Claudio Gutierrez

Translated from Spanish by Inés Gutiérrez

60. Logical Subject and Predicate

Thus far, our analysis has considered the atomic proposition as ultimate structural unit. Such proposition, represented for example by “”, is a complete thought –the linguistically smallest unit of which I can say that it is true or false. Now then: although it is true that there is nothing smaller than the atomic proposition that is true or false, there is something smaller, part of the atomic proposition, which is true of ... or false of .... For example, “… has confidence in himself” can be true of John. Even if I cannot say that they are true or false by themselves, names as “John” and phrases as “… has confidence in himself” turn out to be extremely important to distinguish. Of the former we say they are logical subjects; of the latter, that they are logical attributes or simply predicates ⁽¹⁾.

The same atomic proposition can be considered an indivisible unit, such as “”, but also a copula of two analytical ingredients, logical subject and predicate. The beauty of this analysis is that it allows us to speak about predicates that correspond to a single logical subject, as well as about logical subjects that correspond to a single predicate. If we take the atomic proposition “John is a human being”, we can decompose it into the logical subject “John” and the predicate “… is a human being”. Yet, John has many other attributes, and so has he a corresponding set of predicates: for instance, “… is intelligent”, “… is American”, “… is slender”, “…is 1.80 m tall”, etc. At the same time, “… is a human being” has many other possible logical subjects, for example “Mary”, “Paul”, “Joseph”, etc.

A moment's reflection will make you identify what we are calling here "predicate" with what we introduced in Chapter VI as "collective term"; so much so since what we are calling here "logical subject" corresponds strictly to what we referred to there as "proper name". In fact, the set of all logical subjects suited to a specific predicate corresponds to the extension of the collective term.

61. Singular and General Propositions

The phrase “… is a human being” is not a proposition, because we cannot say that it is true or false. It is only a form (in the sense of an application form). It can be transformed, however, into a full-fledged proposition simply by filling out the form, i.e. writing a proper name over the empty space –over the dots. We will call this operation instantiation, since “John is a human being” can be considered an instance of “… is a human being”. The result of the operation will be a singular proposition namely the attribution a specific predicate to a definite individual.

This is not, however, the only way in which one can transform a predicate into a proposition. Since we humans love to speak in indefinite terms, another powerful recourse is available for doing precisely that: the use of pseudo-names, like "x", "y" or "z". So, we can transform “… is a human being” into “x is a human being”, where “x” occupies the place of the proper name. We will say that “x is a human being” is almost a proposition, because we do have filled the empty space; but the “proper name” used is enigmatical and we still cannot decide whether the expression is true or false. It is almost a proposition but not quite; still, not a genuine proposition. Hence, we are going to call it quasi-proposition.

A quasi-proposition turns itself into a full-fledged proposition as soon as we determine to how many individuals in our universe of discourse (or our data base) it applies. We call this determination “quantifying a quasi-proposition.” The result is, to be sure, a proposition: a general or quantified one. We can say, for instance, that the quasi-proposition applies to one, and only one, individual. Our quasi-proposition would then be transformed into the following: “for exactly one x, (I assert that) x is a human being”, where the indefinite proper name “x” would represent any element, we do not know specifically which, of our universe. Or we could say, “for any x (I assert that) x is a human being” (this would be, probably, a false proposition). These are two different nice ways to convert a quasi-proposition into a genuine proposition. False or true, our expressions would now be propositions, not quasi-propositions, since we can begin trying to decide their logical values: we are dealing here with logically complete thoughts.

62. Universal and Existential Propositions

This second procedure to transform a predicate into a proposition receives the name of generalization, since it is a way of speaking in general terms, without specifying the proper name of our logical subjects. Of the many forms that the operation of generalization can cover, two are especially useful. The first one, which we will call universal generalization, consists in placing before the quasi-proposition the words “for every x it is said that”; the second, which we will call existential generalization, consists in placing before it the words “there is at least one x such that”. Let us see some examples.

If our predicate is “… is good”, our quasi-proposition will be “x is good”, which we do not know whether it is true or false, neither can we find it out. To do that, it is necessary to place some phrase before it. In this manner, an optimist will say: “for all x (I believe that) x is good”, which in ordinary language would be equivalent to “all is good”. A pessimist, in return, will say, “for all x (I believe that) it is not the case that x is good”, which in everyday language is equivalent to “nothing is good”. Sensible people will rather say: “there is at least one x such that it is good" or "there is at least one x such that it is not good”. The first two generalizations are universal; the last two, existential.

Let us now try to represent graphically these new distinctions and analytical concepts. If the atomic proposition, not logically analyzed except in order to be able to tell its truth value, was represented with any of the letters “U”, “V”, “W”, “Y”, we can represent the predicates with the same letters followed by a hyphen⁽²⁾. This hyphen will remind us that something has been taken out from the original proposition, namely, the proper name or logical subject. Finally, we put everything inside parenthesis as a warning to the effect that we are still dealing with atomic propositions, not with molecular ones: “(U-)”, “(V-)”, “(W-)”, and “(Y-)”. If we now want to go back to represent propositions, but in a way that the analysis of them in predicate-plus-subject is conserved, we can agree on representing the subject by an “o”, which (like the “x”, but neither the “y” nor the “z”) has the advantage of not altering its shape by mirror reflection, preserving for us the negation operation. As we will surely want to speak of several logical subjects, in expressions such as “John loves Mary”, let’s convene also that “o” will be the name of a different person or object depending on its color. Thus, “John is a human being” would be rendered “(Uo)", but “Peter is a human being” would be represented as “(Uo)”, and “John loves Mary” as “(Aoo)”, where “A” would be our way to represent the predicate “x loves y”.

