Two Treatments of Definite Descriptions in Intuitionist Negative Free Logic Nils Kürbis Published in the Bulletin of the Section of Logic 48/4, 299–317 https://doi.org/10.18778/0138-0680.48.4.04 Abstract Sentences containing definite descriptions, expressions of the form 'The F', can be formalised using a binary quantifier ι that forms a formula out of two predicates, where ιxrF,Gs is read as 'The F is G'. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INFι of intuitionist negative free logic extended by such a quantifier, which was presented in (Kürbis, 2019), INFι is first compared to a system of Tennant's and an axiomatic treatment of a term forming ι operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INFι in which the G of ιxrF,Gs is restricted to identity. INFι is then compared to an intuitionist version of a system of Lambert's which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion. Keywords: definite descriptions, binary quantifier, term forming operator, Lambert's Law, intuitionist negative free logic, natural deduction. 1 Introduction Sentences of the form 'The F is G' can be formalised by using a binary quantifier ι that forms a formula out of two predicates as ιxrF,Gs. This provides an alternative to the usual way of formalising definite descriptions by means of an operator ι that forms a term out of a predicate, where ιxF is read as 'The F'. This paper is a comparison of the two approaches. The use of the same symbol ι for the binary quantifier and the term-forming operator should not lead to confusion, as context will make clear which one is meant. In (Kürbis, 2019), I presented the system INFι of natural deduction for intuitionist negative free logic extended by the binary quantifier ι and proved a normalisation theorem for it.1 The present paper begins with a brief overview of INFι, so that it can be 1For the proof-theory of term forming ι operators in the context of sequent calculi for classical logic, see Indrzejczak (2018b) and Indrzejczak (2018a). 1 read independently of the previous one. I will then compare INFι to a system of Tennant's sketched in (Tennant, 2004) and (Tennant, 1978). Tennant provides rules of natural deduction for a term-forming ι operator within the version of intuitionist negative free logic used here. After some clarifications related to scope distinctions, it will be shown that Tennant's system is equivalent to the subsystem of INFι in which the G of ιxrF,Gs is restricted to identity. Both systems are also shown to be equivalent to an axiomatic treatment of a term forming ι operator within intuitionist negative free logic. I then compare INFι to an intuitionist version of a system proposed by Lambert in (Lambert, 2001), which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. Both systems are shown to be equivalent by means of a translation between their respective languages. As we go along proving these equivalences, the present paper will also illustrate the workings of the rules for the binary quantifier ι with numerous examples of deductions in INFι, and advantages of the present approach over the usual one will become apparent. In particular, in the formalisation of definite descriptions it is desirable to have a device for scope distinctions. The sole purpose of the abstraction operator in Lambert's system is as an indicator of scope. The formalism of the present system, by contrast, incorporates scope distinctions directly. Thus the formal treatment of definite descriptions with a binary quantifier is in this sense more economical than the approach using a term forming operator.2 2 INFι Let's begin with a review of intuitionist negative free logic INF. The rules for the propositional connectives are just those of intuitionist logic: A B ^I: Â B Â B ^E: A Â B B i A Π B Ñ I: iA Ñ B A Ñ B A Ñ E: B A _I: A_ B B A_ B A_ B i A Π C i B Σ C _E: iC K KE: B where the conclusion of KE is restricted to atomic formulas. The rules for the quantifiers are relativised to an existence predicate: i D!y Π Axy @I : i @xA @xA D!t @E : Axt 2I would like to thank a referee for the Bulletin for the careful and helpful comments. 