Gödel on Syntactical Conventionalism

Question #7: A standard criticism of the linguistic theory of the a priori is that, even if “ordinary” a priori propositions are analytic – that is, follow from definitions by the laws of logic – this account leaves completely unexplained the status of the laws of logic and our knowledge of them. Discuss. Consider the reply to this criticism that the laws of logic follow by the definitions of the connectives, and the counter-reply that the laws of logic would have to follow by the definitions of the connectives and some laws of logic (either the ones being derived or some others). Does anything Godel says bear on this?

The phrase "laws of logic" carries the strong rationalist presumption that logic comprises non-arbitrary laws of thought, not merely conventional rules for the manipulation of symbols. Similarly, when the positivist speaks of the "rules of language" according to which “ordinary” a priori truths follow from definitions, a contingent and perhaps game-like sense of "rule" is assumed, along with an empirical view of language. At issue, among other things, is the epistemic seriousness with which we should take the law-like or rule-based entailment of conclusions from premises. Two kinds of truth appear to be at stake: 1) the soundness of arguments in which, if the method of inference is taken to be correct, the truth of the conclusion depends on the (non-inferential) truth of the premises,¹ and 2) the purely formal (or structural or syntactic) validity of so-called "logical truths.” If we are to ascribe a strong basis for knowledge in (2)—likewise, in the formal component of (1)—it would seem that the rules of deductive inference require a foundation apart from the vagaries of linguistic practice. Otherwise, it seems, logical truths—the simpler examples of which strike us as so obvious—would be true in virtue of nothing; conclusions drawn from true premises would not follow according to any inferential warrant. The pull toward an epistemic foundation for the rules of inference is strong.

Against this view, the logical empiricist might point out that the system of rules of inference is quite spare, once the "syntactic sugar" of the familiar first-order language used in natural-deduction proof has been stripped down to its bare essentials. Only a minimal subset of truth-functional connectives is needed, and the inference rules applied to sentences containing them can perhaps be derived from the connectives, e.g., by gleaning patterns of inference from sample proofs corroborated using truth tables.² The rites of inference, then, will reflect the combinatorial interactions of the connectives, according to their (the connectives’) definitions. Derivation of any proof can then be seen as nothing more than working out the juxtapositions, however complicated, of the defined behaviors of the connectives. In this light, not only the rules of inference but also the connectives themselves can be seen as "syntactic sugar," and the crux of the logical enterprise can be seen as the reduction of complex propositions to one of two values, true or false, whose only requirement is that of utility, viz., that they can be distinguished from each other.

On the “hyper-positivist” view, as we may imagine, the minimalism of the logical system thus denuded is so complete that the logical dimension of truth—the dimension of formal entailment that is typically seen as epistemically on par with empirical fact—drops out entirely. The structure, or syntax, of the system does not drop out, of course. It is pragmatically useful for formulating complex relations among terms, so that equivalent yet interesting propositions about the properties, existence, effects, and so on of referents posited to stand in those contingent factual relations can be worked out. The sole function of the connectives, on this view, is to maintain relational structure according to the definitions of the connectives, and to provide a notation and set of operations (rules of inference) more congenial to human linguistic practice than truth tables. Logical truth itself, then, is not a species of knowledge, merely a way of preserving relations among states of affairs whose factual status may coincide or co- vary—the states of affairs being taken as referents assigned to terms interrelated by the connectives.

Deductive validity, on the standard reading of positivism, is the only alternative to empirical fact as a Source of truth, but we might say, more ambitiously, that empirical facts are the only fundamental "truths," in the stronger sense of "truths about the world," and that logic is merely an apparatus for imparting quasi-linguistic structure to the relations and dependencies that may obtain among them. The intuitionist's response to all this may be that structural considerations—relations, dependencies, and the like—are part of what we mean by "facts" and "states of affairs." It is a fact about table salt, for instance, that if I put it in water (within certain parameters of temperature, ratio of salt to water, etc.) it will dissolve. The conditional form of this fact is no less a "truth about the world" than are the salt and the water. It is no mere coincidence, in other words, that the water-solubility of salt can be expressed as a hypothetical, and that other facts surrounding the proposition may be connected to it using the same logical apparatus, so as to produce further facts, or predictions of fact that can be tested. Yet examples like this seem perfectly consistent with the positivist's pragmatic account given above. On a given occasion, let us say, the salt dissolves. There is nothing logical about this fact. What does involve logic is the network of propositions and entailments concerning bodies of fact, which we construct by way of explanation. Such constructions are not facts;, they seek to explain facts. This broad controversy is related to the particular intuitionist or Platonist reply that the laws of logic do not follow from the definitions of the connectives alone, since definitions are arbitrary and provide no law-like foundation for the rules of inference. Even in the heat of the debate, it seems clear that this argument will be a non-starter without some technical support from the field of logic, since the characterization of logical rules as laws is the very issue at hand. Gödel offers such support. His central thesis in "Is mathematics syntax of language?" is that the syntactical program, as a replacement for mathematical intuition, is self-defeating. The syntactic interpretation of mathematics must be built up out of finitary concepts which are considered to be mere conventions, yet the resulting system must be powerful enough to be applied as comprehensively as classical, intuition-based mathematics. To achieve this, says Gödel, a consistency proof is required for the conventionalist system, because:

it belongs to the concept of a convention that one knows it does not imply any ‘propositions which can be falsified by observation… Without a consistency proof the "convention" itself, since open to disproof, really is an assumption [italics mine]…

Unless the consistency of the system can be proven, that is to say, the world-independence (or self-containment, or "factual impartiality") of its rules cannot be guaranteed. Technical problems arise with the consistency proof. It turns out that the consistency of the mathematical system as a whole cannot be proven without either reproducing its axioms and logical framework or modeling the system in such a way that axioms and inference rules just as powerful as those of the original system have been assumed (p. 345). Since the purpose of the consistency proof is to show that the base system of logic is consistent on its own purely syntactic and conventional terms, Gödel considers this bootstrapping. The syntactical program, on his view, can never rid an axiomatic mathematical system of its logical reliance on its axioms, for it can never explain away the axioms as dispensible to the truth of the system. The axioms may be viewed, not as freely-adopted definitions which will be merely plugged into the logical apparatus, but as knowledge assumptions which are both non-inferential in origin and integral to the consistency of the system.

Without presuming to settle the matter, I will suggest that, although axioms may be viewed as assumptions, they need not be. The question of their precise status—–as knowledge claims, definitions, or substantive concepts—remains (for me) open, despite the powerful intuition (or habit of thought) that mathematical knowledge spans a quasi-scientific domain with its own, privileged content. The positivist's natural continuation would be to hew to the empiricist program: axioms are selected for their applicability to some epistemic domain or other, or for their role in the construction of new, potentially applicable mathematical systems. When the conclusions they yield prove rich enough to be applied to facts about the world, they play their part in successful theory-making. When they fail to "pay off," they may be discarded and replaced with different axioms.³

¹As we shall see below, Gödel will argue that the naive dichotomy between the logical apparatus and its “fodder” does not tell the whole story, since in a priori axiomatic systems of a certain mathematical power the axioms become essential to the consistency of the system.

²Perhaps this is how they were developed in the first place, or perhaps the patterns of inference to be tested were chosen for their resemblance to locutions used in informal mathematical proofs.

³The case of the parallel postulate, for instance, whose replacement fostered surprisingly useful non-Euclidean [original document ends]…