46
Formal Logic
All what? All giraffes, so
(4x)(G(x) … )
Because of the universal quantifier, we expect to use the implication connective,
so now we have
(4x)(G(x) S … )
Thinking of the implication as an “if–then,” we have “if a giraffe, then … .” Then
what? Then it's tall. The final wff is
(4x)(G(x) S T(x))
The second example is
Only giraffes are taller than elephants.
The property of being a giraffe and the property of being an elephant are unary
predicates, and we’ll use G(x) and E(x) to represent them. But “taller than” is a
property that compares two things, so it’s a binary predicate; T(x, y) will mean “x
is taller than y”. There are no obvious quantifier key words, so we understand that
we are talking about all giraffes and all elephants (universal quantifiers). The word
“giraffes” follows the word “only,” so the property of being a giraffe is going to
be the conclusion of an implication and the overall form will be “if xxx, then a
giraffe.” Indeed, if something is taller than an elephant, then it’s a giraffe. Putting
in the universal quantifiers, “if any thing is taller than any elephant, then that thing
is a giraffe,” or (even more tortured English), “for any thing, if it is an elephant,
then for any other thing, if it’s taller than the elephant, then it’s a giraffe.” Now we
can pretty much translate directly into a wff. “For any thing, if it is an elephant,
then” becomes
(4x)(E(x) S … )
and, “for any other thing, if it's taller than the elephant, then,” adds a second implication to the wff:
(4x)(E(x) S (4y)(T( y, x) S … ))
Notice that we introduced y here, a second variable, because we’ve already given
x the elephant property. Also, we’ve written T( y, x), not T(x, y), because we want
this new thing to be taller than the elephant, and our definition of the taller predicate was that the first variable was taller than the second. We are ready for the final
conclusion—this new thing is a giraffe.
(4x)(E(x) S (4y)(T( y, x) S G( y)))
As in Table 1.15(3), a tautology allows us to also write this wff as
(4x)(4y)(E(x) ` T( y, x) S G( y))
“For any two things, if one is an elephant and the other is taller than the elephant,
then the other thing is a giraffe.”
With some practice, you won’t have to go quite this slowly!
Formal Logic
All what? All giraffes, so
(4x)(G(x) … )
Because of the universal quantifier, we expect to use the implication connective,
so now we have
(4x)(G(x) S … )
Thinking of the implication as an “if–then,” we have “if a giraffe, then … .” Then
what? Then it's tall. The final wff is
(4x)(G(x) S T(x))
The second example is
Only giraffes are taller than elephants.
The property of being a giraffe and the property of being an elephant are unary
predicates, and we’ll use G(x) and E(x) to represent them. But “taller than” is a
property that compares two things, so it’s a binary predicate; T(x, y) will mean “x
is taller than y”. There are no obvious quantifier key words, so we understand that
we are talking about all giraffes and all elephants (universal quantifiers). The word
“giraffes” follows the word “only,” so the property of being a giraffe is going to
be the conclusion of an implication and the overall form will be “if xxx, then a
giraffe.” Indeed, if something is taller than an elephant, then it’s a giraffe. Putting
in the universal quantifiers, “if any thing is taller than any elephant, then that thing
is a giraffe,” or (even more tortured English), “for any thing, if it is an elephant,
then for any other thing, if it’s taller than the elephant, then it’s a giraffe.” Now we
can pretty much translate directly into a wff. “For any thing, if it is an elephant,
then” becomes
(4x)(E(x) S … )
and, “for any other thing, if it's taller than the elephant, then,” adds a second implication to the wff:
(4x)(E(x) S (4y)(T( y, x) S … ))
Notice that we introduced y here, a second variable, because we’ve already given
x the elephant property. Also, we’ve written T( y, x), not T(x, y), because we want
this new thing to be taller than the elephant, and our definition of the taller predicate was that the first variable was taller than the second. We are ready for the final
conclusion—this new thing is a giraffe.
(4x)(E(x) S (4y)(T( y, x) S G( y)))
As in Table 1.15(3), a tautology allows us to also write this wff as
(4x)(4y)(E(x) ` T( y, x) S G( y))
“For any two things, if one is an elephant and the other is taller than the elephant,
then the other thing is a giraffe.”
With some practice, you won’t have to go quite this slowly!