LogicandLogicalPhilosophy Volume21(2012),253–269 DOI:10.12775/LLP.2012.013 Jerzy Hanusek ON A NON-REFERENTIAL THEORY OF MEANING FOR SIMPLE NAMES BASED ON AJDUKIEWICZ’S THEORY OF MEANING Abstract. In 1931–1934 Kazimierz Ajdukiewicz formulated two versions of the theory of meaning (A1 and A2). Tarski showed that A2 allows syn- onymousnamestoexistwithdifferentdenotations. TarskiandAjdukiewicz foundthatthisfeaturedisparagesthetheory. TheforceofTarski’sargument rests on the assumption that none of adequate theories of meaning allow synonymous names to exist with different denotations. In the first part of thispaperwepresentanappropriatefragmentofA2andTarski’sargument. Inthesecondpartweconsideranelementaryinterpretedlanguageinwhich individualconstantsoccur,butnotfunctionalsymbols. Forsuchalanguage wedefinesemantically arelation ofsynonymityforsimplenamesandshow that it fulfills syntactical conditions formulated by Ajdukiewicz in A2 and allows synonymousnames to exist with different denotations. Keywords: Ajdukiewicz, Tarski, theory of meaning, simple names, syn- onymity,meaning directives. 1. Historical remarks In1931–1934 Kazimierz Ajdukiewicz formulatedtwoversionsof thethe- ory of meaning ([1, 2]). It isassumed that it was the first attempt to de- fine this term using the methods taken from formal logic. Ajdukiewicz’s theories of meaning have been studied by a number of researches (see e.g. [6, 7, 9, 10, 11, 13, 14]). Despite the great precision of the lan- guage of Kazimierz Ajdukiewicz, his texts leave considerable room for interpretation. This is how this dissertation should be considered. Received March 24,2011;Revised June4,2012 ©NicolausCopernicusUniversity(Toruń)2012 ISSN:1425-3305 254 Jerzy Hanusek The Ajdukiewicz’s idea was to bind an intuitive concept of syn- onymitywithasyntactic-pragmaticrelationofmutualinterchangeability of expressions over a set of so-called meaning directives of language. In these two theories, the definition of the mutual interchangeability of expressions was formulated differently. In [9] I tried to compare both theories exactly for the sake of Ajdukiewicz’s modification of the def- inition. Let A1 represent the theory from 1931 and A2the theory from 1934. It is a well-known fact that the theory from 1934 allows synonymous names to exist with different denotations. It was Alfred Tarski who noticed this fact first and then conveyed his observation to Ajdukiewicz during their conversation. The fact thatthetheory ofmeaning allowssynonymous names toex- ist with different denotations was considered by Ajdukiewicz and Tarski to be an evidence that the theory was incorrect. In fact, the problem of interpreting Tarski’s argument in relation to the theory A2 is more complicated. I discussed it in [9]. I also showed in this work that con- trarytopopularopinion,thetheoryA1alsoallowssynonymousnamesto existwithdifferentdenotations.1 Iexpressedtheopinionbackedupwith arguments that in certain very special cases theories of meaning should allow such names to exist with different denotations. I also framed an openproblemofsemanticdefiningarelationofsynonymityofnames, for which Ajdukiewicz’s theory would provide a syntactic sine qua non and which could be fundamental for a non-referential theory of meaning of names. The following dissertation is a partial solution of this problem. 2. About meaning directives We are going to deliberate here on very special cases of applying the A2 theory. However,wewillpresentitsgeneralprinciples,inordertosituate thesespecial casesevenbetter. Oneofthefundamental concepts ofboth Kazimierz Ajdukiewicz’s theories of meaning is a concept of a meaning directive. LetLbeanyinterpretedlanguage,formalorethnic. Apossible meaning directive of L is an ordered pair hW ,αi, where α is a sentence α inthelanguageL,andW isapossiblecondition(asetofconditions)for α acceptance of α by users of L, regardless of what is the character of the 1 There are two definitions of synonymity in [1], on pp. 132 and 134. In the second one Ajdukiewicz uses the notion of an essential directive and I assume that A1 isbasedonthis definition. On a non-referential theory of meaning ... 