PHIL222 Lecture 8: Quantification and Syntax of MPL

Translations of Predicates

  • Predicate Translations:
  • Gx: x is green
  • Rx: x is red
  • Hx: x is heavy
  • Ex: x is expensive

Basic Logical Statements

  • Existence of Color: Something is both red and green.
  • Logical Form: ∃x(Rx ∧ Gx)
  • Disjunction of Color: Everything is either red or green.
  • Logical Form: ∀x(Rx ∨ Gx)
  • Implication of Heaviness: All red things are heavy.
  • Logical Form: ∀x(Rx → Hx)
  • Mistaken Translation: Would incorrectly imply ∀x(Rx ∧ Hx).
Further Translations
  • Some Red Things are Heavy:
  • Logical Form: ∃x(Rx ∧ Hx)
  • Red but Not Heavy:
  • Logical Form: ∃x(Rx ∧ ¬Hx)
  • Neither Red nor Heavy:
  • Logical Forms: ∃x(¬Rx ∧ ¬Hx) or ∃x¬(Rx ∨ Hx)
  • Conditional on Redness: If something is red, then it is heavy.
  • Logical Form: ∀x(Rx → Hx)

Negations

  • No Red Existence: Nothing is red.
  • Logical Form: ¬∃xRx or ∀x¬Rx
  • No Red Things are Heavy:
  • Logical Form: ∀x(Rx → ¬Hx) or ¬∃x(Rx ∧ Hx)
  • Existence of Not Red: Something isn’t red.
  • Logical Form: ∃x¬Rx or ¬∀xRx
  • Red and Heavy Exist Separately: Something is red and something is heavy.
  • Logical Form: ∃xRx ∧ ∃xHx

Restricted Quantification

  • Defined Terms:
  • Sx: x is special
  • Tx: x is tall
  • Incorrect Translation:
  • 'Everyone is special' should not be translated as ∀xSx.
  • 'Someone is tall' not simply ∃xTx.
  • Correct Translation Using Restricted Quantification:
  • Use Px for person:
  • 'Everyone is special' → ∀x(Px → Sx)
  • 'Someone is tall' → ∃x(Px ∧ Tx)

Application of Restricted Quantification

  • Tour Guide Example:
  • 'Everyone is ready' does not imply ∀xRx.
  • Correct form: ∀x((Px ∧ Tx) → Rx) where Tx indicates being in the tour.

Logical Expressions Involving 'Only'

  • Only Horses Gallop:
  • Logical Form: ∀x(Gx → Hx)
  • All and Only Horses Gallop:
  • Logical Form: ∀x(Hx ↔ Gx)
  • Only Some Smokers Get Cancer:
  • One valid form: ∃x(Sx ∧ Cx) ∧ ∃x(Sx ∧ ¬Cx)

Syntax of Modal Predicate Logic (MPL)

  • Symbols of MPL:
  • Names: a, b, c, …, t
  • Variables: u, v, w, x, y, z
  • Predicates: A, B, C, …, Z
  • Connectives: ¬, ∧, ∨, →, ↔
  • Quantifiers: ∀ (for all) and ∃ (there exists)
  • Additional Notes: Subscripts can be used to distinguish between variables, names, and predicates.

Terms and Well-formed Formulas (wffs)

  • Terms:
  • A name is a term.
  • A variable is a term.
  • Nothing else qualifies as a term.
  • Defining wffs:
  • Atomic formulas are wffs (e.g., Pt).
  • Combining wffs: if α and β are wffs,
    • ¬α, α ∧ β, α ∨ β, α → β, α ↔ β, ∀xα, ∃xα are all wffs.
  • No other expressions qualify as wffs.

Constructing Wffs

  • Example Construction:
  • For ∃x(Fx → Ga):
    1. Fx is wff (Clause 1).
    2. Ga is wff (Clause 1).
    3. (Fx → Ga) is wff (Clause 3.2).
    4. ∃x(Fx → Ga) is wff (Clause 3.2).

Quantifier Scope

  • Quantifier Placement:
  • x is the variable in quantifier ∀x; similarly for y and ∃y.
  • Scope: The subformula denoted as α to which quantifier is prefixed.
  • Example: In ∃x(Fx → Ga), scope is (Fx → Ga). For (∃xFx → Ga), scope is Fx.

Free and Bound Variables

  • Bound Occurrence:
  • Occurrence is bound if it is part of a quantifier or in its scope.
  • Ex: Both occurrences of x in ∀x(Rx) are bound.
  • Free Occurrence:
  • Occurrence is free if it is not bound.
  • Example: x is free in Rx or ∀yRx.

Open and Closed Formulas

  • Closed wff:
  • No free variable occurrences; Ex: ∀xPx ∨ ∃yQy.
  • Open wff:
  • One or more free variable occurrences; Ex: ∀x(Px ∨ Qy).
  • Important note: Open wffs are not true propositions; e.g., Rx means "it is red" without specifying what it refers to.