20th WCP: Metapsychologism In The Philosophy Of Logic

"In the old anti-psychologistic days ...".
W. van O. Quine. "Epistemology Naturalized"

The debate between psychologism and antipsychologism in the XXth century psychology of logic seemed to be solved ultimately in favor of antipsychologism. After G. Frege, E. Husserl, R. Carnap and J. Lukasiewicz it was almost generally recognized that the only true philosophy of symbolic logic is antipsychologism. Antipsychologism was considered as a thesis belonging to the body of symbolic logic itself. In this paper I try to re-examine relations between antipsychologism, psychologism and modern logic.

The problem of psychologism

The re-examination mentioned presupposes an analysis of notions of psychologism and antipsychologism themselves. First, we should state the problem of psychologism clearly. The first point which is evident but nevertheless should be mentioned is that the problem of psychologism itself is not the logical problem but belongs to philosophy of logic. It is a question about logic and in this sense a question external to logic. This is important to mention from the very beginning in order to stress that the changes in the solutions of the problem of psychologism do not influence directly to the solutions of the purely logical tasks. The solution of the problem of psychologism as a problem of philosophy of logic is motivated partly by the developments taking place in logic itself and partly by philosophical considerations. (1)

Usually this problem is treated as a question about the relation between logic, on the one side, and thought or mental processes, on the other side. Thus to understand this question we would need to answer two other questions: What is logic? What is thought? But neither of these questions is easily answered in a general form. (2) For the time being, by the term "logic", I will understand classical first order predicate calculus, (3) and by the term "thought", the reasoning of a cognitive subject when he tries to solve a problem of an arbitrary nature without using any formal logical means. "Thought", then, is reasoning carried on without the conscious application of formalized logical laws and rules of inference. This is still not a very precise understanding of thought, but for the present it is sufficient for a formulation of the problem of psychologism. We shall call this kind of mental processes "natural thought".

Now we can formulate the problem of psychologism in the philosophy of logic. The problem of relation between logic and thought consists of two interconnected questions:

(1) can logic be reduced to psychology? and

(2) can logic be regarded as a model of natural thought?

But these questions in turn need some explication.

(1*) We say that a theory F is reduced to a theory G if the terms of F can be defined in the terms of G and assertions of G can be deduced from assertions of F. This sense of reduction has been worked out in logicist attempts to reduce mathematics to logic. But it has a more general meaning.

(2*) What does it mean to be "a model of natural thought"? A logical structure is a model of a natural thought process, which consists in step by step transformation of a concept or a judgment, if it reproduces a sequence of steps of the course of this transformation. In this case we may say that it reproduces the content of the thought-process, and gives us essential information about this process.

The classical psychologism of the XIXth century (John Stuart Mill is the best example of psychologists of this kind) answered these questions in the following manner:

(P1) Logic is reduced to psychology, i.e. logical notions can be defined in terms of psychological notions and logical laws can be deduced from psychological laws.

The psychologistic answer to question (1) makes question (2) superfluous or, more accurately, yields the trivial answer to it:

(P2) Logic is the model of thought processes, because it is simply a part of these processes.

It seems that there is a weaker conception of psychologism, according to which logic plays a normative role in thought-processes. This idea lies in the basis of Susan Haack's "weak psychologism". (4) However this idea alone is not sufficient for recognition of a conception as psychologistic one. It can not distinguish psychologism and antipsychologism. To obtain a conception of psychologistic type we have to adopt also the thesis of reducibility of norms to facts. We can find the conjunction of the two theses in D. S. Mill's philosophy of logic. We can call a conception of this type a "moderate psychologism".

The brief analysis of different forms of psychologism just undertaken shows that all forms of psychologism share one common feature. They all state a descriptivity of logic in relation to thought-processes.

The classical antipsychologism, as represented in the works of Gottlob Frege and Edmund Husserl, answered these questions in the following manner:

(AP1) Logic cannot be reduced to psychology, it is autonomous,

(AP2) Logic cannot be regarded in any sense as a model of thought.

Frege wrote: "Keinen Vorwurf braucht der Logiker weniger zu scheuen, als den, dass seine Aufstellungen dem naturlichen Denken nicht angemessen seien. ... Das Streben, den natuerlichen Denkvorgang darzustellen, wuerde darum geradeswegs von der Logik abfuehren". (5) The second answer is usually represented as implied by the first one.

Psychologism was subjected to severe criticism at the end of the XIXth century by Frege and Husserl. New applications of logic to Artificial Intelligence and cognitive science show that the paradigm of standard psychologism is also insufficient. Here I propose the new conception of relations between logic and natural thought based on non-psychologistic treatment of logical laws, but establishing the connection between logical structures and thought-processes.

