Philosophy - Philosophy Course Notes
Categorical Reasoning in Predicate Logic
Lecture 12 - Course Notes
June 6, 2007
I. Definitions and Tools
A. Definitions
1. Singular term
A word or phrase which refers to a particular object.
2. Predicate
In a declarative sentence when you say '... is something', without a singular term, that is a predicate. In logic we give the word predicate generally the same meaning as in grammar.
3. Variables
Instead of using '...' to indicate a blank in a predicate, use x,y,z instead.
4. Quantifiers
Specify what we are talking about in numerical sense: all, some, none.
ie. 'for any x', 'there is at least one x such that'
B. Tools
1. Use capital letters to stand for predicates.
2. Use variables to indicate a blank.
3. Use same logical connectives as SL.
II. Quantifiers
Quantifier expressions:
1. For any x
(∀x)
2. There is at least one x such that
(∃x)
(1&2 can be made negative to switch values)
3. Universe of Discourse
UD
A. Scope of Quantified Claims
Works just like in programming languages.
(∀x)Px = scope of quantifier is entire formula
(∀x)Px v Pa = scope is just Px
B. Negating Quantified Claims
use SL negation symbol to negate a value.
ie. at least one thing has P becomes nothing has P
ie. all x have P becomes at least one x does not become P
III. Categorical Reasoning
Same as classical (categorical) reasoning.
A. Universal Affirmation (A)
All x has y
(∀x)(xz => yz)
B. Particular Affirmation (E)
At least one x has y
(∃x)
C. Universal Negation (I)
No x has y
(∀x)
D. Particular Negation (O)
At least one x does not have y
(∃x)
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