255 condition and how it is defined. Let WL represent the set of all possible conditions for acceptance of sentences in the language L, SL – the set of all sentences in the language L and ExpL – the set of all expressions of L. The concept of a possible meaning directive for L becomes a formal concept only when both L and WL are formally defined. Let hW ,αi be a possible meaning directive of L. We say that α hW ,αi is a meaning directive for L, if the following condition is met: α if W is satisfied, a member of community using the language L cannot α reject the sentence α, he has to accept it. Ajdukiewicz understands an act of acceptance in a purely pragmatic way and it does not have to be connected with a belief or knowledge of a language user. It is an expression of community’s language habit and it show how the commu- nity uses the language. Rejecting the sentence α, when the condition W is fulfilled and when the pair hW ,αi is the meaning directive of the α α language L, means that a given person does not belong to the commuity usingthelanguageLandassigningthesamemeaningstoitsexpressions. Ajdukiewicz assumes that, for any expression of L, there are meaning directives for L in which this expression occurs. The condition W , which is the component of the meaning directive α hW ,αi in a special case can consist in accepting some sentences (it can α be the empty set). We will say then, that it has a linguistic character and we will identify it with the set of those sentences. Depending on the type of condition W , Ajdukiewicz distinguishes α threekindsofmeaningdirectives. Axiomaticdirectivesarecharacterized by an empty set of conditions of accepting sentences. We identify such directives with the sentences. In deductive directives the condition of sentences’ acceptance is to accept other sentences; in empirical direc- tives it is a user’s appropriate extralinguistic experience (e.g. sensory, metaphysical). In the last case, weidentify W with the typeof that ex- α perience. The above characteristic of the meaning directives differs a bit fromtheoriginalperspectiveofKazimierzAjdukiewicz. However,Ithink that it properly conveys the nature of his conception; the differences are technical, and the omitted details are of secondary importance.2 Ajdukiewicz assumes that expressions of any interpreted language L, used by a language community for communicative purposes, have meanings that allow users to recognize them and make users understand eachother. Foreverysuchlanguagethereexistsarelationofsynonymity 2 Idiscussedthismatter inmoredetail in[9, pp.128–129]. 256 Jerzy Hanusek betweenexpressions,whichisintuitivelyrecognizedbyusersandputinto int language practice. Let ∼ represent this relation when it is known which language is meant. In his theories of meaning, Ajdukiewicz assumes that the fact that the expressions of the language L have meanings is equivalent to the existence oftheset ofmeaning directivesfor L, and that thesynonymity relation for L depends on the form of that set. Therefore, by referring to meaning directives, we can formulate conditions that describe the int intuitive relation ∼ of synonymity. 3. Basic definitions LetLbeaninterpretedlanguageandD asetofmeaningdirectivesforL. Thesyntacticdefinitionoftherelationofamutualinterchangeability ofexpressionsofLoverthesetD ofmeaningdirectivesforthatlanguage is one of the crucial definitions of the theory A2 (and also A1). Aj- dukiewicz connected this relation with the intuitive synonymity relation int ∼ for L. In his definition he used syntactic operations on expressions, which can be defined in the following way: Definition 1. Let L be any language and α,β,λ be any members of ExpL. We define the operation α (cid:8)β :ExpL 7→ ExpL by putting that λα(cid:8)β is the expression resulting from λ by replacing all occurrences of α with β (and vice versa). It turns out that the transformations defined in Definition 1 have many merits in comparison to analogical transformations defined in [1]. However, they have a serious defect as well, which was not noticed by Kazimierz Ajdukiewicz. They are not always possible to execute. If some expressions α and β are not disjoint, that is to say, α is a part of β or vice versa, then for some expressions γ the operation of the mutual interchange is not possible to execute.3 So Definition 1 defines only partial operations in such cases. Nothing stands in the way of using Definition 1, provided that the parameters of operations α (cid:8) β are going through the set of simple expressions of the given language. 3 Thatwasnoticed byprofessorW. Buszkowskiin[6]. On a non-referential theory of meaning ... 