2 where in @I, y is not free in any undischarged assumption of Π except D!y, and either y is the same as x or y is not free in A; and in @E, t is free for x in A. Axt D!t DI : DxA DxA Axy i , D!y i loooomoooon Π C DE : iC where in DI, t is free for x in A; and in DE, y is not free in C nor in any undischarged assumption of Π except Axy and D!y, and either y is the same as x or y is not free in A. The existence predicate also appears in the premise of the introduction rule for identity; the elimination rule for " is Leibniz' Law: D!t " In : t " t t1 " t2 Axt1 " E: Axt2 where A is an atomic formula and, to exclude vacuous applications of " E, we can require that x occurs in A and that t1 and t2 are different. Finally, there is the rule of atomic denotation: At1 . . . tnAD : D!ti where A is an n-place predicate letter (including identity) and 1 ď i ď n. AD captures the semantic intuition that an atomic sentence can only be true if the terms that occur in it refer. INFι has in addition the binary quantifier ι with the following rules: Fxt G x t D!t Fxz i , D!z i loooomoooon Π z " t ιI : i ιxrF,Gs where t is free for x in F and in G, and z is different from x, not free in t and does not occur free in any undischarged assumption in Π except Fxz and D!z. ιxrF,Gs Fxz i , Gxz i , D!z i looooooooomooooooooon Π C ιE1 : iC where z is not free in C nor in any undischarged assumption of Π except Fxz , Gxz and D!z, and either z is the same as x or it is not free in F nor in G. ιxrF,Gs D!t1 D!t2 Fxt1 F x t2 ιE2 : t1 " t2 where t1 and t2 are free for x in F. INFι formalises a Russellian theory of definite descriptions, as ιxrF,Gs and DxpF^ @ypFxy Ñ x " yq ^ Gq are interderivable. 3 3 Comparison of INFι with Tennant's System To formalise definite descriptions using a term forming ι operator within intuitionist negative free logic, Tennant adds introduction and elimination rules for formulas of the form ιxF " t to INF: D!t iz " t Ξ Fxz Fxz i , D!z i loooomoooon Π z " t ιIT : iιxF " t where in Ξ, z does not occur in any undischarged assumption except z " t, and either z is the same as x or it is not free in F; and in Π, z does not occur in any undischarged assumption except Fxz and D!z. ιxF " t u " t ιE1T : Fxu ιxF " t Fxu D!u ιE2T : u " t ιxF " t ιE3T : D!t where u is free for x in F. It is fairly evident that there are reduction procedures for removing maximal formulas of the form ιxF " t from deductions. ιE3T is a special case of the rule of atomic denotation AD. Notice however that it is more properly regarded as an elimination rule for ι, as there is a reduction procedure for maximal formulas of the form ιxF " t that have been concluded by ιIT and are premise of ιE3T. When negation is applied to GpιxFq, an ambiguity arises: is an internal negation, so that GpιxFqmeans 'The F is not G', or is it an external negation, so that the formula means 'It is not the case that the F is G'? Conventions or a syntactic device are needed to disambiguate. The language of Tennant's system makes no provision for distinguishing different scopes of negation. For this reason, in this section I shall restrict consideration to cases in which terms of the form ιxF occur to the left or right of ". I will consider a more complete system after the comparison of a restricted version of INFι with Tennant's system. It might be worth noting that there is a sense in which it suffices to consider occurrences of ι terms to the left or right of identity. Whenever we are tempted to use a formula GpιxFq, we can introduce a new individual constant c and use Gpcq and ιxF " c instead. Furthermore, in negative free logic, if G is a predicate letter, then GpιxFq can be interpreted as DypGpyq ^ ιxF " yq, and instead of the former, we can use the latter.3 There is also no need to apply the existence predicate to ι terms, as instead of D!