Abstract
Part I presented the basic mechanism of natural communication using the example of a talking robot. Part II explained the complexity-theoretic aspects of syntactic analysis. Part III developed detailed morphological and syntactic analyses of natural language surfaces. Based on these foundations, we turn in Part IV to the semantic and pragmatic interpretation of syntactically analyzed expressions in natural language interpretation and production.
First, traditional approaches to semantic interpretation will be described in Chaps. 19–21, explaining basic notions, goals, methods, and problems. Then a semantic and pragmatic interpretation of LA Grammar within the Slim theory of language will be developed in Chaps. 22–24. The formal interpretation is implemented computationally as a content-addressable database system called a word bank.
Part I presented the basic mechanism of natural communication using the example of a talking robot. Part II explained the complexity-theoretic aspects of syntactic analysis. Part III developed detailed morphological and syntactic analyses of natural language surfaces. Based on these foundations, we turn in Part IV to the semantic and pragmatic interpretation of syntactically analyzed expressions in natural language interpretation and production.
First, traditional approaches to semantic interpretation will be described in Chaps. 19–21, explaining basic notions, goals, methods, and problems. Then a semantic and pragmatic interpretation of LA Grammar within the Slim theory of language will be developed in Chaps. 22–24. The formal interpretation is implemented computationally as a content-addressable database system called a word bank.
Section 19.1 explains the structure common to all systems of semantic interpretation. Section 19.2 compares three different kinds of formal semantics, namely the semantics of the logical languages, the programming languages, and the natural languages. Section 19.3 illustrates the functioning of logical semantics with a simple model-theoretic system and explains the underlying theory of Tarski. Section 19.4 shows why the procedural semantics of the programming languages is independent of Tarski’s hierarchy of metalanguages and details the special conditions a logical calculus must fulfill in order to be suitable for an implementation as a computer program. Section 19.5 explains why a complete interpretation of natural language within logical semantics is impossible and describes Tarski’s argument to this effect, based on the Epimenides paradox.
1 Basic Structure of Semantic Interpretation
The term semantics is used in various fields of science. In linguistics, semantics is a component of grammar which derives representations of meaning from syntactically analyzed natural surfaces. In philosophy, semantics assigns set-theoretic denotations to logical formulas in order to characterize truth and to serve as the basis for certain methods of proof. In computer science, semantics consists in executing commands of a programming language automatically as machine operations .
Even though the semantics in these different fields of science differ substantially in their goals, their methods, their applications, and their form, they all share the same basic two level structure consisting of syntactically analyzed language expressions and associated semantic structures. The two levels are systematically related by means of an assignment algorithm.
1.1 The Two Level Structure of Semantic Interpretation
For purposes of transmission and storage, semantic content is coded into surfaces of language (representation). When the content is needed, any speaker of the language may reconstruct it by analyzing the surface.Footnote 1 The reconstruction consists in (i) syntactic analysis, (ii) retrieval of word meanings from the lexicon, and (iii) deriving the meaning of the whole by assembling the meanings of the parts as specified by the syntactic structure of the surface.
The expressive power of semantically interpreted languages resides in the fact that the inverse procedures of representing and reconstructing content are realized automatically: a semantically interpreted language may be used correctly without the user having to be conscious of these procedures, or even having to know or understand their details.
For example, the programming languages summarize frequently used sequences of elementary operations as higher functions, the names of which the programmer may then combine into complex programs. These work in the manner intended even though the user is not aware of – and does not care about – the complex details of the machine or assembler code operations.
A logical language may likewise be used without the user having to go through the full details of the semantic interpretation. One purpose of a logical syntax is to represent the structural possibilities and restrictions of the semantics in such a way that the user can reason trulyFootnote 2 on the semantic level based solely on the syntactic categories and their combinatorics.
The natural languages are also used by the speaker-hearer without conscious knowledge of the structures and procedures at the level of semantics. In contradistinction to the artificial languages of programming and logic, for which the full details of their semantics are known at least to their designers and other specialists, the exact details of natural language semantics are not known directly even to science.
2 Logical, Programming, and Natural Languages
The two level structure common to all genuine systems of semantics allows one to control the structures on the semantic level by means of syntactically analyzed surfaces. In theory, different semantic interpretations may be defined for one and the same language, using different assignment algorithms. In practice, however, each kind of semantics has its own characteristic syntax in order to achieve optimal control of the semantic level via the combination of language surfaces.
The following three kinds of semantics comply with the basic structure of semantic interpretation shown in 19.1.1:
2.1 Three Different Kinds of Semantic Systems
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1.
Logical languages
These originated in philosophy.Footnote 3 Their complete expressions are called propositions. The logical languages are designed to determine the truth value of arbitrary propositions relative to arbitrary models. They have a metalanguage-based semantics because the correlation between the two levels is based on a metalanguage definition.
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2.
