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Semantic presuppositions in logicalsyntaxYaroslav Kokhan aa Institute of Philosophy, National Academy of Sciences ofUkraine, Triokhsviatytelska str. 4, 01001, Kyiv, UkrainePublished online: 26 Jun 2012.

To cite this article: Yaroslav Kokhan (2012) Semantic presuppositions in logical syntax, Journal ofApplied Non-Classical Logics, 22:1-2, 29-41, DOI: 10.1080/11663081.2012.682437

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Semantic presuppositions in logical syntax

Yaroslav Kokhan*

Institute of Philosophy, National Academy of Sciences of Ukraine, Triokhsviatytelska str. 4, 01001, Kyiv,Ukraine

There are two implicit semantic postulates underlying modern predicate logic. Hence predicatelogic is not semantically neutral. The author proposes to take semantically neutral languages,which have no predicate categorial structure but replace the notion of predicate with generalnotion of function. Some function calculi for different semantics are demonstrated.

Keywords: the standard semantics; non-standard semantics; predicate logic; function logic;choice function

The point of view on logical syntax shared by many logicians consists in that logical syntaxcould be entirely separated from semantics so that it is possible to build and consider anysyntactical objects – formulas, inferences, calculi, theories – without referring to semantics.The aim of this article is to refute the standpoint described.

1. Presuppositions and the standard semantics

As a matter of fact, falsity of the view described is obvious. Indeed, all contemporary formallanguages contain individual designators – constants and variables – i.e. designators for par-ticular individuals. These symbols are introduced immediately as linguistic expressions, whichhave one and only one semantical value-denotatum. Thus, introducing them in syntax refer-ring to semantics is already presupposed. At the present moment, the author has revealed twosemantic postulates implicitly accepted in modern logical syntax. Here they are:

(I) All proper names are single-valued, i.e. each of them could have only one semanti-cal value-interpretant (no more and no less).

(II) Interpretants of linguistic expressions are denotata of these expressions, not theirsenses.

Postulate (II) could be formulated another way: interpretation relation or relation betweenexpression t and its interpretant s is the semantic relation of denoting (t denotes an object s),but not the relation of expressing (t expresses a sense s).

Important consequences follow from these two postulates. Particularly, from (I) follows

(III) Equality is meaningful only for single-valued names.

*Email: yarkaen@gmail.com

Journal of Applied Non-Classical LogicsVol. 22, Nos. 1–2, January–June 2012, 29–41

ISSN 1166-3081 print/ISSN 1958-5780 online� 2012 Taylor & Francishttp://dx.doi.org/10.1080/11663081.2012.682437http://www.tandfonline.com

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Indeed, contraposition to (I) says that there are no non-single valued names of objects. Butwhere equality asserts that two names of objects have the same denotation, this exactly meansthat non-single valued expressions cannot appear in equality expression.

The consequence of (II) is no less principal:

(IV) Existence is a quantifier, not a predicate.

Indeed, as Frege shows in ‘Dialogue with Pünjer’, to be and to exist for denotata is the same.So when we reject a universal proposition ‘all x have property F(x)’, thereby we say thatthere is an individual without this property or, what is the same, that such an individualexists, i.e. we assert that there is (= exists) x without property F(x).

Semantics based on the postulates (I) and (II), will be called the standard semanticsbelow. Thereby, rejection of one or both of these postulates entails different non-standardsemantics.

2. Restrictions of the standard semantics

Although the standard semantics is implicitly accepted in all contemporary logic, its applica-tion area is much more narrow and does not cover even classical mathematical logic. It isevident at least concerning its first postulate. Indeed, this postulate denies the existence ofnon-single valued names for objects, meanwhile such names do exist and regularly appear inlogic as functional terms and descriptions. For example, if a is a proper name and f(x) is apartial function expression, then f(a) may well be an empty name; on the other hand, all g-terms and e-terms are non-single valued by definition. Implicit acceptance of the postulate (I)makes operation with such expressions tangibly more difficult. So, for functional terms, ifpartial functions are admitted, we are forced to introduce in addition to ‘ordinary’ equalityso-called conditional equality for not more than single-valued names (this does not covermany-valued names), or to introduce in metalanguage an additional condition, which equatesall empty names, which is not always acceptable.

