This paper is devoted toa description, with many examples, of one particular framework of this so... more This paper is devoted toa description, with many examples, of one particular framework of this sort: that ofhypersequents. We shall show that this framework is indeed stronger than that of ordinarysequents, but still has properties (1)-(6) above. In addition, the following two pointsconcerning it should be noted:
We present a terminating contraction-free calculus GLC * for the propositional fragment of Gödel-... more We present a terminating contraction-free calculus GLC * for the propositional fragment of Gödel-Dummett Logic LC. GLC * uses hypersequents, and unlike other Gentzen-type calculi for LC, all its rules have at most two premises. These rules are all invertible. Hence it can be used as a basis for a deterministic tableau system for LC. This tableau system is presented
We provide non-deterministic semantics for the 3 basic paraconsistent C-systems C (also known as ... more We provide non-deterministic semantics for the 3 basic paraconsistent C-systems C (also known as bC), Ci, and Cia, as well as to all 9 extensions of them by one or two of the schemata (l) ¬(φΛ¬φ) ⊃ ∘ φ and (e) φ ⊃ ¬¬φ. This includes da Costa’s original C 1 (which is equivalent to Cila). Our semantics is 3-valued for the systems without (l), and infinite-valued for the systems with it. We prove that these results cannot be improved: neither of the systems without (l) has either a finite characteristic ordinary matrix or a two-valued characteristic non-deterministic matrix, and neither of the systems with (l) has a finite characteristic non-deterministic matrix. Still, our semantics suffices for providing decision procedures for all the systems investigated.
Gödel’s main contribution to set theory is his proof that GCH is consistent with ZFC (assuming th... more Gödel’s main contribution to set theory is his proof that GCH is consistent with ZFC (assuming that ZF is consistent). For this proof he has introduced the important ideas of constructibility of sets, and of absoluteness of formulas. In this paper we show how these two ideas of Gödel naturally lead to a simple unified framework for dealing with computability of functions and relations, domain independence of queries in relational databases, and predicative set theory.
We consider a framework for information collecting and processing which con- sists of a set of so... more We consider a framework for information collecting and processing which con- sists of a set of sources providing information about formulas, and a processor. For each formula ' (not only an atomic one, like in (3)!), a source can say that ' is true (1), or false (0), or that it knows nothing about '. The processor collects information from
We present a formalization of the axiomatic set theory ZF which reflects real mathematical practi... more We present a formalization of the axiomatic set theory ZF which reflects real mathematical practice, and is easy for mechanical manipulation and interactive theorem proving. Unlike the standard first-order formalizations, our version provides a rich class of abstraction terms denoting sets on the one hand, and is based on purely syntactical (rather than semantic) considerations on the other hand.
The main goal of the paper is to suggest some analytic proof systems for LCand its nite-valued co... more The main goal of the paper is to suggest some analytic proof systems for LCand its nite-valued counterparts which are suitable for proof-search. This goal isachieved through following the general Rasiowa-Sikorski methodology for constructinganalytic proof systems for semantically-dened logics. All the systems presentedhere are terminating, contraction-free, and based on invertible rules, which have alocal character and at most two premises.
This paper is devoted toa description, with many examples, of one particular framework of this so... more This paper is devoted toa description, with many examples, of one particular framework of this sort: that ofhypersequents. We shall show that this framework is indeed stronger than that of ordinarysequents, but still has properties (1)-(6) above. In addition, the following two pointsconcerning it should be noted:
We present a terminating contraction-free calculus GLC * for the propositional fragment of Gödel-... more We present a terminating contraction-free calculus GLC * for the propositional fragment of Gödel-Dummett Logic LC. GLC * uses hypersequents, and unlike other Gentzen-type calculi for LC, all its rules have at most two premises. These rules are all invertible. Hence it can be used as a basis for a deterministic tableau system for LC. This tableau system is presented
We provide non-deterministic semantics for the 3 basic paraconsistent C-systems C (also known as ... more We provide non-deterministic semantics for the 3 basic paraconsistent C-systems C (also known as bC), Ci, and Cia, as well as to all 9 extensions of them by one or two of the schemata (l) ¬(φΛ¬φ) ⊃ ∘ φ and (e) φ ⊃ ¬¬φ. This includes da Costa’s original C 1 (which is equivalent to Cila). Our semantics is 3-valued for the systems without (l), and infinite-valued for the systems with it. We prove that these results cannot be improved: neither of the systems without (l) has either a finite characteristic ordinary matrix or a two-valued characteristic non-deterministic matrix, and neither of the systems with (l) has a finite characteristic non-deterministic matrix. Still, our semantics suffices for providing decision procedures for all the systems investigated.
Gödel’s main contribution to set theory is his proof that GCH is consistent with ZFC (assuming th... more Gödel’s main contribution to set theory is his proof that GCH is consistent with ZFC (assuming that ZF is consistent). For this proof he has introduced the important ideas of constructibility of sets, and of absoluteness of formulas. In this paper we show how these two ideas of Gödel naturally lead to a simple unified framework for dealing with computability of functions and relations, domain independence of queries in relational databases, and predicative set theory.
We consider a framework for information collecting and processing which con- sists of a set of so... more We consider a framework for information collecting and processing which con- sists of a set of sources providing information about formulas, and a processor. For each formula ' (not only an atomic one, like in (3)!), a source can say that ' is true (1), or false (0), or that it knows nothing about '. The processor collects information from
We present a formalization of the axiomatic set theory ZF which reflects real mathematical practi... more We present a formalization of the axiomatic set theory ZF which reflects real mathematical practice, and is easy for mechanical manipulation and interactive theorem proving. Unlike the standard first-order formalizations, our version provides a rich class of abstraction terms denoting sets on the one hand, and is based on purely syntactical (rather than semantic) considerations on the other hand.
The main goal of the paper is to suggest some analytic proof systems for LCand its nite-valued co... more The main goal of the paper is to suggest some analytic proof systems for LCand its nite-valued counterparts which are suitable for proof-search. This goal isachieved through following the general Rasiowa-Sikorski methodology for constructinganalytic proof systems for semantically-dened logics. All the systems presentedhere are terminating, contraction-free, and based on invertible rules, which have alocal character and at most two premises.
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Papers by Arnon Avron