Programming languages
These are motivated by a practical need to simplify the interaction with computers and the design of software. Their expressions are called commands, which may be combined into complex programs. They have a procedural semantics because the correlation between the levels of syntax and semantics is based on the principle of execution, i.e., the operational realization of commands on an abstract machine which is usually implemented electronically.
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3.
Natural languages
These evolve naturally in their speech communities and are the most powerful and least understood of all semantically interpreted systems. Their expressions are called surfaces. Linguists analyze the preexisting natural languages syntactically by reconstructing the combinatorics of their surfaces explicitly as formal grammars. The associated semantic representations have to be deduced via the general principles of natural communication because the meanings1 (4.3.3), unlike the surfaces, have no concrete manifestation. Natural languages have a convention-based semantics because the word surfaces (types) have their meaning1 assigned by means of conventions (de Saussure’s first law, 6.2.2).
On the one hand, the semantic interpretations of the three kinds of languages all share the same two level structure. On the other hand, their respective components differ in all possible respects: they use (i) different language expressions, (ii) different assignment algorithms, and (iii) different objects on the semantic level:
2.2 Three Kinds of Semantic Interpretation
The most important difference between the semantic interpretation of artificial and natural languages consists in that the interpretation of artificial languages is limited to the two level structure of their semantics, whereas the interpretation of natural languages is based on the [2+1] level structure (4.2.2) of the Slim theory of language.
Thus, in logical and programming languages the interpretation is completed on the semantic level. The natural languages, in contrast, have an additional interpretation step which is as important as the semantic interpretation. This second step is the pragmatic interpretation in communication. It consists in matching the structures of the semantic level with corresponding structures of an appropriate context of use.
In the abstract, six relations may be established between the three basic kinds of semantics:
2.3 Mapping Relations Between the Three Kinds of Semantics
We represent these relations as N→L, N→P, L→N, L→P, P→N, and P→L, where N stands for the natural languages, L for the logical languages, and P for the programming languages.
In reality, there has evolved a complicated diversity of historical, methodological, and functional interactions between the three systems. These relations may be characterized in terms of the notions replication, reconstruction, transfer, and composition:
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(a)
Replication
The logical languages evolved originally as formal replications of selected natural language phenomena (N→L). Programming languages like Lisp and Prolog were designed to replicate selected aspects of the logical languages procedurally (L→P). The programming languages replicate phenomena of natural language also directly, such as the concept of ‘command’ (N→P).
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(b)
Reconstruction
When an artificial language has been established for some time and has achieved an independent existence of its own, it may be used to reconstruct the language it was originally designed to replicate in part. A case in point is the attempt in theoretical linguistics to reconstruct formal fragments of natural language in terms of logic (L→N). Similarly, computational linguistics aims at reconstructing natural languages by means of programming languages (P→N). One may also imagine a reconstruction of programming concepts in a new logical language (P→L).
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(c)
Transfer
The concentrated efforts to transfer methods and results of logical proof theory to programming languages (L→P)Footnote 4 have led to important results. A simple, general transfer is not possible, however, because these two kinds of language use different methods, structures, and ontologies for different purposes.Footnote 5 Attempts in philosophy of language to transfer the model-theoretic method to the semantic analysis of natural language (L→N) have been only partially successful as well.Footnote 6
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(d)
Combination
Computational linguistics aims at modeling natural communication with the help of programming languages (P→N), whereby methods and results of the logical languages play a role in both the construction of programming languages (L→P) and the analysis of natural language (L→N). This requires a functional overall framework for combining the three kinds of language in a way that utilizes their different properties while avoiding redundancy as well as conflict.
The functioning of the three kinds of semantics will be explained below in more detail.
3 Functioning of Logical Semantics
In logical semantics, a simple sentence like Julia sleeps is analyzed as a proposition which is either true or false. Which of these two values is denoted by the proposition depends on the state of the world relative to which the proposition is interpreted. The state of the world, called the model, is defined in terms of sets and set-theoretic operations.
3.1 Interpretation of a Proposition
By analyzing the surface Julia sleeps formally as sleep (Julia) the verb is characterized syntactically as a functor and the name as its argument.
The lexical part of the associated semantic interpretation (word semantics) assigns denotations to the words – here the set of all sleepers to the verb and the individual Julia to the proper name. The compositional part of the semantics derives denotations for complex expressions from the denotations of their parts (4.4.1). In particular, the formal proposition sleep (Julia) is assigned the value true (or 1) relative to the model if the individual denoted by the name is an element of the set denoted by the verb. Otherwise, the proposition denotes the value false (or 0).
A logical language requires definition of (1) a lexicon in which the basic expressions are listed and categorized, and (2) a syntax which provides the rules for combining the basic expressions into well-formed complex expressions. Its semantic interpretation requires in addition definition of (3) a model, (4) the possible denotations of syntactic expressions in the model, and (5) a semantic rule for each syntactic rule.
These five components of model theory are illustrated by the following definition, which in addition to the usual propositional calculus also handles example 19.3.1 in a simple manner.Footnote 7
3.2 Definition of a Minimal Logic
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1.