Let us consider the next example. Suppose we are given two models of the same signa-ture that differ only in one aspect: the domain of the first model consists of residents of somecity in the year k, when the domain of another model consists of residents of that very city inthe year n with respect to the same calendar, and n > k. Suppose then the signature containsthe names not only for alive people, but also for the dead, and individuals a and b belong tothe first model, but in the second model both names ‘a’ and ‘b’ are empty, since individualsa and b have died in the year m, where k\m\n: In this case, in the first model we’ll have atrue sentence ‘a ≠ b’ and a false sentence ‘a = b’. But trying to define truth values for bothsentences in the second model we could come across substantial difficulties. Indeed, one andonly one of them should be true, as follows from the notions of equality and negation. Buthow could a sentence, which includes an empty name, be true? Evidently, this question goesbeyond the limits of the standard semantics, which is why there has been no satisfactory solu-tion until now. Mostly it is stated that the sentences which include empty names are false ormeaningless. Both of these answers, standard for logicians, are clearly inadequate regardingour model example: if two individuals die, any statement about them does not become mean-ingless by this reason, particularly, the statements whether they were different people or thesame person stay meaningful. Therefore, the sentences considered cannot be meaningless. Butthey both also cannot be false in the second model, because they contradict each other. So,we have to accept such semantics that would allow us to determine which of these sentencesis true. There are at least two ways to resolve the empty names problem suggested in logic.

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The well-known method by Gottlob Frege, according to which, all empty names in a givenformal language are given common denotation, being an arbitrary individual. This methodclearly fails with respect to our example: if a and b were different people during their life,they cannot become the same person after their death; so one should not assign one value toall empty names. Another method, proposed by R. Martin, seems not to be widely discussed.It consists of introducing for every model a so-called ‘empty individual’ singled out amongany individuals, like a null among numbers; the empty individual should be assigned toempty names as their denotatum. However, this method does not work with our example forthe same reasons as in the previous case (different people after their death cannot becomeequal and became some ‘null person’).

The obvious fallacy of both methods does not depend on whether there is equality in ourobject language – it is enough for it to have a negation. Indeed, it is always possible to build amodel like that in our example, in the sense it will have predicates determining designated indi-viduals a and b. Suppose predicates F(x) and G(x, y) belong to the signature of both models, sothat these sentences: F(a), :FðbÞ, G(a, b) and :Gðb; aÞ are true in the first model. In this casetheir truth values should not change in the second model as well as truth values of negations ofall these sentences. For example, F(x) can mean ‘to have the daughter called Ada’, and G(x, y) –‘x is older then y’; it is clear, that these facts do not depend on this, whether they are stated aboutdead or alive people, so G(a, b) should be true and its negation :Gða; bÞ should be false, regard-less of whether names ‘a’ and ‘b’ have values-interpretants.

As for the descriptions as a kind of names the situation is even worse, because there is nogeneral method of operating with them. R. Carnap in ‘Meaning and necessity’ describes five(though says about three) methods to interpret definite descriptions in the case when theyprove to be empty. A common feature of all these well-known methods is that they do notfollow from any general theory, but are ad hoc propositions. Carnap himself frankly recognis-es the last fact, stating that no one of these methods is a true or a false theory but they arejust more or less convenient instruments. Let us remind ourselves of all them.

Frege by his first method (he proposed two) assigns the special class of individuals to everynon-single valued description as its denotatum, namely, the class of individuals such that thepredicate being the matrix of description holds for these individuals. Thus, it is proposed to con-sider the class of all kings of France in 1905 (an empty class in this case) as the value of thedescription ‘the king of France in 1905’. Formally, this method is invulnerable, but in the caseof empirical descriptions it is out of synch with intuition. For example, when at the funeral feastone says that ‘the departed man (or woman) was a merciful person’, it means one and only oneindividual (which does not exist now, so the term ‘the departed’ is an empty description) but notthe class of all departed men over the Earth to the moment of that funeral feast.

Another of Frege’s method was described above for the general case of proper names: toassign the same denotatum (it does not matter, what) to all empty descriptions. This methodwe have already discussed.

Martin’s method, which could be taken as a modification of the second of Frege’s meth-ods, also has been discussed above.

Russell’s method provides any formula that includes a definite description that alsoimplicitly includes a statement about the single-valuedness of that description, i.e. it lookslike a conjunction of the formula itself and such statement. This method has two disadvan-tages: firstly, descriptions cease to be a kind of proper name in the sense that the laws ofquantification, particularly the axiom

8xFðxÞ ! FðaÞ; ð1Þ

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do not hold for them (compare with Section 4); secondly, negation of every formula with adescription can be interpreted in two different ways, where in the general case it is impossibleto choose the right one.

Gilbert-Bernaise’s method allows only those descriptions for which their single-valuednessis proven. But in this case the formation rules lose their definiteness: we do not knowwhether any sentence with description is well formed until we prove the single-valuedness ofthat description.

As we can see, all known methods of operating with empty names do not work in thesphere of empirical descriptions. Similarly there are no plausible methods for many-valuednames. There is a wide class of ambiguous names, not expressed in formal languages neitherby functional terms nor by descriptive terms in their modern form, but they are widespread innatural languages; these are proper names of people, geographical and political place names.Really, there has been more than one person on this planet, named ‘Karl Marx’, there is morethan one inhabited locality named ‘Odessa’, at least two rivers named ‘Angara’, etc. If suchnames were single-valued, it could be possible to introduce for every one of them the descrip-tion ixðx ¼ aÞ, where a is a required name.

