PHIL210 -- Slides-16

Page 1: Introduction to Deductive Logic

  • Course: Phil 210, McGill University

  • Instructor: Michael Hallett

Page 2: Readings and Exercises

  • Contents:

    • Extensions to the Language, Part III — Identity

    • The Languages of First-Order Logic (FOL), Definition

    • Interpreting Truth in FOL

Page 3: Readings and Exercises Continuation

  • Reading Assignment:

    • Chapter 26 (Part V)

    • Chapters 29-30 (Part VI) of forall x: Calgary

    • Note: Chapter 30 will be treated more thoroughly than in the textbook.

  • Practice Exercises:

    • Focus on Carnap's work including quantifiers, identity, definite descriptions, and cognate formulations.

Page 4: Table of Contents Reiteration

  • Included Topics:

    • Extensions to the Language, Identity Definitions

    • Languages of FOL

    • Truth Interpretation in FOL

Page 5: Extensions to Language: Definite Descriptions

  • Russell’s Idea:

    • Addresses terms like ‘both’ and ‘neither’ when used as determiners.

    • Example: “Neither villain is younger than Greta” is true under specific conditions involving two villains.

    • Formal Representation:

      [ \exists x \exists y (((V(x) \land V(y)) \land
      eg x = y) \land \forall z(V(z) \rightarrow (z = x \lor z = y))) \land (
      eg Y(x, g) \land
      eg Y(y, g))) ]

    • This shows the complexity involved in working with multiple individuals in FOL.

Page 6: Overview of ‘Neither, Nor’ Condition

  • Explanation:

    • Presents a formula through existential quantifiers outlining conditions related to villains not being younger than Greta.

    • Comparison of treatments for similar propositions.

Page 7: Alternative Formulations of ‘Neither, Nor’

  • Multiple expressions representing the same logical scenarios in different forms:

    [ \exists x \exists y( \forall z(V(z) \leftrightarrow (z = x \lor z = y)) \land (
    eg x = y \land (
    eg Y(x, g) \land
    eg Y(y, g)))) ]

Page 8: Important Translation Points

  • Guidelines:

    1. Use ‘∧’ for existential claims to strengthen claims.

    2. Use ‘→’ for universal claims to weaken claims.

    3. A sentence like ‘∀x(A(x) → B(x))’ emphasizes truth for A under condition B.

    4. Clarifies difference in claims based on assignment of properties.

Page 9: Importance of Quantifier Ordering

  • Different Implications based on Quantifier Order:

    • ∃x∀yA(x, y): There is someone who admires everyone.

    • ∀y∃xA(y, x): Everyone admires someone.

Page 10: Reflection on FOLs

  • Common Skepticism:

    • FOL's cumbersome nature makes it hard to express subtle language.

    • However, it effectively manages to convey complex ideas with simplicity, which raises philosophical questions.

Page 11: Further Recap of Table of Contents

  • Topics Remain:

    • Extensions involving identity and truth in FOL.

Page 12: Description of FOLs

  • Goals:

    • Provide a precise description of FOL syntax.

    • Define truth and validity for arguments in FOLs.

Page 13: Definition of FOL Part I: Syntax

  • Elements:

    1. List of NAMES (e.g., a1, a2)

    2. List of VARIABLES (e.g., x1, x2)

    3. List of PREDICATES based on arity.

    4. CONNECTIVES: ¬, ∧, ∨, →, ↔

    5. Two QUANTIFIERS: ∃, ∀

    6. Equality Predicate: =

    7. Brackets: for grouping terms.

Page 14: Definition of FOL Part II: Expressions & Terms

  1. Expressions: Any finite string of defined symbols.

  2. Terms: Can be a name or variable.

Page 15: Defining FOL Formulas

  • Atomic Formulas:

    1. Form based on predicates (e.g., Pnk(t1, t2,...)).

    2. Equality (e.g., t1 = t2).

    3. Formulas developed using connectives and quantifiers.

    4. Bound vs Free variables are crucial in understanding sentences.

Page 16: Distinction between Formulas and Sentences

  • Hierarchy: Expressions ⇝ Formulas ⇝ Sentences

  • Important Note:

    • Not all formulas are sentences; sentences must have no free variables.

Page 17: Forming Sentences from Formulas

  • Role of Quantified statements in transforming formulas into sentences through variable binding.

Page 18: Names, Predicates, and Atomic Sentences

  • Formation of sentences from atomic elements using logical predicates and the importance of proper variable assignments.

Page 19: Table of Contents Reiteration

  • Recap of topics covered.

Page 20: Interpretations of Atomic Sentences

  1. Discuss the concept of domain selection for interpretation in FOL.

  2. Address how truth-values assign to atomic sentences based on definitions and assignments.

Page 21: Truth Values of Atomic Sentences

  • Example:

    • If ‘m’ names a student in a specific domain, that influences the true/false evaluation of sentences pertaining to that individual.

Page 22: Making Formulas True Overview

  • Importance of building truth values of complex formulas from the truth values of their simpler subformulas and assignments.

Page 23: Assigning Objects to Variables

  • Notation for variable assignments under interpretations and satisfaction functions to verify truth of formulas.

Page 24: Truth of Compounded Formulas

  • Using Truth-Functional Connectives: Truth values for compounded formulas derived from atomic truths.

Page 25: Truth of Quantified Sentences: Existential

  • Definition of success for existential claims requiring a valid object assignment for the truth of a statement.

Page 26: Truth of Quantified Sentences: Universal

  • Definition of universal claims requiring satisfaction across all domain objects under varying assignments.