Introduction How it all Started The earliest discussion of the phenomenon of presuppositions in relation to a formal language probably traces back to the nineteenth century philosopher Gottlob Frege1. Frege’s most significant contribution to modern logic was his ground breaking development of axiomatic predicate logic. Predicate logic uses quantified variables, which makes it far more expressive than Aristotelian syllo- gistic and Stoic propositional logic. Predicate logic could be used to represent inferences involving arbitrarily complex mathematical statements, and success- fully deals with problems which traditional logic is unable to solve, such as the problem of multiple generality2. Amongst his many contributions, Frege is often credited with the principle of the compositionality of meaning, which is the principle that the meaning of a complex expression is determined by the meanings of its constituent expressions and its structure. This principle remains predominant in the field of mathematics, semantics, and natural language pro- cessing, playing a key role when logical representations are constructed from the syntactic structures of a given expression or formula. The details are spelled out formally subsequently, by people such as Tarski (1933; 1944), Montague (1970), ○ Westerstahl (1998), and Janssen (1997; 2001). Presuppositions were not Frege’s central concern. However, according to him, an ideal language should have a truth value for every one of its well- formed sentences. Presuppositions then, make natural language an unfortunate exception to this ideal, because presuppositions must be met in order for the ex- pression containing them to have a truth value. But sometimes presuppositions fail. For example, definite NPs come with a presupposition that they denote properly. Consider the following century old example (Russell 1905): 1Frege, in his article “On Sense and Reference” (1892), in a footnote to a discussion about adverbial clauses, talks about the referential presupposition in the sentence: “after the sepa- ration of Schleswig-Holstein from Denmark, Prussia and Austria quarreled”. Frege points out that the presupposition (Voraussetzung) in which Schleswig-Holstein must be thought of as part of Denmark, is a necessary condition for “the separation” to have a reference. 2The problem of multiple generality is the inability of traditional logic to determine valid inferences of statements involving more than one quantifier. For example: (i) Every number has a successor. (ii) There is a successor preceded by every number. The syntax of traditional logic permits exactly four sentence types: “All A’s are B’s”, “No A’s are B’s”, “Some A’s are B’s” and “Some A’s are not B’s”. Each type is a quantified sentence containing exactly one quantifier. However, (i) and (ii) each involve two quantifiers. The only way for traditional logic to deal with this is to reduce the second quantified expression to a term: ′ (i) Every number is (a-number-which-has-a-successor). (ii)′ Some successor is (a-successor-preceded-by-every-number). But even after this treatment, we cannot determine whether (i) entails (ii). 1 (1) The King of France is bald. The presuppositions in (1) are triggered by the definite NP ‘the King of France’. Frege observed that a singular definite description like this has two constraints associated to its referential function: There is at least one King of France- a presupposition that ‘the King’ must denote an entity in the Model of interpretation; and there is at most one King of France- a presupposition that ‘the King’ is unique in the context. Neither of these are asserted, in other words they are not stated explicitly in (1), and for this reason they are called ‘presuppositions’. These two presuppositions must be satisfied in order that the description can refer. If there is no King, or if there is more than one King of France, (1) will fail to have a proper truth value. Frege saw this as a shortcoming of natural languages when definite descriptions fail to denote some unique entity. The way in which he gets around this deficiency is to include a special nil entity, and whenever a definite description fails to satisfy the aforementioned presuppositions, to assign it assigned to the nil entity. To put this formally, we first designate the iota operator, (an upside down Greek iota), for descriptions (Gamut 1991). Descriptions are an addition to Frege’s predicate logic vocabulary. They are complex terms like the universal quantifier ∀ and the existential quantifier ∃ in that they always come with a variable and are followed by a propositional function which is their scope. Let be a formula and x a variable, in order to interpret descriptions we add the following to the semantics of predicate logic: (2) ⟦ x⟧M,g is the unique individual d ∈ D s.t. VM,g[x~d]() = 1 If there is no such unique individual d ∈ D, then x is