But what to do with many-valued (and also with empty) names? Formally, it is possibleto hold, that ‘Karl Marx’ and ‘Angara’ are names of properties and by this reason considerthe description has a form exFðxÞ, but it is counterintuitive. No sane person would agree that‘Karl Marx’ is not a name of a person but a name of some property of this person. It shouldbe namely the name of an individual. But such names are forbidden in the standard seman-tics, and, as we have seen above, all the attempts to overcome the bounds of this semanticsand to create a theory for ambiguous names have not been successful.

3. Non-standard semantics according to the first postulate

As we ascertained above on our model example, the standard semantics sometimes is inade-quate and should be replaced by some other semantics. So it is important to conceive, howmany alternatives for the standard semantics there are and what they are. Such alternativesare formed by rejection of one or both postulates of the standard semantics.

Firstly, let us reject the first postulate. In this case we have no limitation on the numberof semantic values for proper names, so all the names should be classified somehow. It is nat-ural to classify names in three groups: empty (with no interpretants), single-valued (with onlyone interpretant) and many-valued (with more than one interpretant). It is easy to classify allthe semantics according to this triple division, in the basis of what kinds of noted types ofnames they allow. Generally, there are seven different types of semantics as shown in Table 1.Case 6 here is the class of semantics that are standard according to the first postulate (thevery standard semantics is among them). The classes represented are unequal: thus, semanticsof class 4 are obviously very specific (they are useful perhaps only for depicting the myths,tales and literary characters), whereas semantics of class 1 seems to be the most interesting,as they and only they provide a fully valid operation with descriptions.

Table 1. Types of semantics.

1 2 3 4 5 6 7

Empty names + + + + � � �Single-valued names + + � � + + �Many-valued names + � + � + � +

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Two fundamental questions arise here: what formal languages and what calculi meet therequirements of different classes of semantics from Table 1? We have already seen thatmodern logic has no satisfactory theory to describe semantics of all the classes except forclass 6. Now we are going to show that this is not a historical contingency but a necessaryconsequence of the construction of modern logical systems.

Evidently, from the postulate (I) of the standard semantics it follows that

(V) Proper names are non-empty;

This consequence can be expressed by the formula:

FðaÞ ! 9xFðxÞ: ð2Þ

Indeed, it is enough to take a predicate ‘to have the name c’ as F(x). But formula (2) easilyfollows from (1), which is an axiom in all the calculi with quantifiers (for unknown reasons,textbook authors often ignore this consequence taking (2) as an additional axiom, althoughthe required proof was given by Frege in Begriffsschrift). This means that to these days thelaws of logic were chosen in a way that proper names remain non-empty, which is a neces-sary condition for holding postulate (I). And this means that modern systems of logic are notable to describe semantics of the classes 1–4 from Table 1). As for the description of many-valued names (semantics of classes 1, 3, 5, 7), in modern formal languages there are onlyindefinite descriptions, but operating with them meets the non-emptiness problem as wasshown above. Hence, we need logical systems without axiom (1) to describe non-standardsemantics. But how can it be substituted if that axiom describes the properties of quantifiers?The author does not know a formulation that can be substituted for axiom (1) without violat-ing postulate (II) of the standard semantics. It looks like such a formulation does not exist atall. But if so, there exist some ‘forbidden zones’ in modern logic within the limits of whichwe could not describe anything. This is a rather unpleasant consequence, since it reveals non-universality of modern logical description tools. The reason for this non-universality is thesemantical loading: contemporary logical languages and calculi are not semantically neutralbecause they are constructed holding over the semantical postulates (I) and (II) and as a resultthey cannot describe all semantics.

As a matter of fact, there is a universal method to develop logical languages and calculithat are able to describe semantics of any class. However this method stipulates the changesof the categorical apparatus of logic and building formal languages on principles quitedifferent from those commonly used.

4. Function languages

A categorical system of contemporary logic, on the basis of which formal languages are for-mulated, is grounded on division of all the non-logical objects of investigation into objects(individuals) and predicates. Respectively, in formal languages there appear notations – asbasic – for these two types of objects: individual expressions i.e. terms are divided into twogroups – individual constants (proper names, individual symbols) and individual variables;and predicate expressions are divided into predicate constants and predicate variables. Some-times the notion of term needs to be extended if expressions for functions are introduced intolanguage: then functional expressions together with terms constitute a quasiterm class. Notethat function is a more general category than predicate: the predicates are only a kind offunction, namely a kind of truth-value or propositional function (logical operations also

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belong to both). Nevertheless, function is not a basic logical category, unlike predicate, andmost contemporary formal languages can be called the predicate languages.

