Modal Logic

  • Ted Sider, Fall, 1998

Course Description

This will be an introductory course on systems of logic whose natural semantics are of the possible worlds variety, including propositional and predicate modal logic, counterfactual conditionals, and tense logic. Additional topics may include counterpart theory, multiple indexing and supervenience.


Any standard introductory course in propositional and predicate logic.


Required text: G. E. Hughes and M. J. Cresswell, A New Introduction to Modal Logic. Note not to panic: we will not cover all of the material in Hughes and Cresswell, not even all the material in the assigned portions; this course will be a little less logically "high-powered" than Hughes and Cresswell. Note on symbolism: I will not use the same symbols as used by Hughes and Cresswell, so care must be taken in integrating class discussion with the discussion in the text.

Other reading material: If you're interested, you could get a copy of David Lewis's Counterfactuals, but there's no need to buy this; selections will be placed for copying in the graduate lounge. Similar remarks apply to L. T. F. Gamut, Logic, Language, and Meaning, vol 2. Additionally, some or all of my notes will be made available in the graduate lounge for copying.


Two or three in-class exams; dates and times to be announced. Graduate students may be asked to answer different exam questions than undergraduates. I will hand out homework assignments periodically. These may not be collected or graded; but in any case their completion is highly recommended. I will put answers to selected homework problems on reserve. Exams will consist partly of exercises like those on homework, and partly of short-answer questions.

Course Outline

Note: Hughes and Cresswell readings are listed in the outline; for readings in my notes, simply read the material corresponding to the headings in the outline. Note that since I don't go in exactly the same order as Hughes and Cresswell, sometimes the topics in their readings won't exactly match my presentation.

  1. Intro stuff
  2. Reading: Hughes and Cresswell, pp. 3-13

    1. Logic
      1. Form and abstraction
      2. The "correctness" of logical systems
    2. Modal Logic
    3. Historical remarks on modal logic
    4. Metatheory of propositional logic (PL)
    5. Reading: Elliott Mendelson, Introduction to Mathematical Logic, pp. 27-35.

      1. Language of PL
      2. PL and the axiomatic approach
      3. semantics for PL
      4. soundness and completeness
      5. Limitations of PL
  3. Modal propositional logic (MPL)
  4. Reading: Hughes and Cresswell, pp. 13-17, 23-56

    1. wffs
    2. translations
    3. Axiomatic systems: K, D, T, B, S4, S5
  5. Semantics for MPL
  6. Reading: Hughes and Cresswell, pp. 23-68

    1. A digression: Naive Set Theory
    2. Models
    3. Validity
    4. Semantic validity proofs
    5. Countermodels
    6. Reading: Hughes and Cresswell, chapter 4

    7. Soundness
    8. Completeness
    9. Reading: Hughes and Cresswell, chapter 6

    10. Tense logic
    11. Reading: Hughes and Cresswell, 127-134; L. T. F. Gamut, pp. 32-39

  7. Subjunctive/Counterfactual Conditionals
  8. Reading: David Lewis, Counterfactuals, chapter 1, plus sections 3.4 and 4.2; copy in the graduate lounge

    1. Features of English counterfactuals
    2. The rough idea of the Lewis/Stalnaker view
    3. Stalnaker's system
      1. syntax
      2. semantics
      3. validity proofs
      4. countermodels
      5. examples
      6. features peculiar to MSC
    4. Lewis's System
    5. A problem with both
  9. Quantified Modal Logic
  10. Reading: Hughes and Cresswell, chapter 13

    1. Tarski-style semantics for LPC
      1. syntax
      2. semantics
    2. Modal LPC
      1. syntax
      2. translations
      3. semantics
      4. examples
    3. Inconstant domains
    4. Reading: Hughes and Cresswell, chapters 15-16

    5. Identity and Descriptions
    6. Reading: Hughes and Cresswell, chapter 17

  11. Other systems...