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 Q Vocabulary/Lexicons 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: (, ).

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  yF 2 xy =  x  yFxy But better not  x  y(Fxy→Fa).  x  yF 1 xy,  xF 2 x 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)