Conditional Sentences in Belief Revison Systems

• Author / Creator

  Ozcevik, Ozkan

• The first chapter of the thesis presents Frank P. Ramsey [1960]’s seminal treatment of “If ... , then ...” statements. We also explain how Stalnaker and Thomason [1970] picked up on Ramsey’s idea and undertook the task of giving truth conditions for counterfactual conditionals in contrast to Ramsey’s insistence on rational acceptability conditions. The second chapter is concerned with technicalities related to the proof theory of modern conditional logics. A Fitch-style natural deduction system for the Stalnaker/Thomason sentential conditional logic FCS already exists in the literature [Thomason, 1971]. Here we adjust FCS in a way to arrive at a proof system for Lewis’s “official” conditional logic VC [Lewis, 1973]. We begin with expositions of Stalnaker/Thomason’s CS/FCS and Lewis’s VC. Next, we explain why FCS in its original form is incompatible with VC. Interestingly, it turns out that the strict reiteration rule corresponding to the Uniqueness Assumption (“Stalnaker’s Assumption” in Lewis’s terminology) underlies the incompatibility of FCS and VC. We observe that Stalnaker’s Uniqueness Assumption becomes a very effective proof-theoretic device in natural deduction systems for conditional logics by virtue of allowing us to make use of “indirect conditional proofs”. In FCS, those indirect proofs allow us to derive the VC axioms of centering and rational monotony without need of additional strict reiteration rules.However, the problem we face is that since the characteristic feature of VC is its rejection of Stalnaker’s Assumption, we have no choice but to remove the conditional excluded middle strict reiteration rule and add two new strict reiteration rules (one for centering and one for rational monotony). After making the necessary adjustments to FCS, and thereby transforming it into FVC (that is, a Fitch-style natural deduction system for Lewis’s VC), we prove that FVC and VC are equivalent systems. We remark that Stalnaker/Thomason/Lewis conditional connectives are to be treated as multi-modal connectives (interpreted as “relativized necessity” in the style of Chellas [1980]). We argue that our findings here suggest that int-elim style inferentialism about logical connectives can be problematic in view of multi-modal connectives; that is, introduction and elimination rules alone cannot uniquely determine the “meanings” of such logical connectives. The results seem to show that reiteration rules and restrictions on those reiteration rules also are extremely important for determinations of “meanings” of multi-modal logical connectives. The third chapter of the thesis is largely expository: we introduce the AGM theory of belief change and point out the theory’s close connection with the analysis of conditional statements and the Ramsey Test. The fourth chapter is concerned with the question of whether an alternative doxastic semantics for VC is attainable. The answer is negative: belief change models that can validate Lewis’s VC (called “belief update”) are ontic models. Epistemic semantics for Stalnaker/Thomason/Lewis counterfactual conditional logics seems unattainable. The fifth chapter brings the thesis to a conclusion by summarizing our results.

• Subjects / Keywords

  • Natura deduction
  • Counterfactuals
  • Conditionals
  • Logic
  • Belief update
  • Proof theory
  • Belief revision
  • Ramsey test

• Graduation date

  Fall 2015

• Type of Item

  Thesis

• Degree

  Master of Arts

• DOI

  https://doi.org/10.7939/R38C9RC12

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.