Logics of Variable Inclusion
APPLIED LOGIC, PHILOSOPHY AND HISTORY OF SCIENCE
APPLIED LOGIC, PHILOSOPHY AND HISTORY OF SCIENCE
University of Cagliari, Italy
University of Cagliari, Italy
email: paoli@unica.it
email: paoli@unica.it
Two seemingly unrelated approaches to (propositional) logical consequence will be compared in this tutorial. According to the former one, rooted in Kant'ss notion of analyticity but especially championed by exponents of the C.I. Lewis school at Harvard (Baylis, Nelson, Parry) in the debate about paradoxes of entailment, a conclusion φ validly follows from certain premisses in Γ in case the meaning of φ is "analytically contained" in the meanings of the members of Γ . In this approach, only significant sentences are taken to be of interest to the logician. In the latter perspective, defended among others by Bochvar, Halldén, Kleene, Goddard and Routley, consequence is more traditionally viewed as truth preservation (or as preservation of some other semantic property), but in the presence of possibly non-significant (e.g. meaningless) sentences. The early discussion on these logics of significance has focussed on many-valued propositional logics with infectious truth values, namely, values that are assigned to a sentence φ if and only if they are assigned to some propositional variable occurring in φ.
Two seemingly unrelated approaches to (propositional) logical consequence will be compared in this tutorial. According to the former one, rooted in Kant'ss notion of analyticity but especially championed by exponents of the C.I. Lewis school at Harvard (Baylis, Nelson, Parry) in the debate about paradoxes of entailment, a conclusion φ validly follows from certain premisses in Γ in case the meaning of φ is "analytically contained" in the meanings of the members of Γ . In this approach, only significant sentences are taken to be of interest to the logician. In the latter perspective, defended among others by Bochvar, Halldén, Kleene, Goddard and Routley, consequence is more traditionally viewed as truth preservation (or as preservation of some other semantic property), but in the presence of possibly non-significant (e.g. meaningless) sentences. The early discussion on these logics of significance has focussed on many-valued propositional logics with infectious truth values, namely, values that are assigned to a sentence φ if and only if they are assigned to some propositional variable occurring in φ.
In the last decades, some of the best-known logics of significance have been characterised in terms of variable inclusion constraints between premisses and conclusion. These results have suggested an a¢ nity with some logics of analytic entailment, like Parry's logic, where similar constraints are at work. Moreover, a general pattern has emerged for linking logics of significance and logics of variable inclusion. Within this framework, which makes recourse to universal algebraic tools developed for completely di¤erent motivations (semilattice direct systems and Plonka sums of algebras) and to the machinery of Abstract Algebraic Logic (AAL), it possible to see that logics with infectious truth values are just the tip of the iceberg of a more subtle, more general and potentially much more interesting semantic phenomenon.
In the last decades, some of the best-known logics of significance have been characterised in terms of variable inclusion constraints between premisses and conclusion. These results have suggested an a¢ nity with some logics of analytic entailment, like Parry's logic, where similar constraints are at work. Moreover, a general pattern has emerged for linking logics of significance and logics of variable inclusion. Within this framework, which makes recourse to universal algebraic tools developed for completely di¤erent motivations (semilattice direct systems and Plonka sums of algebras) and to the machinery of Abstract Algebraic Logic (AAL), it possible to see that logics with infectious truth values are just the tip of the iceberg of a more subtle, more general and potentially much more interesting semantic phenomenon.
Prerequisites. This tutorial will be based on the material in [1]. The first class is suitable for anybody who has attended an introductory logic course. The second class requires a basic knowledge of universal algebra. The third class requires a basic knowledge of universal algebra and AAL.
Prerequisites. This tutorial will be based on the material in [1]. The first class is suitable for anybody who has attended an introductory logic course. The second class requires a basic knowledge of universal algebra. The third class requires a basic knowledge of universal algebra and AAL.
1. First class: Philosophy. Different approaches to analytic entailment. Logics of signi
cance and infectiousness. The weak Kleene logics: B3 and PWK. Philosophical interpretation of weak Kleene logics. From analyticity to signifiance via variable inclusion. Beyond infectiousness.
1. First class: Philosophy. Different approaches to analytic entailment. Logics of signi
cance and infectiousness. The weak Kleene logics: B3 and PWK. Philosophical interpretation of weak Kleene logics. From analyticity to signifiance via variable inclusion. Beyond infectiousness.
2. Second class: Algebra. Semilattice direct systems of algebras. Plonka sums of algebras. The Plonka representation theorem. Regular varieties. Generalised involutive bisemilattices.
2. Second class: Algebra. Semilattice direct systems of algebras. Plonka sums of algebras. The Plonka representation theorem. Regular varieties. Generalised involutive bisemilattices.
3. Third class: Logic. Left- and right-direct systems of matrices. Plonka sums of matrices. Left and right variable inclusion logics and their AAL properties. Examples. Logics of pure variable inclusion.
3. Third class: Logic. Left- and right-direct systems of matrices. Plonka sums of matrices. Left and right variable inclusion logics and their AAL properties. Examples. Logics of pure variable inclusion.
References
References
[1] S. Bonzio, F. Paoli, M. Pra Baldi, Logics of Variable Inclusion: A Logico-Algebraic Approach, in preparation.
[1] S. Bonzio, F. Paoli, M. Pra Baldi, Logics of Variable Inclusion: A Logico-Algebraic Approach, in preparation.