by Corry Shores
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[The following is summary. My commentary is in brackets. Boldface is mine.]
Introduction to Logic
Ch. 9 Sets
§9.7 Translating Everyday Language
Formulations of the sort “All ... are ...” as in “All Americans are philosophers” can be translated into set notation as A ⊆ P. In this and in the other cases, either both blanks would be filled by common nouns or the first with a common noun and the second with an adjective. When the second blank is filled by an adjective, we need to convert it to a set name. So “All Americans are mortal” would be translated as “The class of Americans are included in the class of mortal beings” or A ⊆ M. “Some ... are ...” formulations, like, “Some Americans are philosophers”, can be translated to A ∩ P ≠ Λ. (Here Λ is the empty set.) Note, from the “Some S are P” formulations, we can infer that their intersection is not an empty set. However, for “All S are P” formulations, their intersection can in fact be an empty set. We translate “No ... are ...” formulations, like “No Americans are philosophers” as A ∩ P = Λ. And “Some ... are not ...” formulations, like “Some Americans are not philosophers”, are translated A ∩ ∼P ≠ Λ. More complicated expressions need more complicated interpretations. “All Americans are clean and strong” could be A ⊆ C ∩ S. “Fools and drunk men are truth tellers” could be (F ∪ D) ⊆ T. “Some Frenchmen drink wine” could be F ∩ W ≠ Λ. “Some Americans drink both coffee and milk” could be A ∩ C ∩ M ≠ Λ. And “Some Americans who drink tea do not drink either coffee or milk” could be (A ∩ T) ∩ ∼(C ∪ M) ≠ Λ.
We will examine ways to translate everyday language into set notation. This presents difficulties, because we often use the same words for different notions, and also, for the same notion we often use different words or phrases (189).
Our concern here lies merely in sentences that can be translated
into a symbolism consisting just of letters standing for sets, parentheses, and the following symbols:
∩, ∪, ∼, Λ, =, ≠, ⊆ .
This symbolism is fine for statements with one-place predicates, but it is problematic for ones with many-place predicates. And “This symbolism is essentially equivalent to the language of the classical theory of the syllogism” (189). Also note that here we are not using the notion of membership. Instead, we “restrict ourselves to sets all of which are on the same level – subsets of some fixed domain of individuals” (189).
We first consider “All ... are ...” formulations where both blanks are filled with common nouns like “men,” “Americans,” “philosophers,” and so on. Here what is meant is that the “the set of things described by the first noun is a subset of the set of things described by the second noun” (189). Suppes offers this example:
All Americans are philosophers.
Here Americans are a subset of philosophers, so we can reword it a little and substitute the set inclusion symbol:
The set of Americans ⊆ the set of philosophers.
We can further symbolize it by using letters for the sets. We will use A to mean “the set of Americans” and P to mean “the set of philosophers”. We thus obtain:
A ⊆ P
There are other formulas that can express this relation:
A ∪ P = P
[Since A is a subset of P, if we find the union of A and P, there will not be any additional members that are outside of P.] and,
A ∩ ∼P = Λ
[Judging from the formulations that follow, I assume this formula means the intersection of the set of Americans with the set of non-philosophers. We originally said that all Americans are philosophers. This means that there is no American which is not a philosopher. Thus the intersection of the set of Americans and the set of non-philosophers is an empty set.] Suppes continues:
In such discussions a domain of individuals is easily fixed and thus the size of complement sets such as ∼P. Here, for example, V can be taken as the set of all human beings.
[I am not certain, but perhaps he is saying that ∼P should only be for human non-philosophers. I am not sure, but maybe there is worry that unintended members might be implied if we do not clarify this. In this case then, ∼P means the set of all human non-philosophers, but I am not sure if that makes a difference in this example.]
For the next sort of translation, we stick with the “All ... are ...” structure, but this time the second blank is filled with an adjective, like
All Americans are mortal.