ιxA we can use Dy ιxA " y. It is generally agreed that the minimal condition on a formalisation of a term forming ι operator is that it should obey Lambert's Law: (LL) @ypιxF " y Ø @xpF Ø x " yqq 3In positive free logic, only half of the insinuated equivalence holds, if predicates are allowed to form sentences from ι terms: then DypGpyq ^ ιxF " yq implies GpιxFq, but not conversely. 4 Tennant's rules for ι are Lambert's Law cast in the form of natural deduction. Call INF with its language modified to contain a term forming ι operator restricted to occurrences to the left or right of " and augmented by Tennant's rules INFT. Call the same modified system augmented by Lambert's Law as an axiom INFLL. Under the current proposal of treating ι as a binary quantifier, where 'The F is G' is formalised as ιxrF,Gs, formulas of the form ιxF " t employing the term forming ι operator, which intuitively mean 'The F is identical to t', can be rendered as ιxrF, x " ts. Treating ιxrF, x " ts and ιxF " t as notational variants, it is not difficult to show that INFT is equivalent to the fragment of INFι where the G of ιxrF,Gs is restricted to identity. Call the latter system INFιR. For clarity, I will refer to the rules for the binary quantifier ι restricted to suit INFιR by ιIR, ιE1R and ιE2R. It is now convenient to have rules for the biconditionalØ: i A Π B i B Π A Ø I : iA Ø B A Ø B A Ø E1 : B A Ø B B Ø E2 : A For perspicuity, we will mark applications of the rules for the biconditional, of Tennant's rules for ι, and of ιIR, ιE1R and ιE2R in the deductions to follow in the next paragraphs; unmarked inferences are by the more familiar rules of INF. To show that INFT is a subsystem of INFLL, we observe that, treating formulas of the form ιxF " t as atomic, ιE3T is a special case of AD, and that ιE1T and ιE2T are derivable from (LL) byØ E1. The following construction shows that ιIT is also a derived rule of INFLL: 1z " t Ξ Fxz Fxz 1 , D!z 2 looooomooooon Π z " t 1 ØI Fxz Ø z " t 2 @xpF Ø x " tq pLLq D!t ιxF " t Ø @xpF Ø x " tq ØE2 ιxF " t Hence INFT is a subsystem of INFLL. The next three paragraphs show that, if we write ιxF " t for ιxrF, x " ts, the rules ιIR, ιE1R and ιE2R of INFιR are derived rules of INFT. 1. Due to the restriction on INFιR, applications of ιIR are those cases of ιI in which Gxt is an identity. So it can be any identity in which x is replaced by t and the other term is arbitrary, i.e. any identity px " uqxt or t " u for short: Fxt t " u D!t Fxz i , D!z i loooomoooon Π z " t i ιIR ιxrF, x " ts 5 To derive the rule it suffices to change notation and write ιxF " t instead of ιxrF, x " ts, and to observe that Fxt , z " t $ F x z by Leibniz' Law and apply ιIT: D!t Fxt iz " t Fxz Fxz i , D!z i loooomoooon Π z " t i ιIT ιxF " t The premise t " u of ιIR is redundant: a suitable identity can always be provided by deriving px " tqxt , i.e. t " t, from the first premise D!t by " I n. 2. ιE1R is derivable by changing notation and applying DE with the major premise DxpFx ^ x " tq derived from ιxF " t by ιE1T, multiple applications of " In and ιET3, and DI: ιxF " t ιxF " t D!t t " t ιE1T Ft ιxF " t D!t t " t Ft^ t " t ιxF " t D!t DxpFx^ x " tq For a more elegant deduction that does not make the detour through introducing and eliminating DxpFx ^ x " tq, given a deduction Π of C from Fzx, z " t and D!z, replace z with t throughout Π, and add deductions of ιxF " t $ Ft, ιxF " t $ t " t and ιxF " t $ D!t to derive the three open premises. 