44
Formal Logic
3. For any thing, if it is John then, for any other thing, if that thing is Mary,
then John loves it.
(4x)(J(x) S (4y)(M( y) S L(x, y))
In each case, the consequent of the implication is the word following “only” in the
original English statement.
eXAMPLe 22
Given the predicate symbols
D(x) is “x is a dog”
R(x) is “x is a rabbit”
C(x, y) is “x chases y”
Table 1.15 shows examples of an English statement, an intermediate English statement, and a wff translation. Note that in wff 2, the connective associated with E
is ` and the connective associated with 4 is S. In wff 3, the first version shows
two implications associated with the two 4 quantifiers. The second version is
equivalent because of the tautology [A ` B S C ] 4 [A S (B S C )]. This version may appear to violate the rule that universal quantifiers should be used with
implication, not conjunction, but this tautology provides another way to write two
implications. The second version also shows more clearly that “dogs,” the word
following “only,” is the conclusion.
tAbLe 1.15
english Statement Intermediate Statement
Wff
1. All dogs chase
all rabbits.
For any thing, if it is a dog,
then for any other thing, if
that thing is a rabbit, then
the dog chases it.
(4x)[D(x) S (4y)(R( y) S C(x,y))]
2. Some dogs
chase all rabbits.
There is some thing that
is a dog and, for any other
thing, if that thing is a rabbit, then the dog chases it.
(E x)[D(x) ` (4y)(R( y) S C(x,y))]
3. Only dogs chase
rabbits.
For any thing, if it is a rabbit
then, if anything chases it,
that thing is a dog.
For any two things, if one is
a rabbit and the other chases it, then the other is a dog.
(4y)[R( y) S (4x)(C(x,y) S D(x))]
(4y)(4x)[R( y) ` C(x,y) S D(x)]
Often more than one wff exists that is a correct representation of an English statement, as seen with statement (3) in Table 1.15. Also wff (2) is equivalent to
(E x)[D(x) ` (4y)([R( y)]′ ~ C(x, y))]
because of the implication equivalence rule that says (R S C ) 4 (R′ ~ C ), even
though here R and C are predicates instead of just statement letters.
Formal Logic
3. For any thing, if it is John then, for any other thing, if that thing is Mary,
then John loves it.
(4x)(J(x) S (4y)(M( y) S L(x, y))
In each case, the consequent of the implication is the word following “only” in the
original English statement.
eXAMPLe 22
Given the predicate symbols
D(x) is “x is a dog”
R(x) is “x is a rabbit”
C(x, y) is “x chases y”
Table 1.15 shows examples of an English statement, an intermediate English statement, and a wff translation. Note that in wff 2, the connective associated with E
is ` and the connective associated with 4 is S. In wff 3, the first version shows
two implications associated with the two 4 quantifiers. The second version is
equivalent because of the tautology [A ` B S C ] 4 [A S (B S C )]. This version may appear to violate the rule that universal quantifiers should be used with
implication, not conjunction, but this tautology provides another way to write two
implications. The second version also shows more clearly that “dogs,” the word
following “only,” is the conclusion.
tAbLe 1.15
english Statement Intermediate Statement
Wff
1. All dogs chase
all rabbits.
For any thing, if it is a dog,
then for any other thing, if
that thing is a rabbit, then
the dog chases it.
(4x)[D(x) S (4y)(R( y) S C(x,y))]
2. Some dogs
chase all rabbits.
There is some thing that
is a dog and, for any other
thing, if that thing is a rabbit, then the dog chases it.
(E x)[D(x) ` (4y)(R( y) S C(x,y))]
3. Only dogs chase
rabbits.
For any thing, if it is a rabbit
then, if anything chases it,
that thing is a dog.
For any two things, if one is
a rabbit and the other chases it, then the other is a dog.
(4y)[R( y) S (4x)(C(x,y) S D(x))]
(4y)(4x)[R( y) ` C(x,y) S D(x)]
Often more than one wff exists that is a correct representation of an English statement, as seen with statement (3) in Table 1.15. Also wff (2) is equivalent to
(E x)[D(x) ` (4y)([R( y)]′ ~ C(x, y))]
because of the implication equivalence rule that says (R S C ) 4 (R′ ~ C ), even
though here R and C are predicates instead of just statement letters.