Handbook of the History of Logic: - Fordham University Faculty

Relational Logic of Juan Caramuel 657
It is clear that (ii) and (iv) are both negations of the copula, but in each case a
different one. The first stands outside the predicate structure, the second within it.
“Every man does not commit any sin”, añ*i, and “every man non-commits some
sin”, a*ñi, are different, for the first is equivalent to “every man non-commits every
sin”. Caramuel makes the former logical form with outside negation even more
explicit: “añ,q*i”, where “q” stands for the relative pronoun (qui): “every man is
not the one, who commits some sin”. 34
In his relational syllogistics, however, Caramuel settles for another, less conservative
and more innovative approach. The symbolism is simplified. The asterisk
and relative pronoun signs are dropped. Thus “ai” means “every. . . some...” as in
“every man commits some sin”. The key idea in this analysis is the generalization
of the copula from the semantically rather empty verb “to be” to any meaningful
verb, e.g. “commits”, amounting to a major step in the direction towards contemporary
predicate logic. Hence, there is only one copula present in the relational
statement. As for negation, only one symbol for negation is introduced, “n”, either
for quantifier negation (n- - and -n-), or for the negation of the (inner) copula (-
-n). There is no place for the negation of the outer, non-predicate copula, for none
is needed. More precisely, since the form (ii) -ñ*- is equivalent to (iii) -*n-, one
does not need (ii) and can do with (iii), second quantifier negation. This allows
for the simplification, for two different negation symbols are no longer necessary.
The position before or after the quantifier makes all the difference now.
If the verb “to be” is only one copula among many possible ones, the standard
quantifier-subject-copula-predicate upright form is but a special case of relational
statement quantifier-subject term-copula-quantifier-predicate term form. Thus,
grammatically, even though there is no predicate quantifier in “every man is an
animal”, it is just not made explicit, but logically, it is present (“every man is some
animal”). 35 It is precisely this move of generalizing the copula and the consequent
34 “a*ñi” could be rendered as “every man is the one, who does not commit some sin”. Cf.
LO, Pars I, disp. I De Propositione Obliqua in Genere, pp. 407-415.
35 Caramuel’s survey of contemporary logic in Grammatica Audax Pars III, art. 1 shows that
the quantification of the predicate term is not an uncommon feature. Caramuel himself in LO
Pars II quite naturally occasionally rewrites “a” with “ai”. This would mean that the second
from the two squares of opposition presented below is the traditional square of A, E, I and O
statements. Caramuel could thus be credited with discovering another one.
Caramuel also speaks, however, of the extended number of moods in relational logic in comparison
with the classical Aristotelian syllogistics owing to discrimination of two forms within
the universal affirmative statement (A): either ai or aa: J. Caramuel, LO, ParsII,disp. IX,p.
432: Quatuor isti modi constant ex universalibus et affirmativis, et tamen differunt inter se; ut
... intelligatur, quanto sit ditior obliqua Dialectica caeteris; siquidem quatuor aut pluribus modis
praemissas suas disponit, quas omnes rejiceret antiqua ad Bammada, et si non cognosceret hunc
modum, (non enim Aristotelicus sed Platonicus est) ad Barbara. (“These four moods arise from
universal affirmatives; however, they differ among themselves. In order ... to see to what extent
the oblique logic is richer than others, as it disposes its premises in four or more ways, which
ancient logic would all classify as the mood Bammada, or if it did not recognize this mood [for it
is not Aristotelian, but Platonic], as the mood Barbara.”). This would mean that the traditional
square of opposition of A, E, I and O statements contains within itself both squares, as it were.
The first alternative, however, appears to be more likely the case.
The duality pointed out might be related to the two ways of analyzing relational statements, for