Symbolization in Predicate Logic

1. Symbolization in Predicate Logic • In Predicate Logic, statements predicate properties of specific individuals or members of a group. • Singular Statement: A statement that predicates a property of a specific individual. For example: • Sally is happy.

2. General Statement: Statement that predicates a property of the members (every member or just some members) of a group. For example: • All philosophers are happy. • Some philosophers are happy. • Symbolically, a predicate is represented by a single capital letter. For Example: • ___ is happy • H • Symbolically, specific individuals are represented by constants.

3. Constant: One of the first 23 letters of the alphabet in lower case. For Example: • Sally • s • The constant representing the specific individual is always placed to the right of the capital letter representing the predicate. For example: • Sally is happy. • Hs

4. The symbolic statement Hs, is read ‘H of s’ because the property represented by ‘H’ is predicated of the specific individual represented by ‘s.’ • More examples: • Billy is tall. • Tb • Hillary is not wild. • ~Wh • Carl or Alice is leaving now. • Lc v La

5. Not both Rochell and Norbert are coming. • ~(Cr · Cn) • Michael will direct, if Betty won’t; nevertheless, Jolene’s writing is a necessary condition for Ted’s not writing. • (~Db  Dm) · (~Wt  Wj) • Bob or Steve will not win, if, and only if, neither Judy nor Marian will, but Terry does. • (~Wb v ~Ws)  [~(Wj v Wm) · Wt]

6. Symbolizing General Statements • Variable: One of the last three letters of the alphabet in lower case • A variable stands for a non-specific individual. • You can look at a variable as standing for “just any ole thing.” For example, when H=is happy • Hx means • Just any ole thing is happy.

7. Quantifier: Symbol that specifies how many members of a group a property is predicated of. • If a property is predicated of every member of a group, use (x) as the quantifier. • (x) is read ‘For all x . . .’ • If a property is predicated of just some (at least one) members of a group, use (x) as the quantifier. • (x) is read ‘There exists an x, such that . . .’

8. Two types of General Statements • Universal Generalization: Statement in which a property is predicated of every member of a group. • Existential Generalization: Statement in which a property is predicated of just some (at least one) members of a group.

9. How to symbolize a Universal Generalization: • Use two predicates, one to identify the group, the other to identify the predicated property. • Connect the two properties with a , placing the group to the left as the antecedent and the predicated property to the right as the consequent. • Place an x to the right of each predicate.

10. Put the whole statement in (), and put (x) in front. • For example: • All philosophers are happy. • (x) (Px  Hx) • How to symbolize an Existential Generalization: • Use two predicates, one to identify the group, the other to identify the predicated property.

11. Connect the two properties with a ·, placing the group to the left as the left-hand conjunct and the predicated property to the right as the right-hand conjunct. • Place an x to the right of each predicate. • Put the whole statement in (), and put (x) in front. • For example: • Some philosophers are happy. • (x) (Px · Hx)

12. Words indicating Universal Generalization: • All • All dogs are mammals. • (x) (Dx  Mx) • Every • Every cat is four-legged. • (x) (Cx  Fx) • “ever” words (e.g. ‘Whoever,’ ‘Wherever,’ ‘Whenever’) • Whoever is a clown is funny. • (x) (Cx  Fx)

13. Any (when ‘any’ means ‘every’) • Any bull is a male. • (x) (Bx  Mx) • Only (what follows is the predicated property and goes to the right of the ) • Only women are mothers. • (x) (Mx  Wx) • The only (what follows is the group and goes to the left of the ) • The only ones who are fathers are men. • (x) (Fx  Mx)

14. None but (what follows is the predicated property and goes to the right of the ) • None but fools are t-sips. • (x) (Tx  Fx) • Not any (when ‘any’ means ‘some’) • Negate the predicated property, i.e. the predicate to the right of the . • Not any patriots are traitors. • (x) (Px  ~Tx)

15. No • Negate the predicated property, i.e. the predicate to the right of the . • No squares are circles. • (x) (Sx  ~Cx)

16. Words indicating Existential Generalization: • Some • Some birds are robins. • (x) (Bx · Rx) • A few • A few zebras are in the USA. • (x) (Zx · Ux) • There is (are) • There are women in the Army. • (x) (Wx · Ax)

17. Any (when ‘any’ means ‘some’) • Are there any t-sips in College Station? • (x) (Tx · Cx) • Not all • Negate the predicated property, i.e. the predicate to the right of the ·. • Not all t-sips are are bad. • (x) (Tx · ~Bx)

18. Not every • Negate the predicated property, i.e. the predicate to the right of the ·. • Not every Aggie is good. • (x) (Ax · ~Gx) • Not any (when ‘any’ means ‘every’) • Negate the predicated property, i.e. the predicate to the right of the ·. • Not just any actor is a star. • (x) (Ax · ~Sx)

19. Some . . . are not • Negate the predicated property, i.e. the predicate to the right of the ·. • Some fish are not trout. • (x) (Fx · ~Tx) • A few . . . are not • Negate the predicated property, i.e. the predicate to the right of the ·. • A few cowboys are not good sports. • (x) (Cx · ~Gx)