The Cube, the Square and the Problem of Existential Import

Abstract

We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two kinds of import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle's system appears then as a fragment of a broader system which can be developed by using FOL. 1 The case of Frege is more complex; in his Begriffsschrift, he presents a square he assumes to be valid; but he formalizes the propositions in exactly the same way as Russell, and these formalizations do invalidate the square. 2 See for instance Quine 1950, Chapter 15, French translation (p. 96) and Russell 1959, French translation, Chapter VI (p. 83)where both authors assume that the universals do not have an import, when translated by conditionals in the modern symbolism.

Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams

“…12 Note, however, that the second lowest value in the 8-range is reached by some system in S ‡ , since |Π FOL(AX ) (F ‡ )| = 5. The weakest logical system FOL(∅) thus yields the highest bitstring length (16), while the strongest logical system FOL(AX ) yields the lowest (attainable) bitstring length (5). This suggests an inverse correlation between logical strength and bitstring length: stronger logical systems yield shorter bitstrings.…”

“…Informally: even though the fragment parameter is fixed to F, varying the logical system parameter within S suffices to reach all values in the |F|-range. In other words, all bitstring lengths that might theoretically be necessary to represent fragments of the same size as F, are already needed to represent F itself, under the different logical systems in S. 5 To illustrate this account of context-sensitivity, we will consider the case of 4-formula fragments, i.e. the case of Aristotelian squares.…”

“…7 It is shown in [11,Sect. 4] that 5 Note the subtly different quantification patterns corresponding to these two limiting cases: minimal context-sensitivity corresponds to ∃ ∈ R |F | : ∀S ∈ S : |Π S (F)| = , while maximal context-sensitivity corresponds to ∀ ∈ R |F | : ∃S ∈ S : |Π S (F)| = . 6 As is well-known, in the language of first-order logic, these formulas can be formalized as ∀x(Ax → Bx), ∃x(Ax ∧ Bx), ∀x(Ax → ¬Bx) and ∃x(Ax ∧ ¬Bx), respectively.…”

“…Next, the statements A1-A4 can all be seen as (partial) interpretations of the traditional existential import principle. According to its most cautious interpretation [5,40], this principle states that the predicate occuring in the first argument position of a categorical statement should not have an empty extension, which is captured by A1. However, another interpretation is that all predicates should have nonempty extensions, regardless of whether they occur in the categorical statement's first or second argument position [39]; this means that both A1 and A3 should be accepted as axioms.…”

Existential Import and Relations of Categorical and Modal Categorical Statements

“…18 For a brief overview of the debate concerning existential presupposition and categorical statements see [4]. 19 The logical analysis of the predicate "be such an individual that is identical with the King of France" is λxλw […”