Metapsychologism

The central notion of metapsychologism is the notion of a logical procedure. A logical procedure is regarded as a sequence of a cognitive agent's actions implemented in accordance with postulates (axioms or rules of inference) of a given formalized logical system. Formalization requires us to consider at least two languages: an object language L in which the logic in question is formulated, and a metalanguage ML, in which we "speak" about expressions of object language and actions with them and which is usually a part of natural language.

To understand the structure of logical procedures, we need to consider the subject matter of symbolic logic itself. Logic can be defined in the first approximation as a theory of "what follows from what". Then the main business of logic is to formulate statements about entailments or deducibility and to design methods of their justification or refutation. These statements are logical laws, while the justifications and refutations are proofs and disproofs. The problem of discovering of logical relations between statements of a formalized language is formulated and solved in the corresponding metalanguage. An activity of a cognitive agent who deduces some statements form others can be regarded as seeking justifications or refutations of metastatements on deducibility or entailment.

To develop such views of logic I shall use some considerations of J.A. Robinson concerning sequential calculi. He defines a sequent as a meaningful statement about formulas in an object language drawn from the lists and : "... Sequents say something quite specific, and what they say either is so or is not so, therefore they are true or they are false". (6) In such a case logic can be treated as a theory which is involved in establishing of truth and falsity of some meaningful statements of metalanguage: "One might almost say: the task of logic is to separate the true sequents from the false ones, and to find ways of establishing the truth of true sequents and the falsehood of false sequents". (7)

If the metalanguage, in which sequents are formulated, is not formalized, then establishing their truth and falsity is a matter of natural thought. A cognitive agent, of course, uses some metalanguage counterparts of the rules of the logical system, formulated in the object language, but uses them informally. An order of application of logical rules is not fixed and is therefore determined by the natural thought of the cognitive agent.

The crucial fact is that we can also operate with metalinguistic statements also in a purely formal manner. This is what calculi of a sequential type do. Under systems of a sequential type I mean Gentzen's and Kanger's systems, Beth's tableaux, Hintikka's models sets, etc. As we have already seen, a sequent is a metastatement about the deducibility of the list of formulas from the list of formulas in the logic of the object language. We can already treat the classical work of G. Gentzen in such a manner, even though he himself formulated his sequents as expressions in the object language. In his sequential calculi G. Gentzen "has also axiomatized part of the metatheory of his systems for natural deduction. The elementary sentences of his metatheory are of the form U₁,U₂,...,U_n,Z where U₁,U₂,...,U_n, and Z are sentences of the language for which the natural deduction system has been formulated. These new elementary sentences are called sequents and their intended interpretation is that Z can be 'safely' derived from the premises U₁,U₂,...,U_n by means of the rules in his natural deduction system". (8)

The next fact, which is even more important for our purposes, is that formal models of the activities of justifying and refuting of such metastatements have also been invented in logic. These models are represented by proof-search procedures. Sequential systems are well adapted for design of proof-search procedures. Since a proof-search is used for establishing the truth or falsity of metastatements, and since it essentially involves objects of metalanguage (for example, metavariables), it can be labeled as a metaprocess. The possibility of formalizing of the part of metalanguage in which sequents are formulated, and in which the methods of their justification and refutation are developed, creates a new situation in the treatment of logical procedures. We can now proceed from the usual one-level interpretation to a new two-level one. Indeed, a logical procedure can now be represented in its full extension as consisting of two formalized levels and one informal level: (a) the object level, at which we specify a formal system which formalizes a class of valid formulas and their proofs; (b) the metalevel, at which we formalize metastatements on object-level deducibility and methods of their justification and refutation; and (c) the metametalevel which is a level of non-formal reasoning about (a) and (b), and which is similar to a metalanguage in the conventional sense.

Such two-level procedures are usually employed in logic without theoretical and philosophical awareness. One example is given by semantic tableaux in original formulation by E.W. Beth. Here the logic of the object level is constructed in the form of a natural deduction calculus, the logic of the metalevel is formalized in the form of semantic tableaux, and there are some connections between these levels, expressed by the algorithm for reconstructing the results of the metalevel proof-search in semantic tableaux into natural deduction in the object language. Another example is given by the algorithm of proof-search ALPEV-LOMI where the proof-search takes place in the sequential calculus, and the resulting deduction is given in natural form. So we can state that the "two-level" construction of logical procedures is in some sense typical for the organization of proof-search procedures.

In order to understand better such a two-level interpretation, it is essential to attend to the different styles of formalization. Concerning our problem there are two main styles: (a) formal systems that do not include a formalization of the proof-search, and (b) formal systems that include a formalization of (at least some elements of) the proof-search. Axiomatic systems of the Hilbert type and systems of natural deduction are examples of the first kind of system, and the systems of the sequential type are examples of the second kind. The previous considerations allow us to conclude that the formalization of the first kind corresponds to the logic of the object level, and the second kind is appropriate to the logic of the metalevel. (The metametalanguage in this case is a part of natural language, so its logic is not subject to formalization).