257 We always obtain, in such a case, the total operations. If we want to apply Definition 1 without limitation, in the same way as Ajdukiewicz did it, we have to modify the definition, so that it would consider the fact, that the transformation α(cid:8)β is not always possible. Otherwise, the next definition is not correct. In this article we will only consider simple names, so we do not have to worry about thisproblem. It follows from Definition 1 that if α ∈/ λ and β ∈/ λ, then λα(cid:8)β =λ. The notation α ∈/ λ, in the context of expressions, stands for α does not occur in λ. The syntactic operations defined above allow us to define analog- ical operations on conditions W and, in a consequence, on meaning α directives. We assume that if W is a component of an axiomatic or α empirical meaning directive, then Wγ(cid:8)β = W . Let’s now assume that α α W ={δ ,...,δ }, where δ ,...,δ are sentences. Then we set: α 1 n 1 n Wγ(cid:8)β ={δγ(cid:8)β,...,δγ(cid:8)β}. α 1 n Let ∆ be a meaning directive of the form hW ,αi. We set that, α ∆γ(cid:8)β =hWγ(cid:8)β,αγ(cid:8)βi. α Now we can give a definition of a mutual interchangeability of ex- pressions over a set D of meaning directives of a language L. Definition 2. Let L be any language, D a set of meaning directives of L, α,β ∈ ExpL. We say that α and β are mutual interchangeable over D, written α ≈D β, if and only if for any directive ∆∈D, ∆α(cid:8)β ∈D. In D other words: α ≈β if and only if the set of directives D is closed under the operation α (cid:8)β. We noticed, that Tarski showed that the theory A2 allows synony- mous names toexistwithdifferent denotations. Now, wecan present his reasoning. Example 1 (Tarski). Let L be an elementary language, in which the individual constants a and b are the only specific symbols. We assume that a model M is an interpretation of the language L and its universe M M M M consists of two elements a and b , for which we have a 6= b . Let a set of meaning directives D be an elementary theory generated by two sentences : a 6= b and b 6= a.4 As all the tautologies expressible in the 4 Tarski in his example assumed that the set D of meaning directives consists 258 Jerzy Hanusek language L belong to the set D, we can assume that the meaning of the symbols of equality and negation which occur in these two sentences, is themeaningwhichisassigned toclassical logic. Itishardtobelievethat a user of the language L, accepting the set D as the set of the sentences defining the meanings of that language’s symbols, could use the sym- bols of equality and negation in different meanings from the classical. Therefore, the names a and b have different denotations and, moreover, regardless of that fact, the sentences a 6= b and b 6= a should be inter- preted as the sentences saying that a and b have different denotations.5 However, if we take any sentence ∆ ∈ D, it is easy to notice that the sentence ∆a(cid:8)b belongs toD as well. And so, according toDefinition 2.2, D the result is that a ≈b. ⊣ The interpretation of Tarski’s argument is clear, provided that in the theory A2 Ajdukiewicz equated the relation of synonymity with the D relation≈. However, itisnot true. Regardless ofhowTarski’sargument concerns the theory A2, in my view, he attracts attention to an inter- esting problem: whether there are any permissible cases in an accurate theory of meaning when synonymous names have different denotations. It seems that Tarski, as well as Ajdukiewicz, rejected such possibility.6 D If we limit the field of ≈ to simple expressions, it will turn out that it is an equivalence relation. Let δ be any expression of L, α,β,γ any simple expressions of that language. Then ((δα(cid:8)β)β(cid:8)γ)α(cid:8)β =δα(cid:8)γ. D Checkingwhether therelation≈isreflexiveandsymmetricalisnot a D D problem. We assume that α ≈β and β ≈γ. Let ∆ ∈D. Then, by Defi- of all logical tautologies expressible in the language L and of the sentences a 6= b and b6= a. We assumed a slightly different solution. However, it does not influence the interpretation of the example discussed. The above example can be also easily supplemented by a set of logical deductive directives, because the interchange of all namesaandb(andviceversa)transformsanysubstitutioninstanceofmodusponens into a substitution instance of modus ponens and the similar case concerns the rule ofgeneralization. 5 If the set D of directives consisted only of the sentences a 6= b and b 6= a, nothing would entitle us to claim that these sentences express the fact of different denotations of names a and b. In that case, these sentences would express the same ase.g. thesentences R(a,b)andR(b,a). 