proofs with a proof checker operation capable of internalizing its own proofs as terms (cf. [22]) In particular, any proof system for the first order Peano Arithmetic PA (cf. 8] 9] 19] 23] provides a model for LP with Godel numbers of proofs being a instrument of internalizing proofs as terms. Although the soundness ( does not necessarily refer to the arithmetical models of LP , ....

R. Smullyan, Diagonalization and Self-Reference, Oxford University Press, 1994

.... PA (cf. 7] 8] 24] 33] is a natural source of proof systems with Godel numbers of proofs being a natural instrument of internalizing proofs as terms (natural numbers) In principle any system of proofs with a proof checker operation capable of internalizing its own proofs as terms (cf.[32]) can provide a model for LP . If n is a natural number, then n will denote a numeral corresponding to n, i.e. a standard arithmetical term 0 000: where 0 is a successor functional symbol and the number of 0 s equals n. We will use the simplified notation n for a numeral n when it is ....

R. Smullyan, Diagonalization and Self-Reference, Oxford University Press, 1994

....by the boolean propositional logic for the notion of statement. It is shown in [4] 7] that LP enjoys the soundness completeness property: LP F , F is true under any interpretation . Any system of proofs with a proof checker operation capable of internalizing its own proofs as terms (cf. [41]) may be in the scope of LP . In particular, any proof system for the first order Peano Arithmetic PA (cf. 9] 10] 32] 42] provides a model for LP with Godel numbers of proofs being a instrument of internalizing proofs as terms. The soundness ( does not necessarily refer to the ....

R. Smullyan, Diagonalization and Self-Reference, Oxford University Press, 1994

....axioms for computational strategies, and for the internalization of those strategies in a reflective logic are also given. 1 Introduction Reflection is a fundamental idea. In logic it has been vigorously pursued by many researchers since the fundamental work of Godel and Tarski (see the surveys [52, 53]) In computer science it has been present from the beginning in the form of universal Turing machines. Many researchers have recognized its great importance and usefulness in programming languages [54, 51, 60, 56, 23, 19, 30] in theorem proving [62, 7, 48, 20, 2, 29, 14, 16] in concurrent and ....

R. M. Smullyan. Diagonalization and Self-Reference. Oxford University Press, 1994.

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R. Smullyan, Diagonalization and self-reference, Oxford University Press, 1994.

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R.M. Smullyan. Diagonalization and Self-Reference. Oxford, 1994.

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Smullyan, R.: Diagonalization and Self-Reference. Oxford University Press (1994)

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