3. Change of notation and two applications of ιE2T and one of Leibniz' Law derive ιE2R: ιxF " t D!t1 Fxt1 ιE2Tt " t1 ιxF " t D!t2 Fxt2 ιE2Tt " t2 t1 " t2 Thus INFιR is a subsystem of INFT. 6 Finally, we derive (LL) in the version appropriate to INFιR, i.e. with ιxA " y replaced by ιxrA, x " ys: (LL1) Lambert's Law: @ypιxrA, x " ys Ø @xpA Ø x " yqq 1. ιxrA, x " ys $ @xpA Ø x " yq ιxrA, x " ys ιxrA, x " ys 3 D!x 4 D!z 4z " y D!y 4 Axz 4z " y Axy 2 A ιE2Rx " y ιxrA, x " ys 1 Axz 1z " y Axy 2x " y A 1 ιE1R A 2 ØI A Ø x " y 3 @xpA Ø x " yq 4 ιE1R @xpA Ø x " yq 2. @xpA Ø x " yq, D!y $ ιxrA, x " ys @xpA Ø x " yq D!y Axy Ø y " y D!y y " y Axy D!y y " y D!y @xpA Ø x " yq 1 D!z Axz Ø z " y 1 Axz z " y 1 ιIR ιxrA, x " ys Now from 1 and 2 byØ I, we have D!y $ ιxrA, x " ys Ø @xpA Ø x " yq, and so by @I, $ @ypιxrA, x " ys Ø @xpA Ø x " yqq. Hence INFLL is a subsystem of INFιR. This completes the circle, and we have shown: Theorem 1 INFT, INFLL and INFιR are equivalent. 7 4 Comparison of INFι with an Intuitionist Version of a System of Lambert's As noted towards the beginning of the previous section, in the absence of a formal device or a convention for distinguishing two ways of applying negation to GpιxFq, GpιxFq is ambiguous: can either be internal or external negation. To eliminate ambiguity, Lambert introduces an abstraction operator ∆ that forms complex predicate terms ∆xB from open formulas B, and with the formation rule that if ∆xB is a predicate term and t an individual term, then ∆xB, t is a formula. Semantically, ∆xB, t is interpreted as true just in case t exists and Bt is true.4 In this section I will compare INFι to an intuitionist version of Lambert's system. Like Lambert, I will only consider unary predicates and keep the discussion fairly informal.5 In Lambert's system, ∆ is governed by a principle regarded either as an axiom or as a contextual definition: (∆t) ∆xB, t Ø pD!t^ Bxt q (t free for x in B and x not free in t) To formalise a free Russellian theory of definite descriptions, Lambert adds Lambert's Law and the following principle to negative free logic, also regarded either as an axiom or as a contextual definition: (∆ι) ∆xB, ιxA Ø DzpιxA " z^ Bxzq Lambert uses a classical negative free logic, but in this section I will consider adding (LL), (∆t) and (∆ι) to INF. Call the resulting system INFLL∆. In this system, what we may call the primary occurrences of ι terms are those to the left or right of identity and which are governed by Lambert's Law. What we may call the secondary occurrences of ι terms are those introduced on the basis of the primary ones by the contextual definition (∆ι). Lambert notes three characteristically Russellian theorems that are consequences of (LL), (∆t) and (∆ι): (R1) D!ιxA Ø Dy@xpA Ø x " yq (R2) ∆xB, ιyA Ø Dzp@ypA Ø y " zq ^ Bxzq (R3) ιxA " t Ñ Axt (t free for x in A and x not free in t) A further characteristically Russellian thesis mentioned by Morscher and Simons (Morscher and Simons, 2001, 19) is worth listing: (R4) D!ιxA Ñ ApιxAq We will show that INFLL∆ and INFι are equivalent, and then, to take a convenient opportunity to illustrate the workings of the latter system, derive formulas corresponding to (R1) to (R4) in INFι. 4For this and the following, see (Lambert, 2001, 39ff). 5Lambert provides a more general treatment of an abstraction operator in classical positive free logic, but without a description operator, in (Lambert, 1986). A more complete and precise comparison of my treatment of definite description with Lambert's is reserved for sequels to this paper on the binary quantifier ι in intuitionist positive free logic and in negative and positive classical free logic. Fitting and Mendelsohn also employ predicate abstraction as a device for distinguishing scope within modal logic (Fitting and Mendelsohn, 1998, Ch 12). 