Dany Jaspers

The present study describes how three now almost forgotten mid-20th-century logicians, the American Paul Jacoby and the Frenchmen Augustin Sesmat and Robert Blanche, all three ardent Catholics, tried to restore traditional predicate logic to a position of respectability by expanding the classic Square of Opposition to a hexagon of logical relations, showing the logical and cognitive advantages of such an expansion. The nature of these advantages is discussed in the context of modern research regarding the relations between logic, language, and cognition. It is desirable to call attention to these attempts, as they are, though almost totally forgotten, highly relevant against the backdrop of the clash between modern and traditional logic. It is argued that this clash was and is unnecessary, as both forms of predicate logic are legitimate, each in its own right. The attempts by Jacoby, Sesmat, and Blanché are, moreover, of interest to the history of logic in a cultural context in that, in their own idiosyncratic ways, they fit into the general pattern of the Catholic cultural revival that took place roughly between the years 1840 and 1960. The Catholic Church had put up stiff resistance to modern mathematical logic, considering it dehumanizing and a threat to Catholic doctrine. Both the wider cultural context and the specific implications for logic are described and analyzed, in conjunction with the more general philosophical and doctrinal issues involved. © 2016 Elsevier B.V., All rights reserved.

This paper discusses precise quantification by means of number systems on the analogy of Jaspers’ (2005) earlier analysis of the comparatively vague type of quantification expressed by predicate calculus operators {all/every/each, some, no}. It is argued that numbers provide an interesting testing ground for the validity of the Boolean approach to quantifiers in Jaspers (2005). More specifically, this excursion into maths is undertaken to show that a very basic cognitive- logical system of oppositions which underlies natural language logic governs natural mathematics as well. The concrete starting point of the article is Popper’s twin prime problem, which is followed by a discussion of number systems, more specifically the distinction between the natural number system {(0,) 1, 2,...} and the prime number system. The former type of system will be argued to be orders characterized by the operation of addition/subtraction. The prime number sequence is different in that it is multiplicative/ divisional rather than additive. It is generally recognized in mathematical circles that the latter type of sequence is more complex than the former. This fact tallies well with (and hence provides indirect support for) the linguistic findings in Jaspers (2005), whose core was the claim that natural language disjunction – known to be isomorphic with addition in algebra – is cognitively and lexically less complex than conjunction, which is isomorphic with multiplication.

Colour has been on the minds of philosophers, logicians and linguists for a very long time: its connection with logic; the relation between percepts and concepts; the influence of colour language on colour thought, and the question whether colour is in the world or purely mental. The present contribution starts from the age-old observation that the four logical opposition relations (contradiction, (sub)contrariety, entailment) as embodied in the square of opposition and extensions thereof such as the Blanché hexagon are not particularly choosy about the actual conceptual content of the lexical fields they organize. They generalize over modal operators, propositional operators, predicate calculus operators, tense operators, etc. Why would that be? In the first place, it must mean that the relations are located at a level of generality which has no regard for the concrete conceptual differences between those fields, latching on to some very general substrate. Actually, taking that idea further, I have argued (Jaspers, Logica Universalis 6: 227–248, 2012) that the level of abstraction away from concrete incarnations in the vertices of the hexagon even needs to break out of the conceptual realm into that of colour percepts, where a homologous patterning among primary and secondary colour percepts obtains.

In this paper evidence will be provided that Wittgenstein’s intuition about the logic of colour relations is to be taken near-literally. Starting from the Aristotelian oppositions between propositions as represented in the logical square of oppositions on the one hand and oppositions between primary and secondary colors as represented in an octahedron on the other, it will be shown algebraically how definitions for the former carry over to the realm of colour categories and describe very precisely the relations obtaining between the known primary and secondary colours. Linguistic evidence for the reality of the resulting isomorphism will be provided. For example, the vertices that resist natural single-item lexicalization in logic (such as the O-corner, for which there is no natural lexicalization *nall (=not all)) are not naturally lexicalized in the realm of colour terms either. From the perspective of the architecture of cognition, the isomorphism suggests that the foundations of logical oppositions and negation may well be much more deeply rooted in the physiological structure of human cognition than is standardly assumed