Transcript Chapter 5 Quantifiers
Predicate-Argument Structure
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Monadic predicates: Properties Dyadic predicate: Binary relations Polyadic Predicates: Relations with more then two arguments
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Arguments: Individual variables Predicate-argument structures are open, need to be quantified to become statements
5.2 Categorical sentence forms
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Objects and general domain for arguments
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All F are G: For all x, if Fx, then Gx
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Some F are G: There is some x, Fx and Gx
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“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
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Sentence letters: p, q, r, s. (* with/without subscripts). (Italics are used in indicating meta variables) n-ary predicates: F
n , G n , H n , … M n
. * Individual constants: a, b, c, …, o. * Individual variables: t, u, v, w, x, y, z. * Sentential connectives: ¬, →, &, V, ↔.
Quantifiers:
,
.
Grouping indicators: ( , ).
Substitution
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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
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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
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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
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xFx & p,
x(Ax→r) are formulas.
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Convention:
x
yF 2 xy =
x
yFxy
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But better not
x
y(Fxy→Fa).
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x
yF 1 xy,
xF 2 x are not formulas.
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aFa,
pF(p&q) are not formulas .
5.5 Symbolization
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Proper names as constants (Tom, the house)
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Common names as properties monadic predicates (e.g., women, star, player).
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Determiners: Bad discussion (e.g., “a”=any?)
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Adjectives: Monadic predicates for properties.
Symbolization
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Relative clauses Those who (that, where, when) …
x(Fx→Gx) or
x(Fx&Gx)) ?
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Prepositional phrase: in, to, of, about, up, over, from, etc.
x((Fx&Hx)→Gx))
Symbolization
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Verb phrase: Polyadic Predicates
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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)