The bounds of this article do not allow one to explain completely the connectionsbetween described categories and semantics, so we represent only the conclusion (which isclarified below): in order to constitute semantically neutral languages, it is necessary to refusepredicate as a basic logical category. Predicates should be rejected and replaced with func-tions. It has not been done until this moment, because the modern notion of function is toonarrow with respect to expressive power of formal languages. For every n-ary functionf ðnÞðx1; . . . ; xnÞ there is the n + 1-ary predicate Fðnþ1Þðx0; x1; . . . ; xnÞ such that for any a0, a1,..., an from any model an equation

a0 ¼ f ðnÞða1; . . . ; anÞ ð3Þ

implies a proposition

Fðnþ1Þða0; a1; . . . ; anÞ; ð4Þ

but not for every n+1-ary predicate is there the n-ary function such that (4) implies (3). Thereason for this is quite simple: for any given set a1; . . . ; an of arguments of function f therecan be only one value a0 of this function, meanwhile an arbitrary predicate F can not be sin-gle-valued for any of its arguments. The situation that appears can be called the predicate-function paradox: whether predicates are just a kind of function, the expressive abilities ofpredicate atoms appear more rich than expressive tools of propositional atoms, constituted byfunctions.

However, we could extend the notion of function so that function expressive abilitiesexceed expressive abilities of predicates and the predicate-function paradox would disappear.Functions, in the modern sense of the term, are maps. To eliminate the predicate-function par-adox, they should be reinterpreted as partial multimaps, i.e. functions that (i) can have anarbitrary number of values, including a different number of values for different sets of argu-ments, and (ii) can have no arguments at all. Since these functions are ambiguous as func-tions, the relation between function value and the function itself with given arguments (ifthere are any) cannot be an equality. However, the required relation is somewhat a generalisa-tion of an equality relation, because it differs from the last only by that it associates an univo-cally given object, which is the function value, with, generally speaking, an equivocallygiven object, which is the function with given (fixed) arguments (if there are any for thisfunction), because it assigns its every value the same way. On this ground, let us mark therequired relation with the sign of approximate equality ‘�’, because it has no established ter-minological use in modern logic. This relation � we’ll call the representation proceedingfrom the fact that an ambiguous function value is not generated but only represented by it.

After the generalisation mentioned for the notion of function, we immediately notice thatnow for any n + 1-ary predicate Fðnþ1Þðx0; x1; . . . ; xnÞ there can be found the n-ary functionf ðnÞðx1; . . . ; xnÞ such that for any a0; a1; . . . ; an from any given model, formula (4) implies

a0 � f ðnÞða1; . . . ; anÞ: ð5Þ

Particularly, in any model for every 1-ary predicate Fð1ÞðxÞ there is the 0-ary function f (0)

such that for any a a predicate atom F (1)(a) implies a function atom a � f ð0Þ. And this meansthat using functions in this new extended sense makes it possible to express everything for

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whatever is possible to express with predicates. Combining formulas of type (5) with thepropositional and quantification technique, we receive systems that are highly competitivewith familiar predicate systems. But now they are function systems.

Introduction of the representation relation, as a basic category of logic among the categoryof individual and generalised category of function (predicates we have rejected), changes radi-cally the form of propositional atoms of formal languages: this time there will be formulasnot of the kind (4), but formulas

s � t; ð6Þ

which will be called representation formulas and read from left to right as ‘s is a value (oneof the values, if there are any at all) of a function t’ or ‘s is a t-individual (an individual ofkind t)’. We call the formal languages with atoms of form (6) the function languages (to dis-tinguish them from modern predicate languages). In the formation of the rules for the func-tion languages it should be noted that an atomic formula is a line ‘s � t’, when s is a termand t is a quasiterm.

Equality is a partial case of representation (formula (6) expresses equality when t is aterm), so it is defined in function languages – unlike predicate languages. But we are forcedto define equality and inequality separately. Let

s ¼ t�Dfs � t ^ ðr � t ! ðu � t ! r � uÞÞ; ð7Þ

s–t�Df:ðs � tÞ ^ ðr � t ! ðu � t ! r � uÞÞ: ð8Þ

Function languages and calculi unlike predicate ones are semantically neutral. So it is pos-sible to develop calculi in function languages for any semantics. Let us start with the standardsemantics. We will use first-order predicate calculus with equality P=F1 as exemplar (symbol-ism is explained in the next section); we assume the P=F1 contains propositional calculus P,has two lists of variables (free and bound variables differ), axioms (not schemata) and rulesof substitution. Now we can build calculus, which is an exact clone of P=F1 and has the stan-dard semantics, so we will denote it by ISF

