Gert-Jan C. Lokhorst


G.J.C. Lokhorst. Supervenience and truth-functionality in Wittgenstein's Tractatus. In P. Weingartner & G. Schurz, eds., Philosophy and Natural Science: Borderline Questions. Reports of the 13th International Wittgenstein Symposium, pp. 276-278. Hölder-Pichler-Tempsky, Vienna, 1989. ISBN 3-209-00862-0. Reprinted in dissertation.

Abstract. The purpose of this paper is twofold. First, I want to point out that Wittgenstein's Tractatus contains a clear and remarkably modern example of a theory of supervenience. And secondly, I want to argue that this theory of supervenience may be interpreted as a weak form of a principle of truth-functionality--which may exactly be the form of this principle which Wittgenstein himself had in mind.

The purpose of this paper is twofold. First, I want to point out that Wittgenstein's Tractatus contains a clear and remarkably modern example of a theory of supervenience. And secondly, I want to argue that this theory of supervenience may be interpreted as a weak form of a principle of truth-functionality--which may exactly be the form of this principle which Wittgenstein himself had in mind.


"Supervenience" means something like "dependence". When something depends on another thing, we also say that that thing supervenes on the other one. Modern supervenience theorists see a lot of relations of supervenience between various domains. For example, it is often said that the moral supervenes on the non-moral. When two organisms are alike in all non-moral respects, they cannot possibly differ in some moral respect; when someone's biography is exactly the same as that of the present president of Austria in all biological, psychological and other non-moral respects, he or she cannot fail to have the same moral status as that president has (whichever that may be). No moral difference without some other difference. It may be the same with the mental: any exact physical duplicate of me must necessarily have precisely the same mind as I have. No mental difference without some physical difference.

The interesting thing about supervenience is that it is a much weaker notion than reducibility. Moral facts may depend on non-moral ones, but no one may be able to spell out the dependence of the supervenient superstructure on the basis in detail; no one may be able to give a "reduction" in terms of non-moral facts. Similarly, the mental may fully depend on the physical, but no one may ever be able to describe the mechanism (or logic?) of the dependence in detail. This is an advantage of the notion, for physicalists often want to defend only a vague, general form of dependence, without wanting to posit any lawlike relationships.

Haugeland has given a definition of supervenience which nicely fits our purposes:

Two worlds in a class of possible worlds are discernible with a given language just in case there is a sentence of this language which is true at one, and not at the other. [...] A language weakly supervenes on another language (relative to a class of possible worlds) just in case any two worlds in the class of possible worlds which are discernible with the former language are also discernible with the latter. ([1], p. 97.)

Supervenience in the Tractatus

On seeing the above definition, any reader of the Tractatus will immediately be reminded of section 4.26 of this work:

If all true elementary sentences are given, the result is a complete description of the world. The world is completely described by giving all elementary sentences, and adding which of them are true and which false.

In other words, when you take any "possible world" (as Wittgenstein calls it: Notebooks 19.9.1916) different from "the real world", there is always at least one elementary sentence which is true in only one of both worlds. Wittgenstein does not state whether he would want to apply this principle to all possible worlds, but we may safely assume that he did. So let us extend TLP 4.26 to the thesis that worlds may always be discerned by elementary sentences. No difference between worlds without some elementarily describable difference.

Applying Haugeland's definition, it will be clear that we then have a principle of supervenience here. Everything which may be discerned with the whole language may already be discerned by means of the elementary sentences alone, and therefore the whole language supervenes on its subset of elementary sentences. We may extend the definition and likewise say that all facts supervene on elementary facts (Sachverhalte, described by elementary sentences), that all properties supervene on elementary properties (attributed by elementary sentences), etc.

Truth-functionality as supervenience

Wittgenstein states his thesis of truth-functionality in Tractatus 5-5.01:

The sentence is a truth-function of the elementary sentences. (The elementary sentence is a truth-function of itself.) The elementary sentences are the truth-arguments of sentences.

