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)