1. Alphabets of both calculi differ only by predi-cate and function letters (we use F, G, H and f, g, h respectively) and sign for representation.Formation rules are analogous in both calculi except the definition of atomic formulas: inP=F1 atomic formulas have a form u = w or Piðu1; . . . ; unÞ, where P 2 fF;G;Hg andu; u1; . . . ; un;w are free individual variables, meantime in ISF

1 atomic formulas have a formu � w or u � piðu1; . . . ; unÞ, where p 2 ff ; g; hg; formulas with equality are defined in IsF

1

by (7), (8). Axioms of both mentioned calculi are in Table 2. Formulas (aP), (aI) are axiomsof quantifiers, (bP) and (cP) are axioms of equality, (bI) and (cI) are axioms of representation.The existential quantifier is defined in both calculi.

Table 2. Logical axioms in the standard semantics.

P=F1 ISF1

ðaPÞ 8 xFðxÞ!FðaÞ ðaIÞ 8 xðx � f Þ! a � fðbPÞ a ¼ a ðbIÞ a � aðcPÞ a ¼ b!ðFðaÞ!FðbÞÞ ðcIÞ a � b!ða � f!b � f Þ

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Non-standard semantics involve non-single valued terms, so to build calculi for them, weshould introduce into syntax the class of special logical choice functions. If we use general-ised functions, it is necessary to have a possibility to denote particular values of an ambigu-ous function via expression for this function. As any function with fixed arguments (orwithout arguments) does not determinate its values, an additional logical function is requiredin order to choose one of the values of an ambiguous function and represent it. In otherwords we have to introduce notations for choice functions into logical syntax; in this case aconcrete value (if any exists) of any function f, which is selected by a choice function, is asemantical value (interpretant) of the term formed by application of a choice function expres-sion to the expression for the function f as argument. More precisely, given language L, wetreat any choice function as a partial map ch: t� I ! M, where t � L is the set of quasitermsof the language L, I is a set of indices, and M is the domain of the principal interpretation ofL. Practically we need only a denumerable set of indices, so we can take words ′, ″, …, <n> ,... of L as indices (L must contain symbol ‘′’) and use naturals to sign them. We denote thevalue of choice function in L by ch(t, i) or by chi(t) or even by ti, where i 2 I. If t isf ðnÞðx1; . . . ; xnÞ, let ti be ‘f iðnÞðx1; . . . ; xnÞ’. If function f is a partial map or a map, let usdenote its single value by ‘f 0’. We say that any quasiterm of form ti is a marked quasiterm.It is clear that for every proper name a and every i that is true, the marked term ai is equal tothe unmarked term a and ðaiÞj is equal to ai.

Choice functions as well as interpretation maps in the standard model theory are semanticinterpretation functions. Thus every context in a fixed language L, for example a calculus ora theory, can be related with only one choice function. Logical laws don’t depend on interpre-tation, so any calculus with language L can be related with an arbitrary choice function thatcan be used with L; hence we don’t need to define choice function, which we use, anyway.

Introduction of the choice functions allows us to describe various objects with the samename within one context. For example, sentence ‘That dog attacked another dog’ can be for-malised as ‘dog1 � attack (dog2)’; additionally we can assert in this case that :ðdog1 �dog2Þ. Using languages with symbolism for a choice function we even do not need to intro-duce individual designations, because marked quasiterms of a form f i can play their role,where ‘f’ is a name for a 0-ary function. In particular, marked quasiterms can be quantifiedas it happens in natural language, so we can formalise sentences of human speech literally inthe simplest cases. For example, the sentence ‘all stars are shining’ can be formalised as‘8s2ðs2 � shÞ’ where ‘s’ signs the 0-ary function ‘star’, ‘sh’ signs the 0-ary function ‘shining’.Thus we have to determinate free and bound marked quasiterms or free and bound occur-rences of the marked quasiterms; for this purpose we accept the next convention: free occur-rences of the marked quasiterms in formal expressions are tagged only by odd indices of thechoice functions but bound ones only by even.

Let us build as example the calculus Id1F1 for the semantics of class 1 with postulate (II).This is the most interesting and important semantics because it involves names with anynumber of values. So, the corresponding calculus Id1F1 can be treated as the most importantfunction calculus.

Alphabet A (Id1F1) = fA;B;C; f ; g; h;:;!;�;¼;8; 9;0 ; ð; Þg. Here ‘A’, ‘B’, ‘C’ are propo-sitional letters, ‘f ’, ‘g’, ‘h’ are function letters; other signs treated as usual.

Metaalphabet includes signs ‘s’, ‘t’, ‘r’ for quasiterms and signs ‘F’, ‘G’, ‘H’ for quasi-formulas.

Language L(Id1F1) is the set of all words (i.e. lines of letters) in the alphabet A(Id1F1)including empty word K.