The quasi-proposition, i.e. the predicate inasmuch as attributed to an indefinite individual, will still be represented by an “x” following the letter-predicate, that is, instead of the hyphen. In this manner, “x is a human being” would be represented as “(Ux)”, where the color will allow us to indicate that the “x” in one color and the “x” in another do not necessarily refer to the same individual. Thus, the formula “(Axx)” would allow the person who reads it to freely translate the quasi-proposition either to “John loves Mary” or “John loves himself”; but on the other hand, “(Axx)” would compel the reader to the interpretation that, whoever x may be, he or she will love him/herself. We are now ready to ask ourselves how to represent quantified expressions, v.g. “all is good” or “something is good” or “every human being has another human being that loves him”, proposition that, in addition to having a difficult representation, is perhaps definitely false. But to be able to do representations such as these, we still need some preparatory work.

63. Connectives and Quantifiers

As we have seen, we can distinguish in logical analysis between elements of the universe and their predicates or attributes. Elements are also called individuals, or entities, or things, and are the ones that can act as logical subjects in propositions, represented by their proper names. Thus, an individual will be John, or Philadelphia, or this table, that book, or the planet Mars (but, watch out: “Planet” is a predicate of Mars). Predicates are, on their part, properties of individuals, which can be asserted of them using propositions. In each one of these propositions I will say that such or such predicate is attributed or not to the thing in question, is true or not of it.

Let us imagine now that we know the number of individuals in the universe; and that it is, let us say, exactly three individuals. Let us call those individuals o, o, and o. In such a small universe, a universal proposition, for instance “for all x it is the case that x is good”, must be understood as the conjoined attribution of the predicate to each one of the individuals of the universe. Our example will be equivalent to the following proposition of conjunctive character: “(Wo) and (Wo) and (Wo)”.

In a similar way, the existential proposition, for instance “there is at least one x such that x is good”, must be understood as the alternative attribution of the predicate to each one of the different individuals. Our example will be equivalent to the following disjunctive proposition: “(Wo) or (Wo) or (Wo)”. In general, and for a universe of n individuals, the universal proposition will be equivalent to the conjunction “(Wo) and (Wo) ... and ()", and the existential proposition equivalent to the disjunction “(Wo) or (Wo) ... or ()".

Thus, knowing the number of individuals in our universe, it is always be possible to express general propositions by means of conjunctions or disjunctions. In practice, however, either because we do not know that number or because it is too large, we will not be able to dispense of quantifiers and quantified expressions. Thus, the reduction of quantified expressions to expressions with connectives has just a theoretical or explanatory value.

64. Truth and Falsehood of General Propositions

The universal proposition, because it is reducible to conjunctive propositions, will be false if and only if at least one individual from the universe does not bear the predicate, that is, if at least one of the singular propositions –to which it can theoretically be reduced– is false; otherwise it will be true. The existential proposition, because it is reducible to a disjunctive proposition, will be true if and only if at least one individual from the universe bears the predicate, that is, if at least one of the singular propositions –to which it can theoretically be reduced– is true; otherwise, it will be false ⁽³⁾.

If we reduce the universe artificially, for example, to the members of the Republic of Costa Rica’s Congress, we will be able to assert several things about the elements of that universe, which we know are 57. We will be able to say: “for all x it is the case that x was elected by the people”, “there is at least one x such that he/she fulfils his/her duties”, etc., very diverse propositions which will be some true and some false. All of them can be transformed into conjunctions or disjunctions, each one composed of 57 singular propositions. These conjunctions or disjunctions, and similarly their corresponding universal or existential quantifications, will be true or false depending on how the values of truth and falsehood are distributed among the singular propositions that serve them as theoretical base. A simpler example: if our universe is formed by the persons of the Holy Trinity according to the Christian faith, “for all x it is the case that it is God” would translate onto “”. This is true if we consider “”, “” and “” also true. It would be false if we considered “” false, “the Holy Spirit is God”, as was preached by one of the first Christian heresies. “There is at least one x such that he became man” would translate as “”, that would be true if one of the singular propositions, for example “”, is true, but false if all the singular propositions were false.

So, there is a very close relationship between the phrase “for all x it is the case that”, which we call universal quantifier, and the conjunction connective. That relationship consists in the fact that both express the idea of strong truth, of atomic propositions or of a quasi-proposition. Similarly, there is a very close relationship between the phrase “there is at least one x such that”, which we will call existential quantifier, and the disjunction connective. That relationship consists in the fact that both express the idea of weak truth, of atomic propositions or of a quasi-proposition. Such close relationships will orient us at the time of deciding in a general way, whatever the number of individuals in the universe may be, how we will represent graphically universal and existential propositions. This is the subject matter of next chapter.