assigned to d0, where d0 is the nil entity. At this point we simply have to ensure that d0 does not belong to the set of entities satisfying . In the example of (1), if there is no King of France, as long as d0 does not belong to the interpretation of bald, (1) will be false because d0 ∉ bald. This solution however, is ad hoc and merely scratches the surface of the problem of presupposition failure. Consider the negation of (1): (1)′ The King of France is not bald. Applying (2) to (1)′ just as we did for (1), when there is no King of France, the variable representing the King of France is assigned to d0. Since d0 ∉ bald ′ is the same as ¬(d0 ∈ ), (1) must be true. This outcome suggests that the King of France is not bald if there is no King of France, a rather unintuitive reading, since we typically expect both (1) and (1)′ to be false or unknown in circumstances where there is no such King. A possible solution here would require the modification of the negation operator so that its output for those formulae with d0 is either false or undefined. This modification ultimately leads to trivalent (Strawson 1952; Kleene 1952) and multivalent systems (Herzberger 1973; Martin 1977; Bergmann 1981), both of which have consequences on the entire logical framework that are unforseen by Frege. 2 The first notable attempt after Frege to deal with this problem is Russell. In On Denoting (1905), Russell characterizes the notion of nil entity (or in Russell’s own words, the null class3) as something which is used to denote the class of all “unreal individuals”. Russell firmly rejects the notion of unreal individuals, because to commit to the notion of some existing, yet unreal objects, such as ‘the round square’, ‘the King of France’, is “apt to infringe the law of contradiction”. Instead, he proposes an alternative: Let C be a denoting phrase, say ‘the term having the property F’; then the sentence “C has property G” translates to “there is one and only one term that has the property F, and that one has the property G”. Thus (1) according to Russell should really be read follows: (3) There is one and only one entity that is King of France and he is bald. This approach exemplifies the misleading form thesis, according to which the grammatical form of sentences sometimes does not reflect their logical form, and is therefore misleading. The underlying cause for confusion is that definite descriptions and proper names have the same syntactic function, as ‘the King of France’ and ‘Louis XIX’ (who does not exist) are syntactically interchangeable (substitution of one for the other preserves syntactic well-formedness). Russell points out that this grammatical similarity is deceptive. A sentence such as (1) consisting of a singular definite description and a predicate is not logically a subject-predicate sentence at all, despite its grammatical subject. ‘The King of France’ should not be thought of as a normal NP any more than quantified expressions like everyone and no one. Instead, a definite description is a kind of ‘existential proposition’ and the formulae representing it should be re-written to capture the logical form of such proposition4.For every one-place predica- 3Russell relates the null class to Frege in the following passage (Russell 1905): . Another way of taking the same course (so far as our present alternative is concerned) is adopted by Frege, who provides by definition some purely conven- tional denotation for the cases in which otherwise there would be none. Thus ‘the King of France’, is to denote the null-class. But this procedure, though it may not lead to actual logical error, is plainly artificial, and does not give an exact analysis of the matter. Thus if we allow that denoting phrases, in general, have the two sides of meaning and denotation, the cases where there seems to be no denotation cause difficulties both on the assumption that there really is a denotation and on the assumption that there really is none. 4Russell makes the further claim that proper names are in fact, concealed descriptions. Consider the following: (i) Socrates is a man. (ii) All man are mortal. (iii) (Therefore,) Socrates is mortal. Traditional logic is insufficient for the deductive reasoning from (i) to (iii), because it lacks the syntax to represent (ii). Even though (ii) could be represented as Mortal(S) where S=Socrates, Mortal(all man) simply won’t do for us to arrive at (iii). Frege improved the syllogistic logic system by introducing quantifiers which gives (ii): ∀x man(x) → Mortal(x). Russell expanded this further by claiming that proper names too, are quantified expressions: 3 tion containing a description expressed with the iota operator, we can give an equivalent formula expressed using the standard predicate logic quantifiers: (4) G( xFx) = ∃x(Fx ∧ (∀y(Fy → y=x)) ∧ Gx) And this translation can be generalized to arbitrary formulas of predicate logic.