Variables: propositional variables are the words of a form U\m[, where U is a proposi-tional letter and <m> is a word in the alphabet f0g; function variables are the words of a form

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p\n[p\m[, where p is a function letter and <n>, <m> are words in the alphabet f0g (n meansarity of a function variable). We sign the words in the alphabet f0g by naturals and write Um

and pðnÞm instead of U\m[ and p\n[p\m[; also we can omit an index of arity and write pm.Quasiterms: (qt0) the words of a form pKp\m[ or p\n[p\m[t1; . . . ; tn, where

<n> is non-empty, are unmarked quasiterms; ðqt1Þ if s is an unmarked quasiterm containing functionletter p, then p\i[s is a marked quasiterm, where <i> is a non-empty word in the alphabet{′} (i means an index of a choice function); (qt2) the unmarked and the marked quasitermsare quasiterms. As above we write pið0Þm or pim and piðnÞm ðt1; . . . ; tnÞ or pimðt1; . . . ; tnÞ insteadof p\i[pKp\m[ and p\i[p\n[p\m[t1; . . . ; tn; also for a quasiterm with index i of a choicefunction we can sign it as si or ti or ri.

Terms are quasiterms containing only odd indices of a choice function.Atomic quasiformulas: (qf0) propositional variables are atomic quasiformulas; (qf1) if s is

a marked quasiterm and t is a quasiterm, then s � t is an atomic quasiformula.Quasiformulas: (f0) the atomic quasiformulas are quasiformulas; (f1) if F is a quasiformula,

then qF is a quasiformula; (f2) if F;G are quasiformulas, then ðF ! GÞ is a quasiformula; (f3)if FðsiÞ is a quasiformula containing a term si, then 8siþ1Fðsiþ1Þ is a quasiformula. As in predi-cate logic we say that in the last quasiformula the first occurrence of a quasiterm si+1 is anoccurrence in a quantifier but the second occurrence of si+1 is an occurrence in the range of aquantifier; such occurrences of quasiterms are bound, all another occurrences are free.

Formulas are quasiformulas that contain (occurrences of) any marked quasiterm that isnot a term only in some quantifier or in the range of that quantifier (in other words, containno free occurrences of marked quasiterms that are not terms).

Axioms:

ðad1Þ 8 f 2 ðf 2 � gÞ ! f 1 � g;

ðbd1Þ f 1 � f 1;

ðcd1Þ f 1 � g1 ! ð f 1 � h ! g1 � hÞ;ðdd1Þ f 1 � g1 ! f 1 � g;

ðed1Þ 9 g2 ðf 1 � g2Þ ! f 1 � g;

ðf d1Þ 9 h2ðh2 � f 1Þ ! ðf 1 � g ! 9 f 2 ðf 2 � gÞÞ;ðgd1Þ 8 f 2 ðf 2 � gÞ ! :9 f 2: ðf 2 � gÞ;

In addition, axioms of propositional logic and definitions (7), (8) are introduced. Here ðad1Þis the axiom of an universal quantifier, ðbd1Þ � ðdd1Þ are axioms of representation, ðed1Þ; ðf d1Þare axioms of an existential quantifier and axiom ðgd1Þ relates both quantifiers.

Rules of transformation includes one rule of inference – modus ponens, two rules ofquantification:

ðaÞ G ! Fðf iÞ ‘ G ! 8f iþ1Fðf iþ1Þ;ðbÞ 9hjðhj � f iÞ ! ðFðf iÞ ! GÞ ‘ 9f iþ1Fðf iþ1Þ ! G;

where j is even but i is odd, and three rules of replacement:Propositional substitution SGAFðAÞ : FðAÞ ‘ FðGÞ;Replacement of an index of a choice function Rj

iFðsiÞ : FðsiÞ ‘ FðsjÞ;where i, j both are odd or both are even,

Quasiterm replacement StsFðsÞ : FðsÞ ‘ FðtÞ;where t is a term if s is a term or an unmarked quasiterm, and t is a marked quasiterm if s isa marked quasiterm.

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Representational substitution SGðq20;...;q2nÞ½s��pð½t1�;...;½tn�ÞFð½s� � pð½t1�; . . . ; ½tn�Þ : Fð½s� � pð½t1�; . . . ;

½tn�Þ ‘ FðGð½s�; ½t1�; . . . ; ½tn�ÞÞ,where square brackets mean that quasiterms s; t1; . . . ; tn may vary in different occurencies ofsubformula ½s� � epð½t1�; . . . ; ½tn�Þ in F, and n P 0.