Lexicon
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2.
Model
A model \(\mathcal{M}\) is a tuple (A, F), where A is a non-empty set of entities and F a denotation function (see 3).
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3.
Possible Denotations
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(a)
If P1 is a one-place predicate, then a possible denotation of P1 relative to a model \(\mathcal{M}\) is a subset of A. Formally, \(\mathrm{F}(\mathrm{P}_{1})\mathcal {M} \subseteq \mathrm{A}\).
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If α is a name, then the possible denotations of α relative to a model \(\mathcal{M}\) are elements of A. Formally, \(\mathrm{F}(\alpha)\mathcal {M} \in \mathrm{A}\).
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If ϕ is a sentence, then the possible denotations of ϕ relative to a model \(\mathcal{M}\) are the numbers 0 and 1, interpreted as the truth values ‘true’ and ‘false.’ Formally, \(\mathrm{F}(\phi)\mathcal {M}\in \{0,1\}\).
Relative to a model \(\mathcal{M}\) a sentence ϕ is a true sentence if and only if the denotation ϕ in \(\mathcal{M}\) is the value 1.
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4.
Syntax
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If P1 is a one-place predicate and α is a name, then P1(α) is a sentence.
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If ϕ is a sentence, then ¬ϕ is a sentence.
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If ϕ is a sentence and ψ is a sentence, then ϕ & ψ is a sentence.
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If ϕ is a sentence and ψ is a sentence, then ϕ∨ψ is a sentence.
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If ϕ is a sentence and ψ is a sentence, the ϕ→ψ is a sentence.
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If ϕ is a sentence and ψ is a sentence, then ϕ=ψ is a sentence.
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(a)
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5.
Semantics
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(a)
‘P1(α)’ is a true sentence relative to a model \(\mathcal{M}\) if and only if the denotation of α in \(\mathcal{M}\) is an element of the denotation of P1 in \(\mathcal{M}\).
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(b)
‘¬ϕ’ is a true sentence relative to a model \(\mathcal{M}\) if and only if the denotation of ϕ is 0 relative to \(\mathcal{M}\).
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(c)
‘ϕ & ψ’ is a true sentence relative to a model \(\mathcal{M}\) if and only if the denotations of ϕ and of ψ are 1 relative to \(\mathcal{M}\).
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(d)
‘ϕ∨ψ’ is a true sentence relative to a model \(\mathcal{M}\) if and only if the denotation of ϕ or ψ is 1 relative to \(\mathcal{M}\).
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(e)
‘ϕ→ψ’ is a true sentence relative to a model \(\mathcal{M}\) if and only if the denotation of ϕ is 0 relative to \(\mathcal{M}\) or the denotation of ψ is 1 relative to \(\mathcal{M}\).
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(f)
‘ϕ=ψ’ is a true sentence relative to a model \(\mathcal{M}\) if and only if the denotation of ϕ relative to \(\mathcal{M}\) equals the denotation of ψ relative to \(\mathcal{M}\).
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(a)
The rules of syntax (4) define the complex expressions of the logical language; those of the semantics (5) specify the circumstances under which these complex expressions are true.
The simple logic system establishes a semantic relation between the formal language and the world by defining the two levels as well as the relation between them in terms of a metalanguage. The theory behind this method was presented by the Polish-American logician Alfred Tarski (1902–1983) in a form still valid today.
In the formal definition of an interpreted language, Tarski (1935) distinguishes between the object language and the metalanguage. The object language is the language to be semantically interpreted (e.g., quoted expressions like ‘ϕ & ψ’ in 19.3.2, 5), while the definitions of the semantic interpretation are formulated in the metalanguage. It is assumed that the metalanguage is known by author and reader at least as well or even better than their mother tongue because there should be no room at all for differing interpretations.
The metalanguage definitions serve to formally interpret the object language. In logical semantics the task of the interpretation is to specify under what circumstances the expressions of the object language are true. Tarski’s basic metalanguage schema for characterizing truth is the so-called T-condition. According to Tarski, the T stands mnemonically for truth, but it could also be taken for translation or Tarski.
3.3 Schema of Tarski’s T-Condition
T: x is a true sentence if and only if p.
The T-condition as a whole is a sentence of the metalanguage, which quotes the sentence x of the object language and translates it as p. Tarski illustrates this method with the following example:
3.4 Instantiation of Tarski’s T-Condition
‘Es schneit’ is a true sentence if and only if it is snowing.
This example is deceptively simple, and has resulted in misunderstandings by many non-experts.Footnote 8 What the provocative simplicity of 19.3.3 and 19.3.4 does not express when viewed in isolation is the exact nature of the two level structure (19.1.1), which underlies all forms of semantic interpretation and therefore is also exemplified by the particular method proposed by Tarski.
