Claudio Gutierrez

Translated from Spanish by Inés Gutiérrez

65. Quantification Formulae

At the end of last chapter, we were pointing out the close relation that exists between universal quantification and conjunctive proposition; and between existential quantification and disjunctive proposition. Based upon that relation, we have decided to represent universal quantification with the same symbol as conjunction, and existential quantification, with the same symbol as disjunction. Thus, “for all x, x is good” will be “”; and “there is at least one x such that x is good” will then be “”. By doing this we ensure that universally quantified quasipropositions be treated as strong truths, in the same way as any part of a conjunction; and that existentially quantified quasipropositions be treated as weak truths, the same way as any part of a disjunction.

The following issue immediately arises: in the conjunction “” or in the disjunction “” “W” is an atomic proposition, and so is “U”. On the other hand, in the formulae “” and “,” we know that “( Wx )” is an atomic quasiproposition (it will become a proposition when placed under the influence of the quantifier). But then, what is “( x )”? It is precisely the quantification symbol. By itself, it represents nothing; but being within the conjunction brace that precedes a quasiproposition that sign transforms the whole expression into a universal proposition. Something similar happens if it appears as the first element within the disjunction brace, but in this case it transforms the whole into an existential proposition. Thus, the conjunction brace with the symbol “( x )” as its first element, “”, represents the universal quantifier which reads: “for all x, it is the case that”; similarly, the disjunction brace with the symbol “( x )” as its first element, “”, represents the existential quantifier, which reads: “there is at least one x such that”.

66. Opening of a Universal Quantifier

Let us ask ourselves now if we can also apply to quantified formulae the other system of graphic representation, that is, that of boards. First of all, let us ask ourselves if we can open quantified formulae. Nothing prevents it, as long as we do it carefully, understanding well what we are doing, so as not to obtain absurd results. Let us begin with the universal proposition. In accordance with the conjunction rule, we could open the formula “” in a strong board, like this:

( x )      ( Wx )

Now, as we have seen, “(x)” alone does not have propositional value, it is only the quantification sign, part and parcel of the brace which marks as a special kind of conjunction. So, we can freely eliminated it, along with the conjunctive brace turned into a board; by so doing we will be preventing mistakes and confusion. Furthermore, the “(x)” inside the conjunction connective conveys the meaning that the quasiproposition is always true, whatever the proper name taken up by the corresponding predicate, in this case W. This is precisely what we mean when we consider the quasiproposition as a strong truth: all singular derived propositions –for different individuals in our universe– are true, as legitimate components of that big conjunction. Since this is also what it means to be in the main board, being propositions in good standing, the separated “(x)” is no longer necessary and we can dispense of its services. Let’s say, then, in rule form, that the “(x)” disappears in the act itself of opening the conjunction which connects it to the other formula, the object of the quantification. The result of opening up the universal quantification is then simply the following:

But there is more. Because of the nature of universal quantification, to open it up gives us the right to substitute the x that accompanies the predicate, in this case “W,” for any other proper-name mark. Remember that the quantifier tells us that “for all x” something (a predicate) is true. Then it is definitely true of “o,” of “o,” of “o” etc. Hence, any of the formulae “( Wo )” or “( Wo )” or “( Wo )” ... can stand within our privileged board, for instance:

Notice that this formula is no longer a quasiproposition but an authentic atomic proposition, a singular (analyzed) proposition. There is no difference between this proposition and any other atomic (unanalyzed) proposition with respect to truth value.

67. Opening of the Existential Quantifier

Let us now see how to proceed in the case of existential quantification. The result of opening up the formula “”, of course in a weak board, will be:

( x ) ( Wx )

Here, unlike what happens within the strong board, the “(x)” does still perform a function: even if it does not represent anything in particular, it rings a precautionary note. The function it performs is that of a flag reminding us that the formula “(Wx)” is a weak one, and we should not open another formula, in the yard it occupies, with exactly the same flag.