If we want to add individual designations (i.e. single-valued terms) to this calculus, weshould add in it individual’s letters a, b, c, x, y, z, extend the notions of atomic quasiformulaand formula, then add the next two axioms:

ðgIÞ :8xðx � gÞ ! 9x:ðx � gÞ;ðhIÞ 9xðx � aÞ;

and add the rules of substitution and replacement of the individual variables; the authordenotes this new calculus by Id1I F1:

Different semantics cause different sets of logical laws. In particular, in semantics of clas-ses 1–4, which involve empty names, quantifiers are not dual to each other, so formula

:8f 2ð f 2 � gÞ $ 9 f 2:ð f 2 � gÞ

here is not a logical law (indeed, the set of f-individuals can be empty for some f ).Hence quantifiers are not defined one by another, so they must be introduced separatelyand described with the different axioms (see above axioms of types (a) and (f )). Addi-tionally we set an axiom of type (g) describing the relation between both quantifiers; ifcalculus contains individual notations, we need one more axiom ðgIÞ, which together withthe axiom of type (g) guarantees the duality of the quantifiers under individual variables.Because of the absence of duality in the semantics of classes 1–4, the rule ðbÞ does nothave the form

Fðf 1Þ ! G ‘ 9f 2Fðf 2Þ ! G;

which it would have in predicate logic, but gets an additional existential antecedent (seeabove).

In semantics of classes 5–7, where there’s no empty names, the situation is much moreregular: here quantifiers are mutually dual as well as being dual in predicate calculi. So, theyare defined one by another and we don’t need an axiom of types (f ) and (g) but obtain thetheorem

f 1 � g ! 9 f 2ð f 2 � gÞ;

which is completely analogous to theorem (2) from predicate logic.Let us make two important remarks on the functional syntax. The first of these concerns

predicates. There is no separate category of predicate in function languages, but the predicatesare present here as propositional functions. So, every n-ary function f, whose first argumentsare g1-individuals, . . ., n-th ones are gn-individuals, and the values are g0-individuals, isdefined by expression

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f ðnÞðg21; . . . ; g2nÞ; ð9Þ

the analogous expression

g20 � f ðnÞðg21; . . . ; g2nÞ ð10Þ

defines the n+1-ary predicate as a propositional form in the sense of A. Church, and hencedenotes the propositional function. The case of 0-ary function f is the same. If g0; . . . ; gn arethe same function g, then (9) and (10) become f ðnÞðg2; . . . ; g2nÞ and g2 � f ðnÞðg4; . . . ; g2ðnþ1ÞÞrespectively.

The second remark concerns descriptions. We already have some of them: these will befunction expressions of the form f iðnÞðg11; . . . ; g2n�1

n Þ as n P 0. But we can even introducedescriptions of the general form into function languages; these are indefinite descriptions anal-ogous to e-descriptions from predicate languages. Not discussing details, the result is given.Description from the point of view of the function logical system is an unary function with apropositional function as the only argument. In other words description ‘f-individual such thatsatisfy constraint F’ will be written as

f 1ðFðf 1ÞÞ;

in particular, ‘f-individual such that have property g’ can be written as ‘f 1ðf 1 � gÞ’. As far asone can judge, if one extends a function language with descriptive quasiterms, it needs noadditional axiom, because descriptions behave themselves as ordinary general names. Thusany problems with treatment of descriptions shouldn’t emerge.

5. Non-standard semantics according to the second postulate

If the refusal of the first postulate of the standard semantics generates a whole class of alter-natives, then the second postulate has only one alternative:

(VI) Interpretants of linguistic expressions are their senses, but not their denotata.

Both of these alternatives are not equal: postulate (VI) is poorer for the consequences thanpostulate (I), since linguistic expressions as well as senses may play the role of denotata (sofar as the category of denotatum is relative), meanwhile the category of sense is absolute.The most natural technique for empirical descriptions, in particular for descriptions that con-tain empty names, is based on postulate (VI). This is conditioned by the point that if postu-late (VI) is accepted, then existence does not become a quantifier but transforms into apredicate speaking about particular individuals. If we denote the latter by ‘x � E’, the propo-sition ‘Julius Caesar exists’ could be written down as ‘c � E’, where c is ‘Julius Caesar’, but‘not every town exists’ as ‘:8t2ðt2 � EÞ ’, where t is the function-property ‘town’. At thesame time existential quantifier will change from existential state into partial: being con-nected with him will mean existence no more.

Let us call semantics containing postulate (II) denotational semantics or d-semantics, andsemantics containing postulate (VI) – sense semantics or s-semantics. So, let us use super-scripts ‘d’ and ‘s’ in the notation of calculi and their axioms for the sake of indicating thefact of what kind of semantics lies at their basis (index ‘d’ has used in the previous section

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just in this sense). If calculus contains individual notations, let us use subscript ‘I’ in itssymbolism. The author denotes predicate calculi by letters ‘PF’ (meaning: ‘propositionalfunction’), and function calculi – by letters ‘IF’ (so far as it goes concerning the functionsof individuals); superscript after ‘F’ denotes the logic’s order in the sense of theory oftypes.

