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

• • • • • • • •

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 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

• • • • •

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



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 .

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)