With this restriction we avoid confusions such as the following:


It says here, for instance, that something in the universe is tall and that something in the universe is short. If we open up now the first existential quantification, we will have:

( x ) ( Wx )


The formulae current in a yard are considered strong with respect to all the content of that yard; therefore, the formula “” can be opened inside the yard where “(Wx)” is. But that would put us in danger of concluding that there is something in the universe that is tall and short at the same time. However, the flag “( x )” warns us of that danger. So, we will proceed to open the second quantification with a different flag, changing the color of “( x ),” like this:

( x ) ( Wx ) ( x) ( Yx )


making clear that the predicates “W” and “Y” qualify different individuals.

68. Categorical Propositions

Thus far we have considered quantifications where the corresponding quasiproposition contains no connective. Nevertheless, quasipropositions of the majority of general expressions are interpretable as molecular formulae. The most important of these are so called categorical propositions. They are the following:

    All W are Y              (v.g., All men are mortal)            – universal affirmative
    No W is Y             (v.g., No man is perfect)             – universal negative
    Some W are Y         (v.g., Some men are wise)           – particular affirmative
    Some W are not Y   (v.g., Some men are not good)    – particular negative

Applying analysis of structure, we get the following equivalent expressions:

    All W are Y                for all x, if x is W then x is Y                                         – universal affirmative
    No W is Y                  for all x, if x is W then x is not Y                                   – universal negative
    Some W are Y           there is at least one x such that x is W and x is Y            – particular affirmative
    Some W are not Y     there is at least one x such that x is W and x is not Y      – particular negative

Let us formalize these expressions, according to our symbology (“x is not Y” is interpreted as “it is not the case that x is Y”):

    For all x, if then                                                        
    For all x, if then                                                        
    there is at least one x such that                                  
    there is at least one x such that                                  

As you can easily conclude, not everything that is subject in ordinary language continues to be subject in our formalized language: “W” is grammatical subject in the various original propositions, but predicate in the formalized ones. The only logical subject is the “x” that appears inside the quantified propositions in the quasipropositions, as well as the “o”s that appears in the atomic propositions, by themselves or as components of molecular propositions.

69. Opposition Relations

Categorical propositions can be arranged in a graphic or box traditionally called square of propositions, so that both universal propositions are placed on top and both affirmative propositions are placed on the left-hand side, like this:

	affirmative	negative
Universal
Existential

It is interesting to note the relationships that hold between these four propositions. If we observe attentively, we will realize that the universal affirmative formula is the mirror image of the existential negative formula; likewise, the universal negative formula is the mirror image of the existential affirmative formula. Let us remember that this graphic relationship, being mirror twins, represents our negation operation. These formulae, then, are the negation of each other; that is why we call them contradictory. What one says the other denies, and vice versa. If we return to ordinary language, “all W are Y” is contradictory with “some W are not Y”; “no W is Y”, on its part, is contradictory with “some W are Y”. From the truth of one of each pair of contradictories we can infer the falsehood of the other. And from the falsehood of any of them we can infer the truth of its companion.

This gives us basis to propose a different way of expressing these relations, so that specular symmetry (mirror imaging) becomes more conspicuous. To wit:

	affirmative	negative
Universal
Existencial
	negative	affirmative

Existential Content

Some authors affirm that, besides the aforementioned, other logical relations exist in the square of propositions. For instance, that if “all W is Y” is true, “no W is Y” cannot be true, and that if “some W are Y” is false, “some W are not Y” cannot be false. However, these authors work under the assumption that “there is at least one W”, known as the existential content hypothesis. According to this hypothesis, it is assumed that nobody would bother to do assertions without assuming that the grammatical subject of the sentence exists. Nevertheless, this is not the case. We can make assertions of that kind, for instance, as a joke or simply due to superstitious ignorance. In those cases, it is possible that “all W is Y” and “no W is Y”, are both true; for example, if a student goes by bus to visit his girlfriend’s (as he always does), he could factitiously tell her: “All my Mercedes have flat tires”, which would be true, because if something was Mercedes and mine it would be flat, but since nothing is mine nothing is flat, which is exact. And his girlfriend will be thinking, also with truth: “No Mercedes of his is flat”, because she knows with certainty that he does not have any Mercedes. On the other hand, on the understanding that flying saucers do not exist, we would have to consider false the following two propositions: “some flying saucer are extraterrestrial” and “some flying saucer are not extraterrestrial”. For many reasons of operational effectiveness, we prefer –as the great majority of modern logicians– not to work under that assumption of existential content. Ware its use to be essential on occasion, we could express it directly, as an additional premise for our deductions, like this: “there is something that is Mercedes and mine” (which is false, for the great majority of logic students –the author included– anyway).