8 In the present formalisation of ι as a binary quantifier, no conventions or syntactic devices are needed for the disambiguation of complex formulas involving ι. Ambiguity is avoided by the notation for the operator itself, which incorporates the relevant scope distinction. In this sense, the current formalisation of definite descriptions is more versatile than a formalisation using a term forming operator: it does the work of both, the term forming ι operator and the abstraction operator. There is a certain redundancy in Lambert's axioms. D!t^ Bxt is equivalent to Dzpt " z^ Bxzq:6 D!t^ Bxt D!t t " t D!t^ Bxt Bxt t " t^ Bxt D!t^ Bxt D!t Dzpt " z^ Bxzq Dzpt " z^ Bxzq 1 t " z^ Bxz t " z 1 D!z D!t 1 t " z^ Bxz t " z 1 t " z^ Bxz Bxz Bxt D!t^ Bxt 1 D!t^ Bxt This means that there is a uniform treatment of the ∆ operator, irrespective of whether the term a predicate abstract is applied to is an ι term or not, and one axiom suffices to replace (∆t) and (∆ι): (∆t1) ∆xB, t Ø Dzpt " z^ Bxzq (t free for z in B and z not free in t) This works only for a Russellian theory of definite descriptions, however: an alternative theory of definite descriptions within positive free logic may be intended to provide room for the option that ∆xB, ιxA is true even though there is no unique A: such a theory may contain (∆t) but not (∆ι). Furthermore, ∆xB, t is equivalent to ∆xB, ιxpx " tq, both being equivalent to Dzpt " z ^ Bxzq. Thus there is a sense in which nothing is lost from Lambert's system if the formation rules for the abstraction operator were reformulated so as to require a predicate and an ι term to form a formula out of them. The ι symbol, being embedded within the ∆ operator, could then just as well be omitted, so that ∆ forms a formula out of two predicates, which is exactly how the ι operator works in INFι. Of course what is crucial for Lambert's system is Lambert's Law, and in his formulation of it ∆ does not occur. The present system is thus in a sense more economical than Lambert's. We can emulate Lambert's use of both, the abstraction operator and the term forming ι operator, in the present system: ∆xG, ιxF is translated as ιxrF,Gs, and where t is not an ι term, ∆xA, t is translated as ιxrt " x,As: instead of naming an object and applying a predicate to it, we pick out the object by a predicate that is true at most of it. Then what is expressed by ιxA " y in Lambert's system is 6The second deduction is constructed so as not to appeal to any rules of INF that are not also rules of the system IPF of (Kürbis, 2019, Sec 3). The first deduction can be adjusted to IPF by deducing t " t from no premises by " I. 9 expressed in INFι by ιxrA, x " ys, and what is expressed by D!ιxA is expressed by ιxrA, D!xs. A little more precisely, to show that INFLL∆ and INFι are equivalent, observe that their languages differ only in that the former has ∆ and the term forming ι, which the latter lacks, and in that the latter has the binary quantifier ι, which the former lacks. We construct a translation τ from the language of INFLL∆ to the language of INFι. Atomic sentences and those containing operators other than ∆ and ι are translated homophonically: (a) if A is atomic formula not containing any ι terms, then τpAq " A, (b) if the main operator of A is a unary operator   (i.e.   is , D or @), then τp Bq "  τpBq, (c) if   is a binary sentential operator, then τpA   Bq " τpAq   τpBq. Next, the primary occurrences of ι terms: (d.i) τpιxA " tq " ιxrτpAq, x " ts; similarly for t " ιxA (i.e. τpιxA " ιyBq " ιxrτpAq, ιyrτpBq, x " yss). For formulas containing ∆ and the secondary occurrences of ι terms, we need a distinction: (e.i) if t is not an ι term, then τp∆xB, tq " ιxrt " x, τpBqs, (e.ii) if t is an ι term ιxA, then