A closer study of Tarski’s text shows that the purpose of the T-condition is not a redundant repetition of the object language expression in the metalanguage translation. Rather, the T-condition has a twofold function. One is to construct a systematic connection between the object language and the world by means of the metalanguage: thus, the metalanguage serves as the means for realizing the assignment (19.1.1) in logical semantics. The other is to characterize truth: the truth value of x in the object language is to be determined via the interpretation of p in the metalanguage.
Both functions require that the metalanguage can refer directly (i) to the object language and (ii) to the correlated state of affairs in the world (model). The connection between the two levels of the object language and the world established by the metalanguage may be shown schematically as follows:
3.5 Relation Between Object and Metalanguage
The direct relation of the metalanguage to the world requires the possibility of verification, i.e., the ability to actually determine whether p holds or not. For example, in order to determine whether Es schneit is true or not, it must be possible to determine whether or not it is actually snowing. Without the possibility of verifying p, the T-condition is (i) vacuous for the purpose of characterizing truth and (ii) dysfunctional for the purpose of assigning semantic objects.
That Tarski calls the p in the T-condition the ‘translation’ of the x is misleading because translation in the normal sense of the word is not concerned with truth at all. Instead a translation is adequate, if the speaker meaning2 of the source and the target language expressions happen to be equivalent. For example, translating Die Katze ist auf der Matte as The cat is on the mat is adequate simply because the German source and the English target expression mean the same thing. There is obviously neither the need nor even the possibility of verification outside a theory of truth.
Tarski, however, took it to be just as obvious that within a theory of truth the possibility of verification must hold. In contradistinction to 19.3.4, Tarski’s scientific examples of semantically interpreted languages do not use some natural language as the metalanguage – because it does not ensure verification with sufficient certainty. Rather, Tarski insisted on carefully constructing special metalanguages for certain well-defined scientific domains for which the possibility of verification is guaranteed.
According to Tarski, the construction of the metalanguage requires that (i) all its basic expressions are explicitly listed and that (ii) each expression of the metalanguage “has a clear meaning” (op.cit., p. 172). This conscientious formal approach to the metalanguage is exemplified in Tarski’s (1935) analysis of the calculus of classes, which illustrates his method in formal detail. The only expressions used by Tarski in this example are notions like not, and, is contained in, is an element of, individual, class, and relation. The meaning of these expressions is immediately obvious insofar as they refer to the most basic mathematical objects and set-theoretic operations. In other words, Tarski limits the construction of his metalanguage to the elementary notions of a fully developed (meta-)theory, e.g., a certain area in the foundations of mathematics.
The same holds for the semantic rules in our example 19.3.2, for which reason it constitutes a well-defined Tarskian semantics. The semantic definition of the first ruleFootnote 9 as a T-condition like 19.3.5 may shown as follows:
3.6 T-Condition in a Logical Definition
The possibility of verifying the T-condition 19.3.6 is guaranteed by no more and no less than the fact that for any given model \(\mathcal{M}\) anyone who speaks English and has some elementary notion of set theory can see (in the mathematical sense of “unmittelbare Anschauung” or “immediate obviousness”) whether the relation specified in the translation part of T holds in \(\mathcal{M}\) or not.
In the history of mathematics, the appeal to immediate obviousness has always served as the ultimate justification, as formulated by Blaise Pascal (1623–1662):
En l’un les principes sont palpables mais éloignés de l’usage commun de sorte qu’on a peine à tourner la tête de ce côté-la, manque d’habitude: mais pour peu qu’on l’y tourne, on voit les principes à peine; et il faudrait avoir tout à fait l’esprit faux pour mal raisonner sur des principes si gros qu’il est presque impossible qu’ils échappent.
[In [the mathematical mind] the principles are obvious, but remote from ordinary use, such that one has difficulty to turn to them for lack of habit: but as soon as one turns to them, one can see the principles in full; and it would take a thoroughly unsound mind to reason falsely on the basis of principles which are so obvious that they can hardly be missed.]
Pascal, Pensées (1951:340)
In summary, Tarski’s theory of truth and logical semantics is clearly limited to the domains of mathematics, logic, and natural science insofar as only there are sufficiently certain methods of verification available.
4 Metalanguage-Based or Procedural Semantics?
In contrast to the semantic definitions in 19.3.2, which use only immediately obvious logical notions and are therefore legitimate in the sense of Tarski’s method, the following instantiation of the T-conditions violates the precondition of verifiability:
4.1 Example of a Vacuous T-Condition
‘A is red’ is a true sentence if and only if A is red.
The example is formally correct but vacuous because it does not relate the meaning of red in the object language to some verifiable concept of the metalanguage. Instead the expression of the object language is merely repeated in the metalanguage.Footnote 10
Within the boundaries of its set-theoretic foundations, model-theoretic semantics has no way of providing a truth-conditional analysis for a content word like red such that its meaning would be characterized adequately in contradistinction to, for example, blue. There exists, however, the possibility of extending the metatheory by calling in additional sciences such as physics.
