Existential Import

The present paper deals with the semantics of all, some and not in English and related languages. Since these semantics imply a logic, it also deals with the logic of quantification in natural language. To the extent that the linguistic findings are universally valid, the results are relevant to the study of human cognition in general. Since standard quantification theory (SQT) replaced Aristotelian Predicate Calculus (APC), philosophers of language and semanticists have struggled with the fact that APC corresponds better with natural semantic intuitions than SQT. Pragmatics was called in to restore APC on non-logical, pragmatic grounds. The replacement of APC by SQT was motivated on the grounds that (a) APC suffers from improper existential import, and (b) SQT is a straightforward application of Boolean algebra, and as such preserves syllogistic reasoning on a mathematical basis. Logically speaking, SQT differs from APC in that the Aristotelian subaltern entailments are abolished, which makes the Aristotelian Square collapse but saves the equivalences (conversions) between ¬∀¬ and ∃, and ¬∃¬ and ∀. In SQT, the quantifying predicate ∃ over pairs of sets yields truth iff X ∩ Y ≠ Ø and ∀ does so iff X ⊆ Y. The author recently found that if the condition X ≠ Ø is added to the condition for ∀ and the conversions are changed into one-way entailments (i.e. ∀x(Fx,Gx) | – ¬∃x(Fx,¬Gx) and ∀x(Fx,¬Gx) | – ¬∃x(Fx,Gx), but not vice versa), APC is restored without improper existential import (only the subcontraries are lost). The Boolean basis is unaffected since whenever ∀ yields truth, it is still so that X ⊆ Y. The resulting revised Aristotelian predicate calculus (RAPC), is represented in the Hexagon of fig. 1 (arrows stand for entail-ments, 'C' for contraries, 'CD' for contradictories). Note that the tradi-of why the O-corner in APC is never lexicalised: there ís no O-corner to be lexicalized!) The revision of APC is easily shown by means of a valuation space interpretation (VSI) (Van Fraassen 1971). Let the valuation space (VS) of a sentence A, /A/, be the set of situations in a universe U of possible situations in which A is true. To say that A is true now amounts to saying that the actual situation s a ∈ /A/. Clearly, /¬A/ = U – /A/, /A ∧ B/ = /A/ ∩ /B/, and /A ∨ B/ = /A/ ∪ /B/, and the whole of standard …