6 Ajdukiewicz statedthatclearly in [4]. On a non-referential theory of meaning ... 259 nition2,∆α(cid:8)β,(∆α(cid:8)β)β(cid:8)γ alsobelongtoD. Andso((∆α(cid:8)β)β(cid:8)γ)α(cid:8)β = ∆α(cid:8)γ. Hence α ≈D γ.7 The way in which a relation of expression’s interchangeability iscon- nected with a synonymity relation was presented by Ajdukiewicz in the form of the following thesis. Thesis (Ajdukiewicz) Let L be any interpreted language, D a set of meaning directives of that language, α and β its any expressions. If int D α ∼ β, then α ≈β. 4. About theories of meaning of names As a matter of fact, the theories A2 and A1 are universal and refer to any expressions of any languageethnic or interpreted formal lan- guage.8 Therearealsonolimitingconditionsimposedonsetsofmeaning directives. As far as kinds of the languages in question, a shape of the meaning directives’ set and the range of the expressions are concerned, we will further assume far-reaching limitations. As we have mentioned, Ajdukiewicz in his theory does not impose any essential conditions on a language. A set of meaning directives depends on linguistic habits of language users, which are not limited by any conditions. Therefore we canassertthatour findingsarenotexactlymodifyingAjdukiewicz’sthe- ory of meaning but rather focus attention on some special applications. However, it does not mean that Ajdukiewicz’s theory of meaning can be reduced to these particular cases. We assume the following: 1. We limittheclass of considered languages tointerpreted, elementary languages based on classical logic. In order to specify our delibera- tions, we choose the specific system of logic. Let it be the system of elementary logic withthe identitydefined in [12]. In thissystem, the modus ponens and the rule of generalization are rules of inference (we have only the general quantifier in the language of this logic). 7 Iamcopyinghere thereasoningof W. Buszkowskifrom [6]. 8 Inthesecondpartof[2]Ajdukiewiczrestrictedconsideredlanguagestocoherent andclosedlanguages. However,thisassumptionwasconnectedwiththedefinitionof atranslation,thuswithponderingthesynonymyofexpressionsbelongingtodifferent languages. If we limitourselves to ponderingthe synonymyof expressionsbelonging to onelanguage,thatassumptionisnotneeded. 260 Jerzy Hanusek 2. We limit members of sets of meaning directives to axiomatic direc- tives. Furthermore, we assume that a set of axiomatic directives is an elementary theory, true in a distinguished model, being an inter- pretation of a given language. 3. We limit the range of considered expressions to simple names, i.e. individual constants of elementary languages. Thereby, we assume that not all true sentences of a given language L have to be accepted by users of that language on the strength of the meaning of expressions (the knowledge does not have to be analytical), and that users of L have a full ability to deduce. Point 2 requires a justification. Let L be any interpreted elementary language and D a set of meaning directives for L. The assumption that there are no empirical directives in D seems natural. If we assume that the set of deductive directives for L consists of all substitution instances of modus ponens and the rule of generalization, it will turn out that it D does not influence the form of the relation ≈. Let ∆ ∈ D be any substitution instance of modus ponens, α and β any disjoint expressions of L belonging to the same syntactic category. Then ∆α(cid:8)β is also a substitution instance of modus ponens, regardless of whether the expressions α and β are terms, formulas or variables, provided that the operation α(cid:8)β is possible to execute. In such a case for any formula γ and δ we have (γ → δ)α(cid:8)β =γα(cid:8)β → δα(cid:8)β. Hence ∆α(cid:8)β ∈ D. Therefore, these directives do not influence the re- D lation ≈ on the set of simple expressions of these categories. They can influence this relation on the set of logical symbols. However, it is easy to show that even only the axiomatic directives are the reason for which therelationofmutualinterchangeabilitycoincides withtheidentityrela- tion on this set. It would be enough to consider some tautologies in the D language L, inorder tocheckthat≈isalwaysanidentityrelationonthe set of logical symbols. For example: ifα is any axiomatic directive, then ¬α →¬α and α∨¬α are alsoaxiomatic directives, but (¬α →¬α)→(cid:8)∧, (¬α→ ¬α)→(cid:8)∨ and(α∨¬α)∧(cid:8)∨ arenot. Thesimilarsituationisforthe rule of generalization. Therefore, provided the assumptions, deductive D directives do not influence the relation ≈ on the set of logical symbols too, and can be omitted. On a non-referential theory of meaning ... 