Section 1.3 Quantifiers, Predicates, and Validity
43
then it is ugly.” Letting P(x) denote “x is a parrot” and U(x) denote “x is ugly,” the
statement can be symbolized as
(4x)[P(x) S U(x)]
Other English language variations that take the same symbolic form are, “All parrots are ugly,” and, “Each parrot is ugly.” Notice that the quantifier is the universal quantifier and the logical connective is implication; 4 and S almost always
belong together. The wff (4x)[P(x) ` U(x)] is an incorrect translation because it
says that everything in the domain—understood here to be the whole world—
is an ugly parrot. This says something much stronger than the original English
statement.
Similarly, “There is an ugly parrot,” is really saying, “There exists something
that is both a parrot and ugly.” In symbolic form,
(E x)[P(x) ` U(x)]
Variations are, “Some parrots are ugly,” and, “There are ugly parrots.” Here the
quantifier is the existential quantifier and the logical connective is conjunction; E
and ` almost always belong together. The wff (E x)[P(x) S U(x)] is an incorrect
translation. This wff is true as long as there is anything, call it x, in the domain
(the whole world) that is not a parrot, because then P(x) is false and the implication
is true. Indeed, this wff is true if there are no parrots in the world at all!
To translate an English statement into a wff, it may help to first write an intermediate English language statement and then symbolize that statement. We did
this with the parrot examples.
The word “only” seems particularly troublesome in translations because its
placement in a sentence can completely change the meaning. For example, the
English statements
1. John loves only Mary.
2. Only John loves Mary
3. John only loves Mary.
say three entirely different things. Using the predicate symbols J(x) for “x is John,”
M(x) for “x is Mary,” and L(x, y) for “x loves y,” they can be rewritten as
1. If John loves any thing, then that thing is Mary.
or
1. For any thing, if it is John then, if it loves anything, that thing is Mary.
(4x)(J(x) S (4y)(L(x, y) S M( y))
2. If any thing loves Mary, then that thing is John.
or
2. For any thing, if it is Mary then, if anything loves it, that thing is John.
(4x)(M(x) S (4y)(L( y, x) S J( y))
3. If John does any thing to Mary, then that thing is love.
or
ReMIndeR
Think
4 S
and
E `
43
then it is ugly.” Letting P(x) denote “x is a parrot” and U(x) denote “x is ugly,” the
statement can be symbolized as
(4x)[P(x) S U(x)]
Other English language variations that take the same symbolic form are, “All parrots are ugly,” and, “Each parrot is ugly.” Notice that the quantifier is the universal quantifier and the logical connective is implication; 4 and S almost always
belong together. The wff (4x)[P(x) ` U(x)] is an incorrect translation because it
says that everything in the domain—understood here to be the whole world—
is an ugly parrot. This says something much stronger than the original English
statement.
Similarly, “There is an ugly parrot,” is really saying, “There exists something
that is both a parrot and ugly.” In symbolic form,
(E x)[P(x) ` U(x)]
Variations are, “Some parrots are ugly,” and, “There are ugly parrots.” Here the
quantifier is the existential quantifier and the logical connective is conjunction; E
and ` almost always belong together. The wff (E x)[P(x) S U(x)] is an incorrect
translation. This wff is true as long as there is anything, call it x, in the domain
(the whole world) that is not a parrot, because then P(x) is false and the implication
is true. Indeed, this wff is true if there are no parrots in the world at all!
To translate an English statement into a wff, it may help to first write an intermediate English language statement and then symbolize that statement. We did
this with the parrot examples.
The word “only” seems particularly troublesome in translations because its
placement in a sentence can completely change the meaning. For example, the
English statements
1. John loves only Mary.
2. Only John loves Mary
3. John only loves Mary.
say three entirely different things. Using the predicate symbols J(x) for “x is John,”
M(x) for “x is Mary,” and L(x, y) for “x loves y,” they can be rewritten as
1. If John loves any thing, then that thing is Mary.
or
1. For any thing, if it is John then, if it loves anything, that thing is Mary.
(4x)(J(x) S (4y)(L(x, y) S M( y))
2. If any thing loves Mary, then that thing is John.
or
2. For any thing, if it is Mary then, if anything loves it, that thing is John.
(4x)(M(x) S (4y)(L( y, x) S J( y))
3. If John does any thing to Mary, then that thing is love.
or
ReMIndeR
Think
4 S
and
E `