[We need to convert the adjective to a set and thus to a substantive.] We translate this to:
The class of Americans ⊆ the class of mortal beings.
A ⊆ M
Often times for this sort of formulation, people omit the “All”. So they might write
Tyrants are mortal
All tyrants are mortal
which we can symbolize as
T ⊆ M
But this pattern of translation does not work in all cases. So
Men are numerous
does not mean
The set of men ⊆ the set of numerous things.
For we are not saying that every man is himself numerous. So let us see the better way to symbolize this. Let
M be the set of men and N be the set of sets which have numerous members:
M ∈ N.
[Suppes then mentions the distinction between distributive and collective. I will quote it, with the enumerated examples given in brackets, but I am not sure I understand this distinction.]
Corresponding to the distinction which we have made between membership and inclusion, the older logic made a distinction between the “distributive” and “collective” application of the predicate to the subject. Using this terminology, one says that in (1) [All Americans are philosophers], (2) [All Americans are mortal], (3) [All tyrants are mortal], and (4) [All women are fickle] the predicate is applied to the subject distributively, and that in (5) [Men are numerous] and (6) [The apostles are twelve] it is applied to the subject collectively.
We now turn to the formulation “Some ... are ...” in which the blanks contain common nouns, or as we saw above, where the first blank is a common noun and the second one is an adjective (191). [We will see the example, “Some Americans are philosophers”. Here we have the set of Americans and the set of philosophers. We know that the two sets are not equals, that is to say, not all Americans are philosophers and not all philosophers are Americans. But, that does not mean that no Americans are philosophers and no philosophers are Americans. The intersection of the two sets is not empty.]
An English statement of the form ‘Some ... are ...’ , where the blanks are filled by common nouns (or perhaps the second blank is filled by an adjective) means that there exists something which is described by both terms: i.e., that the intersection of the two corresponding sets is not empty. Thus, for instance:
(7) Some Americans are philosophers
means that there exists at least one person who is both an American and a philosopher, and is accordingly translated:
A ∩ P ≠ Λ.
[Suppes then makes a point it seems about inference. It seems that we can infer from the “Some ... are ...” formulation that the set is not empty. But from the “All ... are ...” we cannot infer that it is non-empty. It can in fact name an empty set. This is something in modern logic that was not the case in older logics which allow us to infer “Some S are P” from “All S are P”, in other words, that “All S are P” is not an empty set.]
Although a statement of the form of (7) implies that the sets corresponding to subject and predicate are not empty, no such inference is to be drawn from a statement of the form of (1). Thus, for example, it is true that
All three-headed, six-eyed men are three-headed men,
but it is not true that
Some three-headed, six-eyed men are three-headed men.
(Modem logic differs in this point from the older logic, which allowed the inference of ‘Some S are P’ from ‘All S are P’.)
We now examine “No ... are ...” formulations, where the blanks are filled by common nouns [or the second with an adjective]. Here we are saying that the intersection of the two sets is empty. Thus we would symbolize the sentence
No Americans are philosophers
A ∩ P = Λ
Recall the sentence:
All Americans are mortal.
This would have the same meaning as:
No Americans are immortal.
Which we would symbolize as
A ∩ ∼M = Λ
[Here the ∼ might just be negation. Or maybe it is difference. If it is difference, perhaps it means all those humans that are not in the set of mortals. So we are saying then that there are no American (humans) who lie outside the set of mortals.]
Next we turn to statements “Some ... are not ...” where again the blanks are filled with common nouns. [Let us consider the example first, “Some Americans are not philosophers”. Here we are saying that there are some members of the set of Americans who are not found in the set of philosophers. Thus the intersection of Americans and non-philosophers is not an empty set.] These formulations mean
that there exists something which belongs to the set corresponding to the first noun, and does not be long to the set corresponding to the second noun: i.e., that the intersection of the first set with the complement of the second is not empty. The sentence:
Some Americans are not philosophers
A ∩ ∼P ≠ Λ.
Suppes will now deal with more difficult translations.
