PhDr. Ruzicka

January 23, 1998

Language and Logical Structure

When analyzing the concept of meaning, Charles Sanders Peirce distinguished three levels of which most signs consist:

sign (i.e. the physical part: a chain of letters; sound waves, etc.), which signifies

concept (characterized by a set of properties)

object in the world (if any; it falls outside language), which is referred to by the sign.

If we apply this division to sentences we get:

sentence

proposition

state of affairs.

If we generalize this pattern for our purposes, we may conclude that signs and sentences stand for language, concepts and propositions belong to the level of logical structure of language, and objects and states of affairs are parts of the extra-linguistic reality, the world.

In this essay I shall discuss the relationship between language and its logical form, which might be interesting especially for linguistically oriented people. However, since the discussion has some ontological implications, which might be more challenging to a philosopher, I shall briefly outline the relationship between the logical structure of language and the world and the ontological consequences of the different theories, too.

The essay is based on an article by Max Black that presents a theory of the relationship between language and logical structure. However, before I discuss Black’s argument, I shall briefly outline an earlier theory advocated for by Bertrand Russell and Ludwig Wittgenstein in the period of “logical atomism”.

The basic assumption of the theory is that language is a mirror of the world and its structure corresponds to the structure of the reality. However, it is not the surface structure of language, but the deep structure, the logical structure that corresponds to the structure of reality. Russell and Wittgenstein were convinced that every object could be analyzed to discover the elementary parts, atoms of reality of which the object consists. These are, according to Russell, sense data, and that is as far as we can go in the analysis. The sense data are captured in language by means of what is called logically proper names – the elementary parts of logical structure of language. Unlike the correspondence between the atoms of reality and the atoms of logical structure, which is presupposed, the relationship between language (the surface structure) and the logical structure is questioned. The logical structure manifests itself in language in different ways. Sometimes there is “one to one” correspondence between the elements of the surface structure and the elements of the deep structure, but sometimes we find what Gilbert Ryle calls systematically misleading expressions. These are expressions in which the grammatical structure is different from the logical structure and thus are misleading in the sense that they postulate entities which are scientifically unacceptable. Ryle names several examples which are grammatically subject-predicate expressions (such as: “God exists,” “Infidelity calls out for condemnation,” “The King of France is not Poincar`e.”), but which cannot be regarded subject-predicate also logically, because the subjects of the expressions cannot refer. Ryle concludes that the expressions “infidelity”, “God”, “the King of France” are predicative expressions. The logical structure of the expressions would then be: at least one x is God; for all x where x is unfaithful implies x calls out for condemnation; at least one x is the King of France, the only x that is King of France is wise, respectively.

According to Russell, there is one universal logical structure behind the surface structure of language. This structure is constant for all natural languages. Thus arguments that are valid in Hebrew are valid also in English, Czech and other languages. However, the logical structure must first be discovered.

Before we proceed to Black’s theory, we shall get acquainted with a move in the approach of analytic philosophers which took place in the period between the beginning of the century when Bertrand Russell published his theory and the beginning of the second half of the century when Max Black wrote his article. Russell and his followers were not primarily concerned about natural language, because it was regarded inadequate, ambiguous, vague and suffered from many diseases that could threaten science. Their primary concern was to create a formal language that would be precise. By comparing this logical construct with natural languages, we could, they were convinced, learn a lot about the nature of our language. When they did talk about natural language, they talked again about an idealized language, language as a system, disregarding the fact, that such a system is completely dependent on natural languages.

Beginning with J. L. Austin we enter a branch in analytic philosophy that will be called Oxford School or Ordinary Language Philosophy. Unlike Russell, Carnap and others, who were mathematically oriented philosophers, Austin, Urmson, Black and others were first of all philologists, and their primary concern was about natural language itself. The basic assumption underlying Ordinary Language Philosophy is that ordinary language is a result of long lasting evolution process and thus it must be “sound” otherwise it would not work. The distinctions made by natural languages are ontologically valuable and natural languages are worth studying for its own sake.

