Nissim Francez
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Phone: (+972)(4) 9928444
Address: Office: Faculty of computer science, Technion, Haifa 32000, Israel
Home: 10B Hanita st., Nahariya 22385, Israel
Phone: (+972)(4) 9928444
Address: Office: Faculty of computer science, Technion, Haifa 32000, Israel
Home: 10B Hanita st., Nahariya 22385, Israel
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Papers by Nissim Francez
providing a different way to answer the following question:
what does it mean that some object language formula
does not have a given truth-value.
Along the way, the paper provides a general definition of truth and falsity
in an arbitrary many-valued logic.
and prove soundness and strong completeness theorems. The calculi
are obtained from the truth tables of the logic under consideration in a
straightforward manner and there is a natural duality among these calculi.
We also prove the cut elimination theorems for the sequent-like systems.
truth-value constants: verum and falsum, that are, respectively, true and false under every bivalent
valuation.
We are interested to de ne and study analogical constants
that in an arbitrary multi-valued logic over truth-values
have the truth-value v_i under every (multi-valued) valuation. We Definition ne such constants
proof-theoretically via their associated I/E-rules in
a natural-deduction proof system.
As is well known, the absence or presence of such constants has a significant deductive impact on the logics studied.
In particular, we propose a generalization of the notions of contradiction and explosiveness of a logic to the context of multi-valued logics.
when the embedded conditional is a connexive conditional,
in place of the classical material conditional.
The main tool used is class-definition scheme, standard in classical 1st-order logic, in which:
— class-formation is defined by satisfaction (in models) of an open formula.
— class membership is define by the satisfaction of predication.
— intersection, union and complementation is defined by conjunction, disjunction and negation, respectively.
— boundary classes (empty and universal) is defined by identical falsity and identical truth of an open formula.
— class inclusion is defined by restricted quantification.
The model definition of a connexive class theory is McCall’s connexive algebra CA.
It is shown that:
1. Wansing 1st-order connexive logic QC does not lead to a CA by the above scheme.
2. A first order extension of Priest’s 1st-order QC, based on the null account of
the truth-condition for the conditional, does not lead either to a CA under the above scheme..
3. A variant of the 1st-order extension of Priests connexive logic P_S,
based on a partial account, does lead to a CA under the above scheme.
I/E-rules* in a meaning conferring ND-system.
Examples of such parametric constants are conjunction/disjunction of varying arity, and higher-order quantification of any order.
As a result, it is claimed that there is *one concept* of conjunction, disjunction and quantification.
leading to a family of contra-classical logics. The two main ideas are the following:
1. Instead of shifting points of evaluation (in a frame), shift the evaluated formula.
2. Introduce an incompatibility set for every atomic formula, extended to any compound formula, and impose the condition on valuations that a formula evaluates to true iff all the formulas in its incompatibility set evaluate to false. Thus, atomic sentences are not independent in their truth-values.
The resulting negation, in addition to excluding the negated formula, provides a positive alternative
to the negated formula.
I also present a sound and complete natural deduction proof systems for those logics.
In addition, the kind of negation considered in this paper is shown to provide an innovative notion of grounding negation.
1. A brief exposition of proof-theoretic semantics (PTS), not necessarily in connection to natural language (NL).
2. A review, with a contrastive flavour, of some of the applications of PTS to NL with an indication of advantages of PTS as a theory of meaning for NL.
providing a different way to answer the following question:
what does it mean that some object language formula
does not have a given truth-value.
Along the way, the paper provides a general definition of truth and falsity
in an arbitrary many-valued logic.
and prove soundness and strong completeness theorems. The calculi
are obtained from the truth tables of the logic under consideration in a
straightforward manner and there is a natural duality among these calculi.
We also prove the cut elimination theorems for the sequent-like systems.
truth-value constants: verum and falsum, that are, respectively, true and false under every bivalent
valuation.
We are interested to de ne and study analogical constants
that in an arbitrary multi-valued logic over truth-values
have the truth-value v_i under every (multi-valued) valuation. We Definition ne such constants
proof-theoretically via their associated I/E-rules in
a natural-deduction proof system.
As is well known, the absence or presence of such constants has a significant deductive impact on the logics studied.
In particular, we propose a generalization of the notions of contradiction and explosiveness of a logic to the context of multi-valued logics.
when the embedded conditional is a connexive conditional,
in place of the classical material conditional.
The main tool used is class-definition scheme, standard in classical 1st-order logic, in which:
— class-formation is defined by satisfaction (in models) of an open formula.
— class membership is define by the satisfaction of predication.
— intersection, union and complementation is defined by conjunction, disjunction and negation, respectively.
— boundary classes (empty and universal) is defined by identical falsity and identical truth of an open formula.
— class inclusion is defined by restricted quantification.
The model definition of a connexive class theory is McCall’s connexive algebra CA.
It is shown that:
1. Wansing 1st-order connexive logic QC does not lead to a CA by the above scheme.
2. A first order extension of Priest’s 1st-order QC, based on the null account of
the truth-condition for the conditional, does not lead either to a CA under the above scheme..
3. A variant of the 1st-order extension of Priests connexive logic P_S,
based on a partial account, does lead to a CA under the above scheme.
I/E-rules* in a meaning conferring ND-system.
Examples of such parametric constants are conjunction/disjunction of varying arity, and higher-order quantification of any order.
As a result, it is claimed that there is *one concept* of conjunction, disjunction and quantification.
leading to a family of contra-classical logics. The two main ideas are the following:
1. Instead of shifting points of evaluation (in a frame), shift the evaluated formula.
2. Introduce an incompatibility set for every atomic formula, extended to any compound formula, and impose the condition on valuations that a formula evaluates to true iff all the formulas in its incompatibility set evaluate to false. Thus, atomic sentences are not independent in their truth-values.
The resulting negation, in addition to excluding the negated formula, provides a positive alternative
to the negated formula.
I also present a sound and complete natural deduction proof systems for those logics.
In addition, the kind of negation considered in this paper is shown to provide an innovative notion of grounding negation.
1. A brief exposition of proof-theoretic semantics (PTS), not necessarily in connection to natural language (NL).
2. A review, with a contrastive flavour, of some of the applications of PTS to NL with an indication of advantages of PTS as a theory of meaning for NL.