From such an additional science one may select a small set of new basic notions to serve in the extended metalanguage. The result functions properly if the meaning of the additional expressions is verifiable within the extended metatheory.
In this way we might improve the T-condition 19.4.1 as follows:
4.2 Improved T-Condition for red
‘A is red’ is a true sentence if and only if A refracts light in the electromagnetic frequency interval between 430–480 THz.
Here the metalanguage translation relates the object language expression red to more elementary notions (i.e., the numbers 430 and 480 within an empirically established frequency scale and the notion of refracting light, which is well understood in the domain of physics) and thus succeeds in characterizing the expression in a nonvacuous way which is moreover objectively verifiable.
Example 19.4.1 shows that the object language may contain expressions for which there are only vacuous translations in the metalanguage. This does not mean that a sentence like ‘x is red’ is not meaningful or has no truth value. It only means that the metalanguage is not rich enough to provide a verification of the sentence. This raises the question of how to handle the semantics of the metalanguage, especially regarding the parts which go beyond the elementary notions of its metatheory.
Tarski’s answer is the construction of an infinite hierarchy of metalanguages:
4.3 Hierarchy of Metalanguages
Here, some vacuous T-conditions of the metalanguage, e.g., 19.4.1, are repaired by nonvacuous T-conditions, e.g., 19.4.2, in the metametalanguage. Some holes in the metametalanguage are filled in turn by means of additional basic concepts in the metametametalanguage, and so on. That an infinite hierarchy of metalanguages makes total access to truth ultimately impossible, at least for mankind, is not regarded as a flaw of Tarski’s construction – on the contrary, it constitutes a major part of its philosophical charm.
For the semantics of programming and natural languages, however, a hierarchy of metalanguages is not a suitable foundation.Footnote 11 Consider for example the rules of basic addition, multiplication, etc. The problem is not to provide an adequate metalanguage definition for these rules, similar to 19.3.2. Rather, the road from such a metalanguage definition to a working calculator is quite long and in the end the calculator will function mechanically – without any reference to the metalanguage definitions and without any need to understand the metalanguage.Footnote 12
This simple fact has been called autonomy from the metalanguage.Footnote 13 It is characteristic of all programming languages. Autonomy from the metalanguage does not mean that computers would be limited to uninterpreted, purely syntactic deduction systems, but rather that Tarski’s method is not the only one possible. Instead of the Tarskian method of assigning semantic representations to an object language by means of a metalanguage, computers use an operational method in which the notions of the programming language are executed automatically as electronically realized operations.
Because the semantics of programming languages is procedural (i.e., metalanguage-independent), while the semantics of logical calculi is Tarskian (i.e., metalanguage-dependent), the reconstruction of logical calculi as computer programs is at best difficult.Footnote 14 If it works at all, it usually requires profound compromises on the side of the calculus – as illustrated, for example, by the computational realization of predicate calculus in the form of Prolog.
Accordingly, there are many logical calculi in existence which have not been and never will be realized as computer programs. The reason is that their metalanguage translations contain parts which may be considered immediately obvious by their designers (e.g., quantification over infinite sets of possible worlds in modal logic), but which are nevertheless unsuitable to be realized as empirically meaningful mechanical procedures.
Thus, the preconditions for modeling a logical calculus as a computer program are no different from nonlogical theories such as physics or chemistry: the basic notions and operations of the theory must be sufficiently clear and simple to be realized as electronic procedures which are empirically meaningful and can be computed in a matter of minutes or days rather than centuries (8.2.2).
5 Tarski’s Problem for Natural Language Semantics
Because the practical use of programming languages requires an automatic interpretation in the form of corresponding electronic procedures they cannot be based on a metalanguage-dependent Tarski semantics. But what about using a Tarski semantics for the interpretation of natural languages?
Tarski himself leaves no doubt that a complete analysis of natural language is in principle impossible within logical semantics.
The attempt to set up a structural definition of the term ‘true sentence’ – applicable to colloquial language – is confronted with insuperable difficulties.
Tarski (1935), p. 164
Tarski proves this conclusion on the basis of a classical paradox, called the Epimenides, Eubulides, or liar paradox.
The paradox is based on self-reference. Its original ‘weak’ version has the following form: if a Cretan says All Cretans (always) lie there are two possibilities. Either the Cretan speaks truly, in which case it is false that all Cretans lie – since he is a Cretan himself. Or the Cretan lies, which means that there exists at least one Cretan who does not lie. In both cases the sentence in question is false.Footnote 15
Tarski uses the paradox in the ‘strong’ version designed by Leśniewski and constructs from it the following proof that a complete analysis of natural language within logical semantics is necessarily impossible:
For the sake of greater perspicuity we shall use the symbol ‘c’ as a typological abbreviation of the expression ‘the sentence printed on page 391, line 2 from the bottom.’ Consider now the following sentence:
$$ \mbox{c is not a true sentence} $$Having regard to the meaning of the symbol ‘c’, we can establish empirically:
(a) ‘c is not a true sentence’ is identical with c.
