Possible Worlds Semantics for Indicative and Counterfactual Conditionals?: A Formal Philosophical Inquiry into Chellas-Segerberg Semantics

✍ Scribed by Matthias Unterhuber

• Publisher: De Gruyter
• Year: 2013
• Tongue: English
• Leaves: 356
• Series: Logos; 21
• Category: Library

✦ Synopsis

Conditional structures lie at the heart of the sciences, humanities, and everyday reasoning. This is why conditional logics – logics specifically designed to account for natural language conditionals – are an active, interdisciplinary area. Discussing a wide range of topics, this book gives a formal and a philosophical account of indicative and counterfactual conditionals in terms of Chellas-Segerberg semantics.

✦ Table of Contents

Preface
I Foundational Issues
Arguments for Conditional Logics
The Framework of My Investigation
Indicative Conditionals
Counterfactual Conditionals
Normic Conditionals
The Generic Case
The Qualification Problem
The Propositional Case
The Role of Normic Conditionals
Conversational Implicatures
The Conditional Logic Project in an Interdisciplinary Context and Default Logics
The Conditional Logic Project in an Interdisciplinary Context
The Conditional Logic Project
The Linguistics of Conditionals Project
The Philosophy of Conditionals Project
The Psychology of Reasoning Project
The Non-Monotonic Logic Project
Non-Monotonic Logics, Defaults Logics, and Conditional Logics
A Motivation for the Study of Non-Monotonic Logics
Reiter Defaults
Default Logics, Non-Monotonic Rules, and Inductive Inferences
Types of Non-Derivability Conditions and the Rule of Substitution
Model Theory, Proof Theory, and Axiomatization of Default Logics
Conditional Logics and Default Logics
Possible Worlds Semantics and Probabilistic Semantics for Indicative Conditionals: a Survey and a Defense of Possible Worlds Semantics
Outline of My Defense of Possible Worlds Semantics for Indicative Conditionals and the Core Idea of Chellas-Segerberg Semantics
Possible Worlds Ordering Semantics for Conditionals: D. Lewis' and Kraus et al.'s Semantics and Related Approaches
The Ramsey Test, Stalnaker Semantics, and a General Ramsey Test Requirement
Ramsey's Original Proposal
Stalnaker's Version of the Ramsey Test
Stalnaker Models
Stalnaker Semantics, Set Selection Semantics, and Chellas-Segerberg Semantics
Contrasting Ramsey Test Interpretations of Conditionals and Ordering Semantics
Stalnaker Semantics, Conditional Consistency Criteria, and the Principle of Conditional Excluded Middle
Bennett's Version of the Ramsey Test
A General Ramsey Test Requirement
Indicative and Counterfactual Conditionals and Conditional Logics
Criteria for Distinguishing Indicative and Counterfactual Conditionals
Bridge Principles and Logics for Indicative and Counterfactual Conditionals
Subjective and Objective Interpretations of Indicative and Counterfactual Conditionals and the Ramsey Test
Fundamental Issues of Probabilistic Approaches to Conditional Logic
Subjective and Objective Probabilistic Semantics and the Principle of Total Evidence
The Stalnaker Thesis, the Ramsey Test, and Conditional or Non-Conditional Probabilities as Primitive
Languages of a Probabilistic Conditional Logic
Further Reasons for the Restriction of Languages of Probabilistic Conditional Logics
Propositions, NTV (``No Truth Value''), and Conditional Logic Languages
Adams' Probabilistic P Systems
The P Systems: Systems P, P, and P+
Threshold Semantics
Adams' System P and Schurz's Modification
Possible Worlds Semantics for (Indicative) Conditionals and Quasi Truth Value Assignments in Adams' Probabilistic Semantics
D. Lewis' Triviality Result and Logics for Indicative Conditionals
D. Lewis' Triviality Result
Triviality due to Iterations (and Nestings) of Conditional Formulas
D. Lewis' Triviality Result, Restrictions of Languages, and Truth Value Accounts of Indicative Conditionals
Conclusion
Bennett's Gibbardian Stand-Off Argument
Bennett's Extended Gibbardian Stand-Off Argument
A Criticism of Bennett's Gibbardian Stand-Off Argument
Summary
Conclusion
II Formal Results for Chellas-Segerberg Semantics
Formal Framework
Why Chellas-Segerberg Semantics?
Proof-Theoretic Notions
Languages LCL, LCL-, LrCL, LrCL, and LrrCL
Logics
Non-Monotonicity
Consistency and Maximality
A Propositional Basis for Conditional Logics
System CK
A Further Axiomatization of System CK
Model-Theoretic Notions
Chellas Frames and Chellas Models
Chellas Models and Frames and Kripke Semantics
Segerberg Frames and Segerberg Models
Validity, Logical Consequence, and Satisfiability
Notions of Frame Correspondence
Standard and Non-Standard Chellas Models and Segerberg Frames
Notions of Soundness and Completeness
Frame Correspondence for a Lattice of Conditional Logics
Non-Trivial Frame Conditions for a Lattice of Conditional Logics
The Notions of Trivial and Non-Trivial Frame Conditions
A Translation Procedure from Axiom Schemata to Trivial Frame Conditions
A Non-Triviality Criterion
Chellas Frame Correspondence Proofs
System P
Extensions of System P
Weak Probability Logic (Threshold Logic)
Monotonic Principles
Bridge Principles
Collapse Conditions Material Implication
Traditional Extensions
Iteration Principles
Soundness and Completeness for a Lattice of Conditional Logics
General Overview
Focus of the Completeness Proofs
Proofs for Segerberg Frame Completeness and Chellas Frames Completeness
Singleton Frames for CS Semantics
Soundness w.r.t. Classes of Chellas Frames
Standard Segerberg Frame Completeness
General Principles
Canonical Models
Canonicity Proofs for Individual Principles
Chellas-Segerberg Semantics for Indicative and Counterfactual Conditionals
The Basic Chellas-Segerberg Systems (Systems CK and CKR)
Objective and Subjective Interpretations of Chellas-Segerberg Semantics
Alternative Axiomatizations of System CKR
Conditional Logics without Bridge Principles
System C
System CL
System P
System R
D. Lewis' System V
Monotonic Systems without Bridge Principles (Systems CM and M)
Conditional Logics with Bridge Principles
Adams' System P*
D. Lewis' System VC
Stalnaker's System S
The Material Collapse System MC
Summary
Concluding Remarks
References