Brief notes on Quantificational Logic

(Originally written on November 8, 2006 in Book 9)

Class Notes

Test - 11/15/06

S-Proofs
At least:

• 1 single assumption is valid
• 1 single assumption is invalid
• 1 multiple assumption is valid
• 1 multiple assumption is invalid

LogiCola!! due by 11/17/06 Friday!!!

F's & G's

[half a page of nonsense]

First-Order Logic or Quantificational Logic

Functions (Predicates)

Socrates is Mortal = Ms

M- Function (predicate)

S- Socrates

Ms= Socrates has the function of Mortality

Functions (predicates) - Capital Letters

Individual, constant - small letter

Ea - This aardvark eats ants

aardvark - individual

eats - function

Eb - This baboon eats aardvarks

baboon - individual

eats function

An individual or constant is always referring to one specific entity

1) a specific cat

2) a specific litter of cats

~(Ea⊃Uc) -> it is not the case that if this aardvark eats ants then the Colts are undefeated.

Something is rotten in Denmark

R?

There is something such that it is rotten in Denmark

R?

There is some 'x' such that Rx

x- is a variable

x- can be potentially satisfied with various constants

Variables can be fulfilled by various constants

∃ - the quantifier

The quantifier means "at least one"

There is some 'x' such that Rx = (∃x)Rx 

(∃x)Rx is a particular statement or existential statement

(∃x)Sx - something is square 

~(∃x)Sx - nothing is square

~(∃x)~Sx - something is not a square 

The bound variable (the x associated with the above S's) is governed by the quantifier

without the (∃x) or quantifier, the variables are free variables

There must be a quantifier to make a true/false statement 

(x)Cx -> for all of x, C of X

(x) is the universal qualifier