by Corry Shores
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Summary of
Patrick Suppes
Introduction to Logic
Ch. 3 Symbolizing Everyday Language
§3.4 Quantifiers
Brief summary:
The variables in our propositions can be quantified. If the variable applies to all such things, it is universally quantified. The sentence “Everyone is a miser” could be expressed “For all x, x is a miser” and rendered symbolically [in Suppes’ text] as: (x)(x is a miser). If however the variable applies to some or to just one such thing, it is existentially quantified. The sentence “something is greater than zero” could be expressed “there is an x such that x is greater than 0” and symbolically rendered as: (∃x)(x > 0). More quantifiers can be added, as in (∃x)(∃y)(∃z)(x + y = z + 2). To symbolize common nouns, we can use relational predicates and quantifiers. First consider an example with universal quantification, “All freshmen are intelligent”. It should be written with the conditional as: (x)(Fx → Ix), meaning “For all x, if x is a freshman then x is intelligent”. But the structure is different for existential quantification. “Some freshmen are intelligent” should be rendered with the conjunction as: (∃x)(Fx & Ix).
Summary
Certain formulas with variables are neither true nor false, but they may be so after we substitute terms in for the variables. Thus on their own
x loved y
and
x + y = z + 2
are neither true nor false. But if we substitute to make the first one
Emma loved Mr. Knightley
then we have a true sentence “(true at least in the world of Jane Austen)”. But if for the second formula we substitute to get
2 + 3 = 4 + 2
then we have a false statement (47).
We can also get true and false statements from the above examples by prefixing to such expressions phrases like “for every x”, and “there is an x such that”, and “for all y” (47).
Suppose we begin with the sentence
x is a miser.
Then we can change it to
There exists an x such that x is a miser.
This has the same meaning as
There are misers.
Or, we can formulate this false sentence:
For every x, x is a miser
, which has the same meaning as
Everyone is a miser.
(48)
We call the phrase “for every x” the universal quantifier. Suppes notates the universal quantifier in this way:
(x)(x is a miser).
(Suppes 48)
We call existential quantifiers such phrases as “some”, “there exists an x such that”, and “there is at least one x such that”. So normally the formula
x > 0
is neither true nor false. But it becomes true when we add to it, “there is an x such that”. We notate it in the following way:
(∃x)(x > 0).
Note that not all formula that are neither true nor false will become one or the other merely by adding a quantifier. For, “The appropriate number of quantifiers must be added” (48). So consider again:
x + y = z + 2
Now add the existential quantifier:
(∃x)(x + y = z + 2)
. At this point, it is still neither true nor false [because the other variables are unaccounted for]. Instead, quantify for them too using the existential quantifier:
(∃x)(∃y)(∃z)(x + y = z + 2)
We could write this as,
There are numbers x, y, and z such that x + y = z + 2.
(Suppes 48)
[Recall this sentence from the prior section: For every x, Mx → Ax. (Suppes 46).] We can now completely symbolize one of our sentences from the prior section as:
(x)(Mx → Ax)
(Suppes 49)
Now consider the sentence
All freshmen are intelligent.
Suppose that F stands for “is a freshmen” and I stands for “ is intelligent” (49). We could write this as:
(x)(Fx → Ix)
But how would we render:
No freshmen are intelligent
? First, we obtain the “partial translation” of this sentence, which would be:
For all x, if x is a freshmen then x is not intelligent.
This we translate symbolically as:
(x)(Fx → –Ix)
(Suppes 49)
This formulation “exemplifies the standard form for sentences of the type ‘No such and such are so and so’ ” (49).
In the footnotes, by the way, Suppes mentions that we can also translate the above formulation using existential quantification as
–(∃x)(Fx → Ix)
But we will see the equivalences in the next chapter.
Suppes then turns to sentences with existential quantification. Consider the sentence:
Some freshmen are intelligent.
(49)
We translate it as:
For some x, x is a freshmen and x is intelligent.
This we render symbolically as:
(∃x)(Fx & Ix)
. But now we note something odd. For universal quantification, we translated
All freshmen are intelligent.
as
(x)(Fx → Ix)
. But here with existential quantification, we use the conjunction. So we should not render it as
(∃x)(Fx → Ix)
. This would be translated as
For some x, if x is a freshman then x is intelligent.
Suppes explains why. So take again this formulation, which we are saying is incorrect:
(∃x)(Fx → Ix)
We can replace the (Fx → Ix) with its tautological equivalent (–Fx ∨ Ix) to get:
(∃x)(–Fx ∨ Ix)
. And we may translate this as:
For some x, either x is not a freshman or x is intelligent,
Or as
There is something which is either not a freshman or is intelligent.
(50)
But recall the conjunctive formulation:
Some freshmen are intelligent,
or,
For some x, x is a freshman and x is intelligent.
This means something different from the translation for the disjunction. Consider for example if all freshmen were completely stupid. That would make this sentence false: “Some freshmen are intelligent,” because in fact there would be none who are. However, the sentence “There is something which is either not a freshman or is intelligent” could still be true, even if all freshmen were completely stupid, so long as there is at least one thing in the universe which is not a freshmen.
As you recall, we are thinking critically of the formulation:
(∃x)(Fx → Ix)
Suppes notes another problem with it, namely, it is nearly always true but as well utterly trivial. “Consequently, such statements are very seldom the correct translation of any sentence of ordinary language in which we are interested” (50).
Suppes provides another example for this triviality of the sentence
(∃x)(Fx → Ix)
. We begin with this sentence:
Some men are three-headed.
(50)
The correct way to symbolize it is:
(∃x)(Mx & Tx)
Probably this sentence is false. But consider if we had (incorrectly) rendered it as
(∃x)(Mx → Tx)
. In that case, we are committed also to the logically equivalent sentence
(∃x)(–Mx ∨ Tx)
[We would then translate this as: “For some x, either x is not a man or x has three heads”, or “There is something which is either not a man or has three heads”.] But this is absurd, so long as there are things in the universe other then men: “this trivial condition guarantees the truth of [(∃x) (–Mx ∨ Tx)]” (50).
Now we will try to symbolically formulate the sentence:
Some freshmen are not intelligent.
[We know that “Some freshmen are intelligent” should be rendered: (∃x)(Fx & Ix)] From the above discussion, we can conclude that this sentence should be written as:
(∃x)(Fx & –Ix)
(Suppes 51)
It is also possible to have quantified sentences that include constants. Suppose a = Adams, x is a variable, and M stands for “is married to.” We would render the sentence,
Adams is not married to anyone
as
(x)–(Max)
Here Max is read, “Adams is married to x”. Consider another example. Here, e = Elizabeth, x is a variable, F is a one-place predicating meaning “is a freshman”, and D is a two-place predicate meaning “dates” as in “x dates y”. We then would render the sentence
Some freshmen date Elizabeth
as
(∃x)(Fx & Dxe)
(Suppes 51)
Suppes, Patrick. Introduction to Logic. New York: Van Nostrand Reinhold / Litton Educational, 1957.
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