Now I shall summarize the content of Black’s theory. Since Black is also a philosopher, he looks for philosophical inferences that can be made on the basis of natural languages. He is convinced, however, that we need to abstract from any insignificant grammatical properties of natural languages:

If philosophical inferences from grammar are to be justified, they must not be based on casual, but significant grammatical properties…Metaphysical inferences must be based on the creation of a hypothetical universal grammar which abstracts from the idiomatic particularities of concrete grammars.

In other words, Black seeks for a universal logical structure underlying all natural languages. We can be quite sure that the reason why a grammatical feature that is present in all natural languages is nothing else but the correspondence between the language and reality. This means that we have to decide which features of languages are insignificant and remove them from our construction of the universal grammar. Let us consider some questions:

Is it necessary that there are as many symbols in an expression as there are elements in the state of affairs captured by the expression? Let us imagine a person who is being taught to drive a car. We need two symbols to let him know when to start the car and when to stop it. We can plausibly use the words “start” and “stop”. In this example the structure of the affair is the same as the structure of the expression. Let us now imagine, that we would use a whistle to signalize both activities. Does it mean that we have one symbol for two situations? Some might say it does not, because in this sense the symbol consists of not only the whistle, but also the state of the car; so we have two symbols for two activities anyway. A counterargument might be that the whistle is a symbol for a change of a state whatever the state is, so there is, in fact, only one symbol for more activities. Black claims that it is not possible to decide whether it is necessary that there are as many symbols in an expression as in the state of affair it describes, for we would need precise criteria of identity of activities and identity of symbols, which do not exist. In other words it is up to anyone who is involved in the matter to choose one way or the other.

Is copula a necessary property of the universal grammar? Copula is a means of connecting the subject of a sentence with its predicate. Those who think copula is redundant might claim that there are some languages (such as Chinese or Hebrew for example) which do not use copula and their expressiveness is sufficient. Those who oppose this view claim that the fact that copula is not expressed in a sentence does not prove that there is no relation of juxtaposition between the subject and the predicate. They claim that the relation between the elements is as important (although unexpressed) as the elements themselves. If we put a longer time or space gap in between the sentence elements the relation is destroyed. However, if we adopt Frege’s theory of predication, where predicates are regarded as incomplete functions, we might claim again that copula is redundant, because the predicates are “eager” to be about a subject and to be completed; thus no connection is necessary between the subject and the predicate of a sentence. The decision is again up to a particular person. However pointless may this argument seem, it has some serious implications as far as fully developed alternative logical systems are concerned. If we think that the characteristic relation between the subject and the predicate of a sentence exists, we also accept questions about properties, so predicates or their substitutions can sometimes function as subjects. Frege’s conception on the other hand does not permit questions and statements about properties as subjects.

Should the subject predicate distinction be a part of the universal grammar? First of all, how to distinguish between the subject and the predicate of a sentence? Let us say that the subject is ‘what the expression is about,’ and the predicate is ‘what we say about the subject.’ Is then the sentence: ”Peter is happy,” about Peter or about happiness? Does it mean that an individual Peter has the property of being happy, or that the property ‘happiness’ is exemplified by an individual Peter? Is the subject of the sentence Peter or happiness? Black proceeds to exemplify the redundancy of the subject predicate structure by an analogue with a chess move. A chess move can be taken down in various ways:

by a sentence: “King’s bishop moved to king’s fourth square”

English-speaking players use a notation: “P-K4”

in European notation: ”e2-e4”

by a line drawn on square divided to 64 parts

by a set of two numbers between 1 and 64

by a number smaller than 4096 (=64x64)

by the Morse’s alphabet

by electric waves, etc.

The further we go in the different notations of the same configuration of facts the more difficult it is to recognize a subject predicate structure. This makes Black accept the conclusion that it is a redundant feature that needs to be excluded from the universal grammar. At this point it must be noted that Black’s argument is marked by a very broad concept of language. From my point of view it is very hard to find a subject predicate structure where it is impossible. We might as well try to reach match point or to give an ace in a chess game. I think from the second example on it is foolish to talk about examples of natural language notation of the fact and if we want to build our universal grammar on the basis of natural languages we should work with natural languages and not with electric waves. In my view, Black is inconsistent here in the sense that he confuses semantics with semiotics. He might have used this type of argument even to exclude correspondence, copula, or other questionable features of natural language. If we take down the sentence “Peter is happy” by means of sound waves, we find neither anything that would suggest that there is a relation of juxtaposition between the subject and the predicate, nor would we recognize any correspondence between the elements of the notation and the state of affairs it captures, for the simple reason that they are incompatible. Having considered these facts I regard his argument invalid.