Therefore a complete logical formalization of the notion of logical procedure presupposes (I) an object level, with a system of a Hilbert or natural type, which formalizes notions of a valid formula and a proof, (II) a metalevel with a formal system of a sequential type, which formalizes metastatements on deducibility at the object level and the methods of processing them; and (III) a metametalevel at which informal reasonings about the first two levels are carried on. (9)

The crucial difference between the object level and metalevel of formal systems is that the latter formalize at least several elements of a proof-search while the former do not. Since a sequential type system is, as we have already seen, a formalization of the metalevel of Hilbert type axiomatic or natural deduction system, we shall understand what is formalized at the object level if we consider what occurs at the corresponding metalevel. In such a system, a logical procedure develops, as it were, in two dimensions: in the formal system itself, where the deduction is written out, and in the "mind" of the cognitive agent operating with this formal system. Everyone, who has any experience in constructing deductions in such systems knows that the cognitive agent has to try many substitutions into axioms, or must form different concrete axioms from axiom schemata, introduce auxiliary assumptions, choose relevant values of terms etc., and also eliminate any errors he has made. This process - which can be improved with experience, with working out in the "cognitive agent's mind" a supply of admissible rules that allow to find the substitutions and assumptions needed more quickly - never can be completely eliminated. All such mental actions of a cognitive agent are actions of informal proof-search, which include building a structure of possible ways of constructing deductions, eliminating of the ways which do not promise to give the deduction needed, and at last choosing a way that can become a valid logical deduction (if one exists). But in object level systems we have no means of expressing such mental actions. Only the results of a proof-search can be expressed in them. Since these actions are directed to the formulas and terms of the object language, they belong to a non-formal metalevel of logical procedure. This feature of object level systems is connected with the fact that in such systems their rules do not include any instructions on how to analyse the structure of a given formula or of the premises and conclusion of a given deduction, so as to find the deduction itself. The analysis is carried out by cognitive agent in his "mind". Since the metalanguage is not formalized, such mental actions of deduction-search belong to the natural thought processes of the cognitive agent.

I have mentioned earlier that for the metalevel of a logical procedure, the sequential style of formalization is appropriate. The main feature of sequential systems is that they create a possibility of formalizing a proof-search by finding principles of analysis for the sequent proved, the rules governing the substitutions of terms, and some means permitting one to choose among ways of proof-search. Consequently, I can state that the complete formalization of the notion of logical procedure should include two interconnected levels: an object level and a metalevel of a logical system. At the object level, a subsystem of proof (inference, deduction) is formalized, while at the metalevel a subsystem of proof-search is formalized, and a description of this procedures takes place at the unformalized metametalevel.

Thus I can state that procedures of proof-search, which are realized at the metalevel, replace the mental actions of a cognitive agent, directed at searching for the intended deduction. The thesis of metapsychologism follows from these considerations: processes of natural thought are simulated by processes of proof-search, which take place at the metalevel of logical procedure.

Metapsychologism raises the usual psychologistic considerations by one level in the hierarchy of logical procedures, while the possibility of a non-psychologist justification of logical relations remains at the object level. It is important that the processes which take place at the metalevel (metaprocesses) depend on the properties of cognitive agent (the supply of knowledge, the logical competence, even the speed of implementation of logical actions) and therefore the structure of his actions and the process of their implementation give us essential information about the essential features of the natural thought-processes involved in reasoning. Thus the thesis of metapsychologism is a synthesis of conventional psychologism and antipsychologism, which restores at the metalevel the most essential features of the psychologistic interpretation of logical procedures (the possibility of regarding information on thought-processes conveyed by logical procedures) and which preserves the rational content of antipsychologistic thesis, i.e. non-psychologistic program of justification of logical relations among statements at the object level.

Metapsychologism differs essentially from conventional psychologism through the fact that it replaces the relation of identity between logical structures and structures of natural thought with the relation of simulation. Therefore, we can speak only about a partial reproduction of a simulated system through a model advanced. There is, moreover, the possibility of a plurality of models for one and the same thought-process, which simulate different features of this process and, as it were, complement each other. Thus at the metalevel we meet with a pluralism of logical models of thought, since a great many different proof-search procedures can be associated with a given logical system - for instance, with the classical first order predicate calculus. This eliminates the rigorous and unrealistic requirement of naive psychologism, that some specific structures defined in object language of predicate calculus (formulas, inferences, etc.) should reproduce some specific structures of thought.