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)