Transcript Chapter 5 Quantifiers

Predicate-Argument Structure
• Monadic predicates: Properties
• Dyadic predicate: Binary relations
• Polyadic Predicates: Relations with more
then two arguments
• Arguments: Individual variables
• Predicate-argument structures are open,
need to be quantified to become statements
5.2 Categorical sentence forms
• Objects and general domain for arguments
• All F are G: For all x, if Fx, then Gx
• Some F are G: There is some x, Fx and Gx
• “The” vs. Truth conditions
5.3 Polyadic Predicates
“Trust” as an example
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Everyone trusts Tom:
Somebody trusts somebody:
Somebody is trusted by somebody:
Somebody trusts everybody:
Everybody trusts somebody:
Everybody trusts everybody:
Somebody trusts herself/himself:
Everybody trusts him/herself:
xTxs
xyTxy
yxTyx
xyTxy
x yTxy
xyTxy
xTxx
xTxx
5.4 The Language Q
Vocabulary/Lexicons
• Sentence letters: p, q, r, s. (* with/without
subscripts). (Italics are used in indicating metavariables)
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n-ary predicates: Fn, Gn, Hn, … Mn. *
Individual constants: a, b, c, …, o. *
Individual variables: t, u, v, w, x, y, z. *
Sentential connectives: ¬, →, &, V, ↔.
Quantifiers: , .
• Grouping indicators: ( , ).
Substitution
• Consider an expression A(d), where d is a constant.
• A(c) is a new expression by replacing every
occurrence of d with an occurrence of c.
• A(x) is a new expression by replacing every
occurrence of d with an occurrence of variable x.
• A(y) is a new expression by replacing every
occurrence of x with an occurrence of y.
• Note the phrase here: “every occurrence of”.
Formation Rules
• Any sentence letter is a formula.
• An n-ary predicate followed by n constants is a
formula.
• If A is a formula,, then ¬A is a formula.
• If A and B are formulas, then A→B, A&B, AVB,
and A↔B are formulas.
• If A(c) is a formula, and v is a variable, then
vA(v/c), and vA(v/c) are formulas.
• Every formula can be constructed by a finite
number of application of these rules (nothing else).
Notes
• The lexicons of sentential logic are included in Q.
• Is A(x) a formula? Depends on the systems.
• It can be treated as an atomic formula, whose
truth values has to be determined by the so-called
value-assignment semantics.
• But in this book, it has to be xA(x/c) for A(c); no
free variables in this book. This is convenient to
Truth-tree method.
• Scope: Usually what next to the quantifier. But in
this book, means the whole: x(Fx→Gx).
Examples
• xFx & p, x(Ax→r) are formulas.
• Convention: xyF2xy = xyFxy
• But better not xy(Fxy→Fa).
• xyF1xy, xF2x are not formulas.
• aFa, pF(p&q) are not formulas.
5.5 Symbolization
• Proper names as constants (Tom, the house)
• Common names as properties monadic predicates
(e.g., women, star, player).
• Determiners: Bad discussion (e.g., “a”=any?)
• Adjectives: Monadic predicates for properties.
Symbolization
• Relative clauses
Those who (that, where, when) …
x(Fx→Gx) or x(Fx&Gx)) ?
• Prepositional phrase: in, to, of, about, up,
over, from, etc.
x((Fx&Hx)→Gx))
Symbolization
• Verb phrase: Polyadic Predicates
• Connectives:
All the beads are either red or blue:
x(Rx V Bx)
All the beads are red or all the beads are blue:
(xRx)V(xBx)