We first consider the use of “and”. [Let us look at the example first, “All Americans are clean and strong”. We first need to convert the adjectives to set names. So perhaps “The set of Americans are included in the set of men who are clean and strong.” But notice we are not saying that all Americans are either clean or strong. This would mean that you could have some Americans who are clean but not strong, and vice versa. But we are not saying that. Instead, we are saying that the set of Americans is included in the intersection of the sets of men who are strong and men who are clean.] .
The word ‘and’ often corresponds to the intersection of sets. Thus:
All Americans are clean and strong
is translated (using obvious abbreviations):
A ⊆ C ∩ S.
The word “but” functions the same way as “and”. So for example, the sentence
Freshmen are ignorant but enthusiastic
would be translated symbolically as:
F ⊆ I ∩ E
But now we wonder, what happens when “and” is found in the subject of the sentence rather than in the predicate? [Let us look first at the example, “Fools and drunk men are truth tellers”. Here we are saying whether one is a fool or a drunk person, in both cases they are a truth teller. We are not saying, all those who are both fools and also drunk persons are truth tellers.]
The situation is quite different, however, when the ‘and’ occurs in the subject rather than in the predicate. Thus:
(9) Fools and drunk men are truth tellers |
is translated, not by:
(10) (F ∩ D) ⊆ T
but rather by:
(11) (F ∪ D) ⊆ T
For (9) means that both the following statements are true:
(12) All fools are truth tellers
(13) All drunk men are truth tellers;
and (12) and (13) are translated, respectively, by:
(14) F ⊆ T
(15) D ⊆ T;
and (14) and (15) are together equivalent to (11).
We might also encounter statements without the verb “to be” but which can still be translated into our set notation. So consider the sentence:
Some Frenchmen drink wine
We will need to translate the predicate into a set. [So this could perhaps be: “Some from the set of men who are French are in the set of men who are wine drinkers”.] If we make F be the set of Frenchmen and W be the set of wine drinkers, we can translate this symbolically as:
F ∩ W ≠ Λ.
Or consider the sentence
Some Americans drink both coffee and milk.
Let us make A be the set of Americans, C be the set of coffee drinkers, and M be the set of milk drinkers. We would symbolize this as:
A ∩ C ∩ M ≠ Λ
[Suppes notes that we are dropping the parentheses that would normally go around C and M. But we return to this in section 9.9. (Suppes 194a)]
Suppes then mentions some even more complicated instances. So consider for example the sentence:
Some Americans who drink tea do not drink either coffee or milk.
We first notice that this sentence’s general form is “Some S are not P.” But let us look first at the subject, “Some Americans who drink tea”. [Here we are not saying “some who are Americans or some who drink tea”. So we are not talking about the union of these two sets. Rather, we are speaking of the intersection of the sets.] We would translate the subject as:
A ∩ T
[The predicate is “do not drink either coffee or milk”. We are not here saying “are not included in the set of people who drink both coffee and milk” and thus not with the intersection of the set of people who drink and coffee and the set of people who drink milk. Rather, we are referring either to the set of those who drink milk or the set of those who drink coffee, and thus to the union of the sets.] We then translate the predicate as:
C ∪ M
[Recall from before that we translated “Some Americans are not philosophers” as A ∩ ∼P ≠ Λ. Here we will use the same structure, but this time we substitute our more complex subjects and predicates.] We can thus translate the whole sentence as:
(A ∩ T) ∩ ∼(C ∪ M) ≠ Λ
[In the following Suppes uses some equivalences that he says we discuss more in section 9.9, so for now I will just quote them.] This formulation above is equivalent to:
A ∩ T ∩ ∼C ∪ ∼M ≠ Λ
since, corresponding to De Morgan’s laws for the sentential connectives, we have:
∼(C ∪ M) = ∼C ∪ ∼M.
Suppes, Patrick. Introduction to Logic. New York: Van Nostrand Reinhold / Litton Educational, 1957.