A Paraconsistent Conditional Logic
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A Paraconsistent Conditional Logic Minghui Ma1 · Chun-Ting Wong1 Received: 3 April 2019 / Accepted: 25 November 2019 / © Springer Nature B.V. 2019
Abstract We develop a paraconsistent logic by introducing new models for conditionals with acceptive and rejective selection functions which are variants of Chellas’ conditional models. The acceptance and rejection conditions are substituted for truth conditions of conditionals. The paraconsistent conditional logic is axiomatized by a sequent system C which is an extension of the Belnap-Dunn four-valued logic with a conditional operator. Some acceptive extensions of C are shown to be sound and complete. We also show the finite acceptive model property and decidability of these logics. Keywords Paraconsistency · Conditional · Polarity semantics
1 Introduction Conditional logic has been a name for logical theories concerning the nature of conditionality. The contemporary development of conditional logic can be traced back to a famous footnote known today as the Ramsey test: If two people are arguing ‘If p, then q?’ and are both in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q; so that in a sense ‘If p, q’ and ‘If p, q’ are contradictories. We can say that they are fixing their degree of belief in q given p. If p turns out false, these degrees of belief are rendered void. If either party believes not p for certain, the question ceases to mean anything to him except as a question about what follows from certain laws or hypotheses. (cf. [17])
Minghui Ma
[email protected] Chun-Ting Wong [email protected] 1
Department of Philosophy, Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China
M. Ma, C.-T. Wong
Ramsey suggested a way to evaluate conditionals. We suppose firstly that the antecedent of a conditional is true and see whether it has any effects upon the consequent. That is the two people in the argument are fixing their degree of belief in the consequent given the antecedent. Edgington [7] claimed that the idea comes from Ramsey’s 1926 work on fundamental laws of probabilistic belief [15], where degree of belief was interpreted by probability, e.g., degree of belief in (p and q) = degree of belief in p × degree of belief in q given p. According to Ramsey’s 1927 work [16], conditionals are not truth bearers but having rational acceptance conditions. Stalnaker [18] noticed that Ramsey’s test covers only the case in which the agent has no opinion on the truth value of the antecedent. This motivates Stanaker to generalize Ramsey’s test to cover the remaining cases. The case that the agent believes the antecedent to be true is trivial; in this case, the stock of belief will not be changed. For the case that the agent believes the antecedent to be false, these beliefs in the stock that conflict with the antecedent should be adjusted to maintain consistency. In this case, Stalnaker suggested the following evaluation of a conditional: First, add the antecedent (hypothetically) t
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