70. Quantification in Ordinary Language

Categorical propositions are a sort of model of what we would like all quantified propositions to be, since its management is very simple. However, this model is not realistic. Most propositions from ordinary language have a more complicated logical rendering. Unfortunately, we will not cover them in this course; we will be limiting our study to categorical propositions, and all those which can in some way be transformed into propositions of this kind. We will be giving some rules for that transformation.

In the first place, many propositions do not contain the verb “to be”, but some other verb. Our categorical propositions always contain it; we have analyzed the atomic proposition as the attribution of a predicate (that expresses a property) to a proper name (which points to an individual) and this attribution is done precisely with the help of the verb “to be”. In our symbology, it is indicated by simple juxtaposition of predicate and proper name (as in “Wx”). Propositions which do not contain the verb “to be” can be easily converted into propositions that contain it, through minimal alterations and some violence to linguistic beauty. For instance: “John is always presuming can be turned into “John is presumptuous”; “Mary studies all year long,” perhaps into “Mary is a nerd”. ⁽¹⁾ The results of some of these transformations are not very pretty, and the English professor would probably lament them; but from the stand point of logical analysis they are totally satisfactory. And inasmuch as they are categorical propositions, these versions are preferred to the original propositions, which are not. We will have to dispense in many cases with literary quality and, sometimes, even with grammatical correctness, although we should try –of course– to avoid all those transgressions whenever possible.

Another type of categorical proposition achieved by transformation from one that is not, results from the elimination of a singular proposition. Thus, “Socrates is mortal”, a singular proposition, can be normally represented as “Yo”. In order to have just “x” and not “o” in our proposition, we can resort to an artificial though reasonable procedure. It consists of introducing a second predicate of our own selection (or invention) representing all properties, even the most minimal, of Socrates'; let assign the symbol "W" to it; the name of the predicate in ordinary language would in this case be, of course, "Socratic", on the condition that its intension be the sum-total of properties that we associate with this particular and unique historical individual, Socrates. Now we can express the idea that Socrates is mortal with a categorical universal affirmative proposition, namely: “Everything Socratic is mortal”; in our symbols: . We can do the same with any other singular atomic proposition, affirmative or negative, which can always be transformed by this strategy into a categorical proposition. Awkward but useful if in fact you are limited to work exclusively with this kind of propositions.

To be sure, other propositions contain more specific quantifiers than “all” or “some”, such as “many”, “a few”, “365”, etc. In all these cases we will have to consider that the proposition has existential character, since the respective quasiproposition is not being affirmed universally. If by doing this we lose some valuable information, that would mean that the proposition is not reducible to elemental logic and must be analyzed through a superior branch of logic (which we do not teach here or maybe has not even been invented yet) or through mathematics (which is also a branch of logic, exceedingly sophisticated, that you cover regularly in other courses). Some quantifiers are mixed: for example, “almost all”, “not all”, “all except a few”; they really express two propositions instead of one. Thus, “almost all representatives agreed” means “some representatives agreed” and “it is not the case that all representatives agreed”.

Finally, it is worth mentioning the universal categorical propositions hidden under the words “just” or “only”. For example, “only the violent will conquer the Reign”, must transform into “all that will conquer the Reign are violent”. We must change the word “only” for the quantifier “all” and, at the same time, interchange the position of the atomic predicative propositions.

Note 1: The proposition “Mary loves John” can be formalized in either of two different ways: “Mary is the lover of John” or else “John is the loved one of Mary”. That is, we can consider “the lover of John” or “the loved one of Mary” as the logical predicates, alternating accordingly the logical subjects. However, the very best formalization – which we will not use in this course – is the one we mentioned before, which takes as predicate simply the relation “to love” that demands two logical subjects (for example, “John” and “Mary”). Logic of relations, where predicates can have two or more logical subjects, is regrettably a more advanced branch of formal logic.