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Predicate-Argument Structure

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.

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Predicate-Argument Structure

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  1. 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

  2. 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

  3. 5.3 Polyadic Predicates“Trust” as an example • Everyone trusts Tom: xTxs • Somebody trusts somebody: xyTxy • Somebody is trusted by somebody: yxTyx • Somebody trusts everybody: xyTxy • Everybody trusts somebody: x yTxy • Everybody trusts everybody: xyTxy • Somebody trusts herself/himself: xTxx • Everybody trusts him/herself: xTxx

  4. 5.4 The Language QVocabulary/Lexicons • Sentence letters: p, q, r, s. (* with/without subscripts). (Italics are used in indicating meta-variables) • 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: ( , ).

  5. 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”.

  6. 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).

  7. 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).

  8. 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.

  9. 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.

  10. 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))

  11. 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)

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