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A Short Introduction to Modal Logic

Grigori Mints

Modal logic can be characterized as the logic of necessity and possibility- of must be and may be - and related concepts. This is a succinct introduction to modern modal logic suitable for introductory courses. Mints presents both semantic and syntactic features of modal logic and illustrates them by detailed analysis of the three best-known and most-often-sited modal systems, S5, S4, and T. The text presupposes some knowledge of classical propositional logic and its notation, however the necessary information is recapitulated at the outset. To avoid a potential source of difficulty for beginners, the axiomatic treatment of the systems is postponed to the final chapter. Mints concentrates on the logical aspects of the subject and provides philosophical motivations to show the point of the formal work. A set of exercises is included.

Grigori Mints (1939–2014) was a professor of philosophy and computer science at Stanford University and was previously a principal research associate at the Institute of Cybernetics of the Estonian Academy of Sciences.


  • Introduction
  • 1 Classical Propositional Logic
    • 1.1 Syntax
    • 1.2 Semantics: Truth Tables
    • 1.3 The Refutation Procedure
    • 1.4 Soundness and Completeness
    • 1.5 Substitution of Equivalents
    • 1.6 The Refutation Procedure Procedure and Disjunctive Normal Form

  • 2 Classical Monadic Predicate Logic
    • 2.1 Syntax
    • 2.2 Semantics: Valuation Rules
    • 2.3 The Refutation Procedure
    • 2.4 Termination Proof for Modal-like Formulas
    • 2.5 Soundness and Completeness

  • 3 The System
    • 3.1 The Syntax of Modal Logic
    • 3.2 Semantics: Possible Worlds and Valuations
    • 3.3 Comparison
    • 3.4 Indexed Formulas and Deduction Rules
    • 3.5 Deduction Rules
    • 3.6 Soundness and Completeness
    • 3.7 Substitution of Equivalents
    • 3.8 Reduction of Modalities

  • 4 System T
    • 4.1 Accessibility of Worlds and Valuation Rules
    • 4.2 Indexed Formulas, the System GT, and the Refutation Procedure
    • 4.3 Termination Proof, Soundness and Completeness
    • 4.4 First Degree Formulas, Substitution of Equivalents, and the Reduction of Modalities

  • 5 System S4
    • 5.1 System GS4 and the Refutation Procedure
    • 5.2 The Disjunction Property, Strict Implication, and Reduction of Modalities

  • 6 Hilbert-type Axiomization
  • Excercises


ISBN (Paperback): 9780937073759 (093707375X)
ISBN (Cloth): 9780937073766 (0937073768)
Subject: Logic; Modality

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