τp∆xB, tq " ιxrτpAq, τpBqs. To construct a translation υ from the language of INFι to the language of INFLL∆, we recycle clauses (a) to (c) of τ and add only υpιxrA,Bsq " ∆xυpBq, ιxυpAq, letting the contextual definitions (∆t) and (∆ι) do the rest. Let τpΓq, υpΓq be the set of formulas in Γ translated by τ, υ. We have: Theorem 2 INFι is equivalent to INFLL∆: (a) if Γ $ A in INFι, then υpΓq $ υpAq in INFLL∆; (b) if Γ $ A in INFLL∆, then τpΓq $ τpAq in INFι. Proof. (a) It suffices to observe that the introduction and elimination rules for ι of INFι remain valid under the translation υ, due to the equivalence of ιxrF,Gswith DxpF^ @ypFxy Ñ y " xq ^ Gq and (R2). (b) It suffices to prove the translations of (LL), (∆t) and (∆ι) under τ in INFι: (LLτ) @ypιxrτpAq, x " ys Ø @xpτpAq Ø x " yqq (∆tτ) ιxrx " t, τpAqs Ø pD!t^ τpAqxt q (t free for x in τpAq and x not free in t) (∆ιτ) ιxrτpAq, τpBqs Ø DzpιxrτpAq, x " zs ^ τpBqxzq For readability I will prove these equivalences 'schematically', it being understood that the formulas A and B in the deductions to follow are translations under τ.7 Then (LLτ) is (LL1), which we proved earlier. The other two we prove next. (∆tτ) ιxrx " t,As Ø pD!t^ Axt q (t free for x in A and x not free in t) 1. ιxrx " t,As $ D!t^ Axt 7From an alternative perspective, the provability of these equivalences shows that adding (LL), (∆t) and (∆ι) to INFι does not increase its expressive power, as for each formula containing the term forming ι operator and ∆, there is a provably equivalent one containing only the binary quantifier ι. 10 ιxrt " x,As 1 D!x 1t " x D!t 1 A 1t " x Axt D!t^ Axt 1 ιE1 D!t^ Axt 2. D!t^ Axt $ ιxrx " t,As D!t^ Axt D!t t " t D!t^ Axt Axt D!t^ Axt D!t 1z " t 1 ιI ιxrx " t,As This is a correct application of ιI: Fxt is px " tq x t , i.e. t " t, and F x z is px " tqxz , i.e. z " t. D!z is discharged vacuously. 11 (∆ιτ) ιxrA,Bs Ø DzpιxrA, x " zs ^ Bxzq 1. ιxrA,Bs $ DzpιxrA, x " zs ^ Bxzq ιxrA,Bs 2 Axz 2 D!z z " z 2 D!z ιxrA,Bs 1 D!x 2 D!z 1 A 2 Axz ιE2x " z 1 ιI ιxrA, x " zs 2 Bxz ιxrA, x " zs ^ Bxz 2 D!z DzpιxrA, x " zs ^ Bxzq 2 ιE1 DzpιxrA, x " zs ^ Bxzq 2. DzpιxrA, x " zs ^ Bxzq $ ιxrA,Bs First, ιxrA, x " zs,Bxz $ ιxrA,Bs: ιxrA, x " zs 2 A 2x " z Axz Bxz 2 D!z ιxrA, x " zs 1 D!y 2 D!z 1 Axy 2 A 2x " z Axz ιE2y " z 1 ιI ιxrA,Bs 2 ιE1 ιxrA,Bs Thus ιxrA, x " zs ^ Bxz $ ιxrA,Bs, and so DzpιxrA, x " zs ^ Bxzq $ ιxrA,Bs. In this last application of DE, D!z is discharged vacuously. Notice that it would have been possible to discharge only one (or indeed none) of the D!z by ιE2, and the discharge the other (or both) by the application of DE. This completes the proof of Theorem 2. 12 Under translation τ, (R1), (R2), (R3) and (R4) become: (R1τ) ιxrτpAq, D!xs Ø Dy@xpτpAq Ø x " yq (R2τ) ιxrτpAq, τpBqs Ø Dzp@ypτpAq Ø y " zq ^ τpBqxzq (R3τ) ιxrτpAq, x " ts Ñ τpAqxt (t free for x in τpAq and x not free in t) (R4τ) ιxrτpAq, D!xs Ñ ιxrτpAq, τpAqs (R2τ) follows from the interderivability of DxpA ^ @ypAxy Ñ x " yq ^ Bq with ιxrA,Bs (see (Kürbis, 2019, 90f)). The rest are proved on the following pages, once more 'schematically' and with τ suppressed for readability. The proofs presuppose a judicious choice of variables. 13 (R1τ) ιxrA, D!xs Ø Dy@xpA Ø x " yq 1. ιxrA, D!xs $ Dy@xpA Ø x " yq ιxrA, D!xs ιxrA, D!xs 2 D!x 3 D!y 1 A 3 Axy ιE2x " y 3 Axy 1x " y A 1 ØI A Ø x " y 2 @xpA Ø x " yq 3 D!y Dy@xpA Ø x " yq 3 ιE1 Dy@xpA Ø x " yq 2. Dy@xpA Ø x " yq $ ιxrA, D!xs Dy@xpA Ø x " yq 2 @xpA Ø x " yq 2 D!y Axy Ø y " y 2 D!y y " y ØE2 Axy 2 D!xy 2 D!y 2 @xpA Ø x " yq 1 D!v Axv Ø v " y 1 Axv ØE1v " y 1 ιI ιxrA, D!xs 2 ιxrA, D!xs 14 (R3τ) ιxrA, x " ts Ñ Axt (t free for x in A and x not free in t) ιxrA, x " ts 1 A 1x " t Axt 1 ιE1 Axt We also have ιxrA, x " ts Ñ D!t (x not free in t): ιxrA, x " ts 1 D!x 1x " t D!t 1 ιE1 D!t Hence ιxrA, x " ts Ñ pD!t^Axt