As we announced, we will proceed in accordance with a semi-formal regime.
Consider an elementary predicative statement of the type P(x), to be read as: ‘x has the property P’; suppose moreover that this statement is referred back to a determinate world, whose transcendental is T. Here the letter x designates a variable, a being of the world taken at random. Since we are dealing with a variable, we do not know whether it possesses the property in question or not. That is why an expression such as P(x) will be called open: its value (true, false, probable, not very probable, etc.) effectively depends on the determinate term which is substituted for x. The variable x will instead be called free.
Now, if the letter ‘a’ designates a determinate being-there, in other words if ‘a’ is the proper name of an apparent, then we should be able to know whether this apparent possesses the property P or not. In this case, we say that P(a) is a closed expression. As for a, it is called (by contrast with the variable x) a constant. Therein lies the entire difference between the out-of-context phrase ‘this thing is red’—whose truth-value is unknow-able in the absence of information about what thing we’re dealing with—
and the phrase referred to the autumnal world ‘the ivy is red’, which is true.
Let us now suppose that we are in possession of a language with variables (x, y, . . .), constants (a, b, c, . . .) and predicates (P, Q, R, . . .).
We can interpret the statements constructed in this language in a transcendental T as follows:
1. If P(a) is true, we will assign it the value M (the maximum in T ).
2. If P(a) is false, we will assign it the value µ (the minimum in T).
3. If there exist, in the transcendental of the world in question, elements other than µ and M, let’s say p, then P(a) = p signifies that the statement, neither true nor false, has an ‘intermediate’ value, for example ‘a strong possibility of being true’, ‘true in some particular cases, but more often false’ and so on.
This is the case for example with ‘the gravel of the path is grey’ which, in absolute terms, is neither true nor false, since, though this gravel is white, the statement can be true if it has rained, if I see the path in the mist and so on.
2. SYNTAX: CONJUNCTION (‘AND’), IMPLICATION (‘IF . . . THEN’), NEGATION, ALTERNATIVE (‘OR’)
The structure of the transcendental, as expounded in Sections 1 and 3 of Book II, will help us to interpret logical connections.
1. What is the value of [P(a) and Q(b)], which consists in simultaneously affirming P(a) and Q(b)? Intuitively, it is clear that [P(a) and Q(b)] is true insofar as a possesses the property P and b also possesses the property Q. If even one of the two does not clearly possess the property, for instance if Q(b) is false, then [P (a) and Q(b)] is certainly false. Generalizing, we can say that [P(a) and Q(b)] cannot be more true than the one of the two which has the weakest truth-value, if these values are comparable. Thus, if P(a) is true but Q(b) is only probable, the conjunction of the two is merely probable.
So it is entirely reasonable to interpret the value of [P(a) and Q(b)] as being, in the transcendental, the conjunction of the presumed values of P(a) and Q(b). In effect, the conjunction of p and q, that is p ∩ q, is the greatest of all those which are lesser than or equal to p and q. If, for example, P(a) = M and Q(b) = M (both are true), then [P(a) and Q(b)] will have the value M ∩ M = M. If P(a) = M and Q(b) = µ, then [P(a) and Q(b)] = M ∩ µ = µ, because M denotes the true and µ the false.
Now, if P(a) = M (true) and Q(b) = p (probable), then [P(a) and Q(b)] = M ∩ p = p, because p ≤ M in every transcendental (applying P.0).
2. The question of implication follows the same pattern and naturally leads to its interpretation in terms of dependence, as defined above.
Intuitively, the fact that P(a) implies Q(b) signifies only that the truth of P(a) compulsorily entails the truth of Q(b). In natural language, this is said
‘if P(a), then Q(b)’. This point is validated by the operator p ⇒ q (the dependence) of a transcendental.
Let’s suppose that P(A) = M and that [P(a) ⇒ Q(b)] = M. We will verify that Q(b) = M, and we will then have the interpretation of dependence in terms of implication.
We know (see subsection 11 above) that if [P(a) ⇒ Q(b)] = M, we neces-sarily have P(a) ≤ Q(b). But it is then required, since P(a) = M, that Q(b) = M.
The mediaevals already noted that ex falso sequitur quodlibet (anything whatsoever follows from the false), meaning that if P(a) is false, the implication of Q(b) by P(a) is always true, whatever Q(b) may be.
This is also valid in a transcendental. It is easily shown that if p = µ, then (p ⇒ q) = M, whatever q may be: if p = µ, then p ≤ q, from which we infer that (p ⇒ q) = M.
To cover the general case, we will pose from the get-go that the value of ‘P(a) implies Q(b)’ is p ⇒ q, if p is the value of P(a) and q that of Q(b).
We could say that (transcendental) dependence interprets (logical) implication.
3. Let us now deal with negation, or the value ‘non-P(a)’ for the world in question. The reader will have already understood that we will interpret it using the reverse of the value of P(a). This does not throw up any particu-lar problem. The value of ‘the ivy is not red’ will be the value of the reverse of the degree of appearance assigned to the red of the ivy. If, as we assumed, in the autumnal world the ivy is effectively red, its degree of appearance being maximal, that is M, and since, as we established, the reverse of M is µ, the final value of ‘the ivy is not red’ is minimal. We will therefore conclude that in this world the statement is false. It is worth noting however that, unless we know the particulars of a transcendental, we cannot predict what follows from the negation of statements whose value is intermediate. For example, ‘the gravel is not grey’ will indeed be worth the reverse of the value assigned to ‘the gravel is grey’. But if we suppose that this value is p, we have no general rule allowing us to know the value of ¬ p. The only certainty is that the conjunction of p and ¬ p is worth the minimum µ, as we demonstrated in Section 3.
4. Let us briefly discuss the alternative, the connector ‘or’, whose clas-sical interpretation is that ‘A or B’ is true if A is true, or B is true, or both. In fact, we can consider it as a particular (finite) case of the envelope. Take, for
example, two apparents, say the ivy and the tile roof of the house. Consider the property ‘being of a colour that contains violet’. The two apparents in question neither truly validate this property nor absolutely reject it. We could say that if a is the ivy, b is the tile roof and P the property in question, the truth-value of P(a) will be intermediate, say p, and the value of P(b) too, say q. What can we say then about the value of ‘P(a) or P(b)’? It is entirely reasonable to assign it the value immediately superior to that of P(a) and of P(b), or simultaneously equal to both if they are equal to one another. This value is provided by the envelope of the set constituted by the degrees p and q. In effect, the connector ‘or’ designates a complex phrase which is true ‘in measure’ of the highest value of its components. We will then pose that when P(a) has the value p and P(b) the value q, the value of
‘P(a) or P(b)’ is Σ{p, q}. We can easily recognize here the classical case in which if either p or q has the value M—meaning ‘true’—then Σ{p, q} cer-tainly has the value M, because the envelope must be greater than or equal to that of which it is the envelope.
To conform with classical notation we have already decided to write Σ{p, q} in the form p ∪ q and to call it the union of the degrees p and q.
Just as we remarked that it is not always true, in the transcendental of a given world m, that the negation of the negation is the same thing as affirmation, it is important here to note that it is no more true in general, in a given world, that the union of a degree and its reverse is always worth the maximum M. In other words, the equation p ∪ ¬ p = M is not a transcendental law. This means that we cannot take for granted that, in every world, the statement ‘P(a) or non-P(a)’ is true. This is only the case for classical worlds, of which we will speak in Section 5.