Introduction to Logic - Ch. 5

(Originally written November 13, 2006 in Book 9)

x isn't evil -> ~Ex
x is either crazy or evil
(x) (Cx v Ex)
(∃x) (Cx·~(Ex·Lx)
(x) (Lx)

??

"Knowledge is meaningless without love"

Love. Joy. Happy.

Introduction to Logic
Harry Gensler

Chapter 5: Basic Quantificational Logic

Quantificational Logic - studies arguments whose validity depends on 'all', 'no', 'some' and similar words.

Easier Translations

Use capital letters for general terms: descriptive terms or categories. ex:
I - an Italian
C - charming
R - drives a Rolls Royce

Use small letters for singular terms: particulars, individuals. ex.
i - the richest Italian
c - this child
r - Romeo

A capital letter alone represents a statement.

A capital letter followed by a single small letters represents a general term -> Ir

A capital letter followed by two or more lame letters represents a relation -> Lrs

Proofs in Quantificational Logic

1. (x)(Gx⊃Wx)
2. (x)(Px⊃Gx)
Therefore, (x)(Px⊃Wx).

3. asm: ~(x)(Px⊃Wx)

Step 1. Eliminate Negated Qualifiers

4. (∃x)~(Px⊃Wx)
5. ~(Pa⊃Wa) 4 (E.I.)
6. (Ga⊃Wa) 1 (U.I)
7. (Pa⊃Ga) 2 (U.I)
8. Pa 5
9 ~Wa 5
10. Ga 8,7
11. ~Ga 9,6 and 10
12. Therefore (x)(Px⊃Wx)

Step 2. eliminate variables with U.I. and E.I.

U.I. - universal instantiation - if a universal is always true then any thing that is a member of that class will make it true

E.I. - Existential Instantiation - if there are multiple/competing (∃x) statements different variables are used. Since a (∃x) statement means 'some' there can be contradictory statements.

·⊃∃