We have seen Black’s effort to show the difficulties with deciding what features of natural languages are necessary features and thus should belong among the principles of the universal grammar. Black comes to the conclusion that there are no criteria for such decisions and so we cannot hold out much hope for the construction of one universal grammar underlying all natural languages. In other words, natural languages are not based on the same logical structure, but instead there are as many structures behind our languages as there are interpretations of what are and what are not substantial features of languages. It is interesting that Black does not question the relationship between the logical structures and reality naither. Once we get to one of the underlying logical structures the correspondence with reality is presupposed. Does it mean that there are not only different views of reality, but also various or even rival realities? This is a purely ontological question which can (or cannot) be answered again only through the analysis of language, but I do not hold out much hope for bringing the solution of this problem in this essay.

For the present it might be plausible to move to the field of logic and outline the different approaches of this problem from the logical point of view. The basic question is: Is there just one uniquely correct logical system, or could there be several which are equally correct? In logic we find three kinds of response to this question:

monism - there is just one correct system of logic

pluralism - there is more than one correct system of logic, even contradictory ones

instrumentalism - there is no correct system of logic, the notion of correctness is inappropriate

For our purposes we will concentrate on the first two approaches. Generally speaking, the controversy is based on the distinction between classical logic and deviant logics. Susan Haack counts among deviant logics the following: many-valued logics, Intuitionist logics, quantum logics, free logics. A monist would claim that these rival classical logic in the sense that they claim to be as correct as classical logic is, which is impossible. Only one logic can be correct. A relativist would claim that deviant logics do not rival classical logic, because it is possible that more than one logical system is correct.

Let us consider a wff ‘p or ~p’. In classical logic it is a tautology. In many-valued logics it is not. Does it mean that the formula is logically true and at the same time not logically true? Certainly not. The problem is that in many-valued logics the meanings of the connectives are defined differently from the way they are defined in classical logic. If a proponent of a many-valued logic claims that ‘p or ~p’ is not logically true, he or she does not contradict the fact that excluded middle is a logical truth, because he does not speak about excluded middle. So far we have shown that deviant logics do not contradict classical logic (in fact we have not, because we have considered only one wff, but we have shown a line along which such a proof might go), but we have not disprove the relativist assumption that there are more logical systems that are equally correct. But if deviant logics do not contradict classical logic, they must be complementary to it. Reality can be captured by many logical systems, the question is how much a particular system bites off. This may be an argument in favor of the monist assumption. There is only one correct logical system based on set of axioms/rules (we probably do not know the whole of the system) and which consists of many subsystems that are based on different interpretations of the set of axioms and thus are complementary. The possibility of a contradiction is excluded.

However, there is a way how to avoid the controversy about correctness of logical systems and that is by rejecting the whole idea of correctness. This is an approach advocated for by instrumentalists. Before I present my arguments for instrumentalism I shall explicate what it is that they reject. Susan Haack offers a definition of correctness of a logical system: “A logical system is correct if the formal arguments which are valid in that system correspond to informal arguments which are valid in the extra-systematic sense, and the wffs which are logically true in the system correspond to statements which are logically true in the extra-systematic sense.” In her book Haack makes an effort to dismiss instrumentalism in favor of a version of pluralism (i.e. global pluralism). But her arguments are not that convincing. The problem is centered around the question: Are there extra-systematic conceptions of validity/logical truth by means of which to characterize what it is for a logic to be correct? Haack is convinced there are:

When, intuitively, we judge some ordinary, informal arguments good and others bad, something like this conception of validity is probably being deployed….I indicated early on that I do take there to be an extra-systematic idea of validity to which formal logical systems aim to give precise expression. It is clear enough from the history of formal logic (consider Aristotle, for instance, or Frege) that the motivation for the construction of formal systems has been, on the basis of an initial conception of some arguments as good and others as bad, to sort out logical from other, e.g. rhetorical, features of good arguments, and to give rules which would permit only the logically good arguments, and to exclude the bad.