For the quotation-mark name of the sentence c we set up an explanation of type (2) [i.e., the T-condition 19.3.3]:
(b) ‘c is not a true sentence’ is a true sentence if and only if c is not a true sentence.
The premise (a) and (b) together at once give a contradiction:
c is a true sentence if and only if c is not a true sentence.
Tarski (1935)
In this construction, self-reference is based on two preconditions. First, a sentence located in a certain line on a certain page, i.e., line 2 from the bottom on page 391 in the current Chap. 19, is abbreviated as ‘c’.Footnote 16
Second, the letter ‘c’ with which the sentence in line 2 from the bottom on page 391 is abbreviated also occurs in the unabridged version of the sentence in question. This permits one to substitute the c in the sentence by the expression which the ‘other’ c abbreviates. The substitution may be described schematically as follows:
5.1 Leśniewski’s Reconstruction of the Epimenides Paradox
If sentence c in line X is true, then what sentence c says, namely that ‘c is not a true sentence’, holds. Thus, ‘c is not a true sentence’ (as the original statement) and ‘c is a true sentence’ (as obtained via substitution and its interpretation) both hold.
To prove that a logical semantics for natural language is impossible, Tarski combines Leśniewski’s version of the Epimenides paradox with his T-condition. In this way he turns an isolated paradox into a contradiction of logical semantics.
5.2 Inconsistent T-Condition Using Epimenides Paradox
There are three options to avoid this contradiction in logical semantics.
The first consists in forbidding the abbreviation and the substitution based on it. This possibility is rejected by Tarski because “no rational ground can be given why substitution should be forbidden in general.”
The second possibility consists in distinguishing between the predicates ‘trueo’ of the object language and ‘truem’ of the metalanguage.Footnote 17 Accordingly, the formulation
c is truem if and only if c is not trueo
is not contradictory because truem≠trueo. Tarski does not consider this possibility, presumably because the use of more than one truth predicate runs counter to the most fundamental goal of logical semantics, namely a formal characterization of the truth.
The third possibility, chosen by Tarski, is to forbid the use of truth predicates in the object language. For the original goals of logical semantics the third option poses no problem. Characterizing scientific theories like physics as true relations between logical propositions and states of affairs does not require a truth predicate in the object language. The same holds for formal theories like mathematics.
Furthermore, for many mathematical logicians the development of semantically interpreted logical calculi was motivated by the desire to avoid what they called the vagueness and contradictions of natural language. For example, Frege (1892) expresses this sentiment as follows:
Der Grund, weshalb die Wortsprachen zu diesem Zweck [d.h. Schlüsse nur nach rein logischen Gesetzen zu ziehen] wenig geeignet sind, liegt nicht nur an der vorkommenden Vieldeutigkeit der Ausdrücke, sondern vor allem in dem Mangel fester Formen für das Schließen. Wörter wie >also<, >folglich<, >weil< deuten zwar darauf hin, daßgeschlossen wird, sagen aber nichts über das Gesetz, nach dem geschlossen wird, und können ohne Sprachfehler auch gebraucht werden, wo gar kein logisch gerechtfertigter Schlußvorliegt.
[The reason why the ‘word-languages’ are little suited for this purpose [i.e., draw inferences based on purely logical laws] is not only the existing ambiguity of the expressions, but above all the lack of clear forms of inference. Even though words like ‘therefore,’ ‘consequently,’ ‘because’ indicate inferencing, they do not specify the rule on which the inference is based, and they may be used without violating the well-formedness of the language even if there is no logically justified inference.]
This long-standing, widely held view provides additional support to refraining from any attempt to apply a Tarskian semantics to natural language.
If logical semantics is nevertheless applied to natural language, however, the third option does pose a serious problem. This is because the natural languages must Footnote 18 contain the words true and false. Therefore a logical semantic interpretation of a natural (object) language in its entirety will unavoidably result in a contradiction.
Tarski’s student Richard Montague (1930–1970), however, was undaunted by Tarski’s proof and insisted on applying logical semantics to natural language:
I reject the contention that an important theoretical difference exists between formal and natural languages. …Like Donald Davidson I regard the construction of a theory of truth – or rather the more general notion of truth under an arbitrary interpretation – as the basic goal of serious syntax and semantics.
Montague (1970), “English as a formal language” Footnote 19
We must assume that Montague knew the Epimenides paradox and Tarski’s related work. But in his papers on the semantics of natural languages Montague does not mention this topic at all. Only Davidson, whom Montague appeals to in the above quotation, is explicit:
Tarski’s …point is that we should have to reform natural language out of all recognition before we could apply formal semantic methods. If this is true, it is fatal to my project.
