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.

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.

– 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

– 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

– 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]

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.

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...’

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.

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.

– 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.

– 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)

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)

– 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)

– 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)

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

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)

– 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)

– 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)

– 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)