Calculi with sense semantics from classes 1–3 and 5–7, containing notations for thepredicate of existence, have the following additional axiom:

ðexs13Þ f 1 � E ! 9h2ðh2 � f 1Þ;

from which, in particular, the next theorem can be derived:

f 1 � E ! 9 f 2ðf 2 � EÞ;

(note that in s-semantics of the classes 1–3 formulas ‘9f 2ð f 2 � f 1Þ’ and ‘9 f 2ð f 2 � EÞ’ aremutually independent, in s-semantics of class 4 they are both identically false, and in s-semantics of classes 5–7 the second formula is logically stronger than the first one). If weremove the predicate of existence and an axiom of type (ex) from any calculus under discus-sion, we obtain the system of logic without existential presuppositions.

6. Notes on model theory

Rejection of the standard semantics leads to changes in model theory. First of all, we wantsignature maps r : E ! M to be completely defined (here E is the set of all well-formedexpressions of a given language L, and M is the domain of a given model M). If we rejectpostulate (V), then E may contain empty names, so we need to introduce a fictional ana-logue of interpretant for every such name. Let us call such nonentities semantic nulls; by N

denote a set of semantic nulls, by A denote a set of ‘ordinary’ interpretants; thenM ¼ A [N; A \N ¼ ø: Unlike the unique Martin’s empty individual, semantic nulls canbe any number. Indeed, putting semantic nulls is necessary if we want to describe casessuch as discussed above in Section 2, where a and b on the second model are fictions,although a ≠ b. It should be emphasised that the idea of semantic nulls is not a hypothesisad hoc, but occurs naturally within the theory of a semantic triangle. It will not be discussedhere this time.

Further, a model M for every calculus C has the single domain M only if C contains indi-vidual notations as, for example, calculi Id1I F1 and ISF1 do. In this case individual variablesset a total class of individuals, i.e. universe; its presence is significant because all the func-tional terms behave as variables of different sorts (a sort determined by a function construct-ing a term), so every term has in the model a corresponding set of individuals. If calculus Cdoesn’t contain individual notations, every model M of C does not have a single domain buta set of domains. As we know, in predicate logic many-sorted calculi are trivially reducible tosingle-sorted ones; in function logic if calculus doesn’t contain individual notations, suchreducibility is impossible. Thus function calculi without individual notations (for example,Id1F1) are calculi with irreducible sorts. Models for such calculi are not relational systems buthave a more general form

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M ¼\fMig; fRjg > i2I ;j2J ;

where I, J are arbitrary sets of indices. Relations Rj are interpretations for function expres-sions ‘f ’, i.e. if a signature map has the form r : E ! Mi, predicate (10) corresponds to func-tion f, and rðg0Þ ¼ M0, …, rðgnÞ ¼ Mn, then r ðf ðg21 ; . . . ; g2nÞÞ ¼ R#M1 � . . .�Mn �M0;rðf iÞ 2 M0 for all i.

In s-semantics, if we introduce the predicate of existence, we have to take pairs\Mi;M�

i[ instead of single sets Mi, where Mi are sets of senses (and, possible, semanticnulls – pseudo senses corresponding to them), M�

i are sets of denotata (and, possible, seman-tic nulls – nonentities corresponding to them; in this case M�

i ¼ A�i [N�

i ; A�i \N�

i ¼ øÞ. Forthe sake of interpreting existence formulas we need to introduce partial maps si :Mi ! M�

i ifwe assume postulate (V), then every si is completely defined on Mi; if we reject (V), thenevery si is completely defined on Ai and completely undefined on Ni. Models of calculi withsemantics under discussion have a form of the following mathematical structures:

M ¼\fMig; fM�i g; fRjg >i2I ;j2J ;

where R is defined as above.

7. Summary

There are two implicit semantic postulates (I) and (II), which form the standard semantics, asit was been called in this paper; they underlie modern predicate logic. Contexts that are notsatisfactorily described in the standard semantics occur in logical applications from time totime. A striking example of semantically non-standard objects are descriptions, for which westill have no satisfactory theory, as all theorists have tried to build it based just on the(implicitly assumed) standard semantics. However, the possibility of building the predicatesyntax based on non-standard semantics has not been currently confirmed. The author isinclined to believe that standard semantics implicitly underlies the categorical distinctionbetween individual and predicate, so predicate languages as such are not semantically neutral.

At the same time there is a semantically neutral alternative to predicate languages – func-tion languages. These new languages are always built on the basis of explicitly specifiedsemantics, and a formal function language could be constructed for each of the semantics.Metatheory of such languages – syntax as well as model theory – is a generalisation of themetatheory of predicate languages in the case of arbitrary semantics.

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