Davidson (1967)
A logical paradox is fatal because it destroys a semantic system. Depending on which part of the contradiction an induction starts with, one can always prove both a theorem and its negation. This is not acceptable for a theory of truth.Footnote 20
Without paying much attention to Tarski’s argument, Montague, Davidson, and many others insist on using logical semantics for the analysis of natural language. This is motivated by the following parochial prejudices and misunderstandings:
For one, the advocates of logical semantics for natural language have long been convinced that their method is the best-founded form of semantic interpretation. Because they see no convincing alternatives to their metalanguage-dependent method – despite calculators and computers – they apply logical semantics in order to arrive at least at a partial analysis of natural language meaning.
A second reason is the fact that the development of logic began with the description of selected natural language examples. After a long independent evolution of logical systems it is intriguing to apply them once more to natural languages,Footnote 21 in order to show which aspects of natural language semantics can be easily modeled within logic.
A third reason is that natural languages are often viewed as defective because they can be misunderstood and – in contrast to the logical calculi – implicitly contradictory. Therefore, the logical analysis of natural language has long been motivated by the goal of systematically exposing erroneous conclusions in rhetorical arguments in order to arrive at truth. What is usually overlooked, however, is that the interpretation of natural languages works quite differently (19.2.2) from the interpretation of metalanguage-dependent logical languages.
Notes
- 1.
For example, it is much easier to handle the surfaces of an expression like 36⋅124 than to execute the corresponding operation of multiplication semantically by using an abacus. Without the language surfaces one would have to slide the counters on the abacus 36 times 124 ‘semantically’ each time this content is to be communicated. This would be tedious, and even if the persons communicating were to fully concentrate on the procedure it would be extremely susceptible to error.
- 2.
In the sense of always going from true premises to true conclusions.
- 3.
An early highlight is the writing of Aristotle, in which logical variables are used for the first time.
- 4.
Scott and Strachey (1971).
- 5.
The transfer of logical proof theory to an automatic theorem prover necessitates that each step – especially those considered ‘obvious’ – be realized in terms of explicit computer operations (Weyhrauch 1980). This requirement has already modified modern approaches to proof theory profoundly (P→L reconstruction).
- 6.
Sect. 19.4.
- 7.
For simplicity, we do not use here a recursive definition of syntactic categories with systematically associated semantic types à la Montague (CoL, pp. 344–349).
- 8.
Tarski (1944) complains about these misunderstandings and devotes the second half of his paper to a detailed critique of his critics.
- 9.
Compared to 19.3.5, 19.3.6 is more precise because the interpretation is explicitly restricted to a specific state of affairs, specified formally by the model \(\mathcal{M}\). In a world where it is snowing only at certain times and certain places, on the other hand, 19.3.5 will work only if the interpretation of the sentence is restricted – at least implicitly – to an intended location and moment of time.
- 10.
Tarski’s own example 19.3.4 is only slightly less vacuous. This is because the metalanguage translation in Tarski’s example is in a natural language different from the object language. The metalanguage translation into another natural language is misleading, however, because it omits the aspect of verification, which is central to a theory of truth. The frequent misunderstandings which Tarski (1944) so eloquently bewails may well have been caused in large part by the ‘intuitive’ choice of his examples.
- 11.
The discussion of Tarski’s semantics in CoL, pp. 289–295, 305–310, and 319–323, was aimed at bringing out as many similarities between the semantics of logical, programming, and natural languages as possible. For example, all three kinds of semantic interpretation were analyzed from the viewpoint of truth: whereas logical semantics checks whether a formula is true relative to a model or not, the procedural semantics of a programming language constructs machine states which ‘make the formula true’, – and similarly in the case of natural semantics. Accordingly, the reconstruction of logical calculi on the computer was euphemistically called ‘operationalizing the metalanguage’.
Further reflection led to the conclusion, however, that emphasizing the similarities was not really justified: because of the differing goals and underlying intuitions of the three kinds of semantics a general transfer from one system to another is ultimately impossible. For this reason the current analysis first presents what all semantically interpreted systems have in common, namely the basic two level structure, and then concentrates on bringing out the formal and conceptual differences between the three systems.
- 12.
See in this connection also 3.4.5.
- 13.
CoL, pp. 307f.
- 14.
With the notable exception of propositional calculus. See also transfer in 19.2.3.
- 15.
For a detailed analysis of the weak version(s) see Thiel (1995), pp. 325–327.
- 16.
The page and line numbers have been adjusted here from Tarski’s original text to fit those of this chapter. This adjustment is crucial in order for self-reference to work.
- 17.
- 18.
This follows from the role of natural languages as the pretheoretical metalanguage of the logical languages. Without the words true and false in the natural languages a logical semantics couldn’t be defined in the first place.
- 19.
Montague (1974), p. 188.
- 20.
As a compromise, Davidson suggested limiting the logical semantic analysis of natural language to suitable consistent fragments of natural language. This means, however, that the project of a complete logical semantic analysis of natural languages is doomed to fail.
