Bypassing Lewis’s Triviality Results. A Kripke-Style Partial Semantics for Compounds of Adams’s Conditionals
Academic Article
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Short description:
Bypassing Lewis’s Triviality Results. A Kripke-Style Partial Semantics for Compounds of Adams’s Conditionals / Mura, Alberto Mario. - In: LOGICA UNIVERSALIS. - ISSN 1661-8300. - (In corso di stampa).
abstract:
Abstract According to Lewis’s Triviality Results (LTR), conditionals cannot
satisfy the equation (E) P(C if A) = P(C | A), except in trivial cases. Since
1935 [9], de Finetti showed that by three-valued truth-tables, the equation
might be satisfied in a general way (at least for indicative conditionals). This
result is not at odds with LTR, because one of its premises, namely that conditionals
are two-valued sentences, is dropped. Unfortunately, de Finetti did
not equip his truth-tables by a proper notion of logical consequence. Ernst
Adams [1], however, provided a probabilistic semantics for the so-called, simple
conditionals (that is for those sentences of the form ‘if A then C’ where
both A and C are two-valued ordinary sentences) that also satisfies equation
(E) and provides a probabilistic counterpart of logical consequence (called pentailment).
Adams’s logic is coextensive to Stalnaker’s and Lewis’s logic as
far as simple conditionals are concerned. A theorem, proved by McGee [22],
shows that no truth-functional many-valued logic allows a relation of logical
consequence coextensive with Adams’s p-entailment. This result applies to
de Finetti’s truth-functional semantics.
This paper presents amodifiedmodal (Kripke-style) version of de Finetti’s
semantics that escapesMcGee’s result and provides a general truth-conditional
semantics for indicative conditionals, without being affected by LTR. The
new framework encompasses and extends Adams’s probabilistic semantics
(APS) to compounds of conditionals. Like APS, the present truth-conditional
semantics does not deal with counterfactual conditionals. From the philosophical
side, this theory challenges the view (endorsed by many authors)
that indicative conditionals lack truth-conditions and show that a truth conditional
semantics bypassing LTR is possible. Moreover, a comparison of
the present theory with Stalnaker’s theories about compounds of conditionals
shows that the present theory presents inherent advantages, especially
regarding the import-export law.
satisfy the equation (E) P(C if A) = P(C | A), except in trivial cases. Since
1935 [9], de Finetti showed that by three-valued truth-tables, the equation
might be satisfied in a general way (at least for indicative conditionals). This
result is not at odds with LTR, because one of its premises, namely that conditionals
are two-valued sentences, is dropped. Unfortunately, de Finetti did
not equip his truth-tables by a proper notion of logical consequence. Ernst
Adams [1], however, provided a probabilistic semantics for the so-called, simple
conditionals (that is for those sentences of the form ‘if A then C’ where
both A and C are two-valued ordinary sentences) that also satisfies equation
(E) and provides a probabilistic counterpart of logical consequence (called pentailment).
Adams’s logic is coextensive to Stalnaker’s and Lewis’s logic as
far as simple conditionals are concerned. A theorem, proved by McGee [22],
shows that no truth-functional many-valued logic allows a relation of logical
consequence coextensive with Adams’s p-entailment. This result applies to
de Finetti’s truth-functional semantics.
This paper presents amodifiedmodal (Kripke-style) version of de Finetti’s
semantics that escapesMcGee’s result and provides a general truth-conditional
semantics for indicative conditionals, without being affected by LTR. The
new framework encompasses and extends Adams’s probabilistic semantics
(APS) to compounds of conditionals. Like APS, the present truth-conditional
semantics does not deal with counterfactual conditionals. From the philosophical
side, this theory challenges the view (endorsed by many authors)
that indicative conditionals lack truth-conditions and show that a truth conditional
semantics bypassing LTR is possible. Moreover, a comparison of
the present theory with Stalnaker’s theories about compounds of conditionals
shows that the present theory presents inherent advantages, especially
regarding the import-export law.
Iris type:
1.1 Articolo in rivista
Keywords:
Indicative conditionals, Lewis's Triviality Results, Adams's Probabilistic Logic, Partial
modal semantics, de Finetti, Tri-events, Compound of conditionals, Import-Export
Law, Conditional Semantics
List of contributors:
Mura, Alberto Mario
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