261 The fact that deductive directives do not influence the relation of mutual interchangeability for non-logical expressions is the advantage of the theory A2. I suppose it was one of the causes for changing the definition by Ajdukiewicz. Thus, if some meaning directives cause that onthesetofthelogicalsymbolstherelationofmutualinterchangeability coincides with the identity relation, then all the remaining deductive directives can be omitted, without changing the form of the relation on the set of all expressions. Therefore we can assume that sets of meaning directives consist only of axiomatic directives. The sentences which are elements of sets of axiomatic meaning directives we shall simply call the meaning directives. Let us sum up our settlements. We consider interpreted elementary languages. Everysetofmeaning directivesconsistsoftruesentences and is closed under the logical consequence. The topic of our interest is the possibility of formal defining for any interpreted, elementary language L a semantic relation (on the set of simplenamesofL),whichcouldbeidentifiedwithintuitivelyunderstood int relation ∼ occurring in Ajdukiewicz’s Thesis. However, that is not all. We would like the definition of a synonymity relation to be the base of the non-referential theory of meaning for simple names, i.e. the theory allowingnamestoexistwithdifferentmeaningsandthesamedenotation as well as names with different denotations and the same meaning. We shall define the above postulates in more detail for a bit more general case, inwhichwetakeintoaccount anyelementary language and all constant terms, not only simple names. Let TrL denote the algebra of constant terms of L. We define it in standard manner. TrL has as its universe the set TrL of all constant terms of L. The signature of the algebra consists of all individual constants and all function symbols of L. They are interpreted as follows: if c is an individual constant, then TrL c = c; if F is an n-argument function symbol and t ,...,t are any 1 n elements of the universe, then TrL F (t ,...,t )=F(t ,...,t ). 1 n 1 n Let us denote the family of all consistent theories in L by Th(L). Any consistent theory in L can serve as the set of axiomatic directives for L, provided that the interpretation of L is a model of that theory. In every such case, our desired theory of meaning should allow to define 262 Jerzy Hanusek with semantic means the relation R⊆TrL×TrL×Th(L) satisfying for any T ∈Th(L) the following conditions: 1. ∼T ={ht ,t i :ht ,t ,Ti∈R} is a congruence relation on TrL, 1 2 1 2 T T 2. if t ∼t , then t ≈t . 1 2 1 2 It follows from the above conditions that the synonymity relation T ∼ over the set T of meaning directives determined on the set of con- stant terms of L should be a congruence on TrL and it should meet Ajdukiewicz’s Thesis. Certainly, when there are no function symbols in L, every equivalence relation on the universe of TrL is a congruence. If M is an interpretation of L, then there exists a distinct congru- ence relation on TrL, connected with that interpretation. This is the M congruence relation ≍ defined as follows : M M M t ≍ t if and only if t =t . 1 2 1 2 Let us denote the congruence lattice of TrL by Con(TrL). We can divide theories of meaning of names into three kinds, depending on an area of the universe of Con(TrL) in which synonymity relations deter- mined by a given theory, relative to various sets T ⊆Th(M) of meaning directives, can occur. 1. Thetheoriesinwhichsynonymityrelationsalwayscomeoutbelow M the relation ≍. We will call them connotative theories. 2. Thetheoriesinwhichsynonymityrelationsalwayscomeoutabove M the relation ≍. We do not have a good name for these theories, besides, they do not occur in practice. 3. The theories in which synonymity relations can occur incompara- M ble with relation ≍. We will call them non-referential theories.9 5. Languages, in which only simple names occur Inthischapterwewillconsiderelementarylanguagesinwhichindividual constantsoccur, butnotfunctionalsymbols. Wesaythatsuchlanguages are functionless. 9 The term was borrowed from [11]. Of course non-referential theories are not extensional,butthequestionwhetherameaningtheoryhavetobeextensionalseems to beopen. I discussedthis problemin [9].