Attempts to avoid the Epimenides paradox in logical semantics are Kripke (1975), Gupta (1982), and Herzberger (1982). These systems each define an artificial object language (first-order predicate calculus) with truth predicates. That this object language is nevertheless consistent is based on defining the truth predicates as recursive valuation schemata.
Recursive valuation schemata are based on a large number of valuations (transfinitely many in the case of Kripke 1975). As a purely technical trick, they miss the point of the Epimenides paradox, which is essentially a problem of reference: a symbol may refer on the basis of its meaning and at the same time serve as a referent on the basis of its form.
- 21.
As an L→N reconstruction (19.2.3).
References
Davidson, D. (1967) “Truth and Meaning,” Synthese VII:304–323
Frege, G. (1892) “Über Sinn und Bedeutung,” Zeitschrift für Philosophie und philosophische Kritik 100:25–50
Gupta, A. (1982) “Truth and Paradox,” Journal of Philosophical Logic 11:1–60
Herzberger, H. (1982) “Notes on Naive Semantics,” Journal of Philosophical Logic 11: 61–102
Kripke, S. (1975) “Outline of a Theory of Truth,” The Journal of Philosophy 72:690–715
Montague, R. (1974) Formal Philosophy, New Haven: Yale University Press
Pascal, B. (1951) Pensées sur la Religion et sur Quelques Autre Sujets, Paris: Éditions du Luxembourg
Scott, D., and C. Strachey (1971) “Toward a Mathematical Semantics of Computer Languages,” Technical Monograph PRG-6, Oxford University Computing Laboratory, Programming Research Group, 45 Branbury Road, Oxford, UK
Tarski, A. (1935) “Der Wahrheitsbegriff in den formalisierten Sprachen,” Studia Philosophica I:262–405.
Tarski, A. (1944) “The Semantic Concept of Truth,” Philosophy and Phenomenological Research 4:341–375
Thiel, C. (1995) Philosophie und Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft
Weyhrauch, R. (1980) “Prolegomena to a Formal Theory of Mechanical Reasoning,” Artificial Intelligence, reprint in Webber and Nilsson (eds.), 1981
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Exercises
Exercises
Section 19.1
-
1.
What areas of science deal with semantic interpretation?
-
2.
Explain the basic structure common to all systems of semantic interpretation.
-
3.
Name two practical reasons for building semantic structures indirectly via the interpretation of syntactically analyzed surfaces.
-
4.
Explain the inverse procedures of representation and reconstruction.
-
5.
How many kinds of semantic interpretation can be assigned to a given language?
-
6.
Explain in what sense axiomatic systems of deduction are not true systems of semantic interpretation.
Section 19.2
-
1.
Describe three different kinds of formal semantics.
-
2.
What is the function of syntactically analyzed surfaces in the programming languages?
-
3.
What principle is the semantics of the programming languages based on and how does it differ from that of the logical languages?
-
4.
What is the basic difference between the semantics of the natural languages, on the one hand, and the semantics of the logical and the programming languages, on the other?
-
5.
Why is a syntactical analysis presupposed by a formal semantic analysis?
-
6.
Discuss six possible relations between different kinds of semantics.
-
7.
What kinds of difficulties arise in the replication of logical proof theory in the form of computer programs for automatic theorem proving?
Section 19.3
-
1.
Name the components of a model theoretically interpreted logic and explain their functions.
-
2.
What are the goals of logical semantics?
-
3.
What is Tarski’s T-condition and what is its purpose for semantic interpretation?
-
4.
Why is verification a central part of Tarski’s theory of truth?
-
5.
What is the role of translation in Tarski’s T-condition?
-
6.
Why does Tarski construct the metalanguage in his example of the calculus of classes? What notions does he use in this construction?
-
7.
What does immediate obviousness do for verification in mathematical logic?
Section 19.4
-
1.
Explain a vacuous T-condition with an example.
-
2.
What is the potential role of non-mathematical sciences in Tarski’s theory of truth?
-
3.
For what purpose does Tarski construct an infinite hierarchy of metalanguages?
-
4.
Why is the method of metalanguages unsuitable for the semantic interpretation of programming languages?
-
5.
What is the precondition for realizing a logical calculus as a computer program?
-
6.
In what sense does Tarski’s requirement that only immediately obvious notions be used in the metalanguage have a counterpart in the procedural semantics of the programming languages?
Section 19.5
-
1.
How does Tarski view the application of logical semantics to the analysis of natural languages?
-
2.
Explain the Epimenides paradox.
-
3.
Explain the three options for avoiding the inconsistency of logical semantics caused by the Epimenides paradox.
-
4.
What is the difference between a false logical proposition like ‘A & ¬A’ and a logical inconsistency caused by a paradox?
-
5.
What difference does Montague see between the artificial and the natural languages, and what is his goal in the analysis of natural languages?
-
6.
Give three reasons for applying logical semantics to natural languages. Are they valid?
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Hausser, R. (2014). Three Kinds of Semantics. In: Foundations of Computational Linguistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41431-2_19
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