q, and so by (∆t τ), ιxrA, x " ts Ñ ιxrx " t,As. We do not, however, have the converse. pD!t^Axt q Ñ ιxrA, x " ts is not true. ιxrA, x " tsmeans 'The A is identical to t', and this does not follow from the existence of a t which is A, i.e. D!t^ Axt . (R4τ) ιxrA, D!xs $ ιxrA,As ιxrA, D!xs 2 Axz 2 Axz 2 D!z ιxrA, D!xs 1 D!v 2 D!z 1 Axv 2 Axz ιE2v " z 1 ιI ιxrA,As 2 ιE1 ιxrA,As 15 To close this section, a few words about @E, DI and " E. In systems where ι is a term forming operator, ι terms can be used as terms instantiating universal generalisations, as terms over which to generalise existentially, and as terms to the left or right of identity in Leibniz's Law. To establish that the current system is as versatile as a system in which this is possible, it remains to be shown that these uses of ι terms can be reconstructed in the present formalism. In other words, we need to show: (@ι) @xB, ιxrA, D!xs $ ιxrA,Bs (Dι) ιxrA,Bs, ιxrA, D!xs $ DxB (" ι) Bxt , ιxrA, x " ts $ ιxrA,Bs An inference concluding the existence of an ι term by AD is a special case of ιxrF,Gs $ ιxrF, D!xs, which holds by (R2τ), (R1τ) and general logic. I will only show that (@ι) and (" ι) hold, the proof of (Dι) being similar. (@ι) @xB, ιxrA, D!xs $ ιxrA,Bs ιxrA, D!xs 2 Axz @xB 2 D!z Bxz 2 D!z ιxrA, D!xs 1 D!y 2 D!z 1 Axy 2 Axz ιE2y " z 1 ιI ιxrA,Bs 2 ιE1 ιxrA,Bs (" ι) Bxt , ιxrA, x " ts $ ιxrA,Bs ιxrA, x " ts 2 x " t 2 A Axt B x t 2 D!x 2 x " t D!t ιxrA, x " ts 1 D!v 2x " t 2 D!x D!t 1 Axv 2x " t 2 A Axt ιE2v " t 1 ιI ιxrA,Bs 2 ιE1 ιxrA,Bs 16 5 Conclusion and Further Work The present formalism has certain advantages over the use of ι as a term forming operator. It incorporates scope distinctions within the notation, without the need for an abstraction operator or other syntactic devices or conventions. It provides a natural formalisation of a theory of definite descriptions, here developed within intuitionist negative free logic. The resulting system has desirable proof-theoretic properties, as deductions in it normalise, and it is equivalent to well known axiomatic theories of definite descriptions. Scope distinctions are of particular interest to the development of a theory of definite descriptions within modal logic. Fitting and Mendelsohn, for instance, provide a detailed account of definite descriptions within quantified modal logic (Fitting and Mendelsohn, 1998, Ch 12), which uses an abstraction operator for scope distinction. They observe that scope distinctions are already needed for formulas containing individual constants, if they are not interpreted rigidly, and so they introduce predicate abstraction well before definite descriptions. However, in their system, as in Lambert's, predicate abstraction does not appear to play any further role than marking scope distinctions. The present notation provides a perspicuous way of distinguishing the scope of modal operators that is independent of abstraction operators: It is possible that the F is G: ^ιxrF,Gs The F is possibly G: ιxrF,^Gs. The possible F is G: ιxr^F,Gs For scope distinctions with regard to non-rigidly interpreted individual constants, we can use the technique of simulating the use of a constant t by a predicate x " t introduced earlier. It would be worth comparing the approach proposed here with Fitting's and Mendelsohn's, but this must wait for another occasion. References Fitting, M. and R. L. Mendelsohn (1998). First-Order Modal Logic. Dordrecht, Boston, London: Kluwer. Indrzejczak, A. (2018a). Cut-free modal theory of definite descriptions. In G. M. G. Bezhanishvili, G. D'Agostino and T. Studer (Eds.), Advances in Modal Logic, Volume 12, pp. 359–378. London: College Publications. Indrzejczak, A. (2018b). Fregean description theory in proof-theoretical setting. 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