The principle of truth-functionality as stated here is weaker and more liberal than the definitions which we have become accustomed to nowadays. According to present-day definitions, a sentence cannot be truth-functional unless its truth-value is some function of the truth-values of the subsentences it contains and of the way it is built up from these. The truth-values of other sentences do not matter. However, the Tractatus does not say that only the truth-values of the elementary subsentences of a sentence matter as to its truth-value. The latter truth-value may as well be a function of the truth-values of all elementary sentences.

Let us, for the moment, interpret the Tractarian principle of truth-functionality in the latter way. Thus, this principle asserts that the truth-values of all elementary sentences (not necessarily only the ones contained in the sentence as subsentences) jointly determine the truth-values of all sentences. Given what Carnap called a state-description--a set which, for each elementary sentence, contains either this sentence or its negation, and no other elements--any sentence may assume only one truth-value.

When we accept this weak formulation, the principle of truth-functionality is easily seen to follow from the supervenience principle we have just mentioned. For if the thesis of truth-functionality did not hold, two worlds could verify the same elementary sentences and yet differ as regards the truth-value of some other sentence. These worlds would hence not be completely described by elementary sentences and violate the principle of supervenience.

The converse implication does, of course, not hold: we may conceive of a language which is purely truth-functional but unable to describe any one world completely. Such a language would, however, not be in accordance with the Tractatus.

The Tractarian principle of truth-functionality

Now did Wittgenstein really have such a weak principle of truth-functionality in mind, or did he accept the principle in its stronger, full-blown modern version? I think it is hard to find evidence for the latter view. Wittgenstein never explicitly banishes modal and doxastic constructions from the ideal language he had in mind. They seem to be perfectly in order, provided they are truth-functional; for otherwise the principle of supervenience of language on elementary language would be violated. And why should they be prohibited, after all? This would not only lead to a drastic impoverishment of language (which is nowhere explicitly advocated in the Tractatus), there is also no justification for it on syntactic grounds. Modalizing a sentence is no more mysterious than negating it. And indeed, most of Wittgenstein's remarks on syntactic operators seem to apply equally well to all operators, including modal and other ones which we do not longer call truth-functional nowadays.

Thus, I believe Wittgenstein's principle of truth-functionality, which would be unduly restrictive otherways, may best be regarded as being only such a rather weak claim. This also explains why Wittgenstein himself is so silent about the truth-functionality of, e.g., intentional ascriptions. They are simply no exception to the rule. They differ from what we still call truth-functional compounds nowadays in that their truth-values are not fully determined by the subsentences they contain and the way they are built up from these alone; the additional factors that play a role (according to the Tractatus) are spelled out in [2]. But they are truth-functions of the elementary sentences nonetheless, so there is no need to pay special attention to them. It is the same with modal sentences: their truth-values are the same in all worlds and in all interpretations which are in accord with the demands the Tractatus puts on such interpretations, as I explain in [2], and so they are trivially truth-functional. (Unlike the case of intentional ascriptions, their truth-values are, however, determined by their structure alone).

In defense of the Tractarian view of truth-functionality

Far from being a drawback, the broad, general nature of the Tractarian principle of truth-functionality is in fact a point in its favour. It is an enviable position to be able to claim that there are some sentences which are basic from an epistemological or scientific point of view, and which jointly determine all truths and falsehoods, without being obliged to say how they manage to do just that. The long history of failed attempts at giving explicit truth-functional definitions of modal and intentional language may indeed suggest that it is well-nigh impossible to do the latter, while the thesis yet remains attractive in its own right. Thus, Wittgenstein may have been wise in going no further.

Modern theories of supervenience, which also argue for dependence without committing themselves to reducibility, stem from the same motivation and share the same appeal. These theories show that it is possible to be precise and yet not too specific when giving physicalistic accounts of morality and mind. The Tractatus shows that the same may be done when one is formulating sweeping statements on the nature of the semantical relations between various kinds of sentences.


J. Haugeland. Weak supervenience. American Philosophical Quarterly, 19:93-103, 1982.
G. J. C. Lokhorst. Ontology, semantics and philosophy of mind in wittgenstein's Tractatus: A formal reconstruction. Erkenntnis, 29:35-75, 1988.

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