On Truth

This is the articulation of what it is for a statement (propositionally structured content) to be true within the framework of the D-Lewisian possible world semantics. A viable theory of truth must entail the T-Schema: “P” is true ↔ P. Here is the formalism of language.

Formalism: Language is a function from sentences to meanings, i.e., a set of ordered pairs of sentences and meanings.

L: language

s: sentence

L(s): the meaning of s

L: {(s₁, L(s₁)), (s₂, L(s₂)), …}

W: possible world

w: fact, state of affairs

The meaning of a set of sentences is a possible world that serves as a specific model for the interpretation of the set. As Wittgenstein remarks, a world is a totality of all facts, or states of affairs. A theory is a function L that pairs one set of sentences with some model in a particular way. As such, a theory is a relation between two structures (sentences and world); although the relation can be surjective, I will focus on bijective relations for simplicity. For states of affairs consist in constellations of objects and properties, sentences are statements, attributed with propositional contents, in being given semantic interpretations. Since the meaning-relation is bi-structural, concepts (cognitive proxies for linguistic items) are metaphorical.

A proposition (the semantic content of a sentence) is a set of possible worlds (each of which is a set of facts). Thus, the proposition, e.g., that snow is white is a set of possible worlds in which the fact that snow is white is an element.

Given this set-up, what is it for, e.g., “snow is white” to be true iff snow is white? More precisely, what is it for the truth-value of “snow is white” is 1 iff snow is white? Truth here is a relation between a sentence and the fact it represents. The truth of a sentence is specific to a particular world, i.e., truth is indexical. Whether the truth-value of sentence is 1 or 0 is tested in respect to a possible world. If snow is white in W₁, then “snow is white” is true in respect to W₁. What it is for “P” to be true tout court is that there is a set of possible worlds in which P. (In the case of an empty set, the sentence is vacuously true.)

v(s) = 1 in L at W ↔ W belongs to the set of worlds L(s).

v(s) = 1 in L ↔ W_actual belongs to L(s).

v(s) = 1 in L tout court ↔ L(s) = ∅ or ∃W (W ⊂ L(s)).

This account of truth is the correspondence theory of truth. The problem with this account is that the truth-relation between sentences and facts as reconstructed in this theory is either logically trivial or epistemically problematic. As Davidson points out, one cannot get out of his skin in order to compare his thought (or statement) and the fact of a matter as to see whether the thought is true. If we can somehow take a vantage point of view from which we can compare two structures (our thoughts-statements and the world), any representative device such as language and cognition is redundant to begin with. But if we always perceive the world through some representative device, then it is quite trivial for us to say that “snow is white” is true. Either we say that snow is white (i.e., utter the right-hand part of the T-schema) or we don’t. If so, what it is for the sentence to be true is simply that it is uttered (or utterable). Taking this problem seriously, various versions of the disquotational (or deflationary) theory of truth attempt to determine the condition of propriety for uttering, asserting, or diquotating a sentence within the practice of a language.

To determine the specific condition of propriety is one thing, and to conceptualize this condition in terms of correspondence is another. Despite the challenges, the merit of the correspondence theory is that it provides an intuitive account of truth because, in my opinion, no condition of propriety can account for the sense in which truth bears necessity. If all there is to truth is some (implicit or explicit) rules for making assertions or manipulating linguistic symbols, then that “snow is white” is true seems to be contingent on what (language-)game one is disposed or decides to play unless the rules themselves are somehow constrained.

The notion of (possible) world as interpretation (semantics) is supposed to provide the constraints for the rules of assertion or symbolic manipulation (syntax). As such, truth is a regulative concept, world is a normative notion, and intentionality (aboutness) is a limiting idea. The myth of truth as a ceremonial ideal is what underwrites every actual condition of propriety in (social) practice.

Just as one could formulate any syntactic rules, one could relate sentences to worlds in whichever way conceivable. Thus, to devise a language-game is to carve nature at some joints. If so, to say that <s_w: snow is white> is true and <s_b: snow is black> is false is to say that L: {(s_w, w₁), (s_b, w₂), …} is the right way to carve nature (i.e., categorize impressions) for W such that w₁ ∈, w₂ ∉ W. As Putnam’s Paradox dictates, one could always make any sentence true by assigning different meanings, i.e., construct different intentional relations. For instance, given the set Lʹ: {(s_b, w₁), (s_w, w₂), …}, s_w would be false and s_b, true. Thus, the effect of saying that s_w is true is that L is authoritative. The actual way L carries or enforces its authority may be merely social. However, L maintains its authority by appealing to something transcendental.

Truth (or true language L_t) as accounted here is therefore an exclusive set of ordered pairs. The elements within the range of L_t constitute an elite set which David Lewis calls natural properties. This analysis of the correspondence theory of truth implicated by the D-Lewisian possible world semantics by no means addresses the logical triviality or the epistemic problem of the theory. Nevertheless, it suggests a direction for the inquiry. What is needed is a rigorous account of the religiosity of truth that is central to language-use and rational agency. What is the nature of this shrine of Alethia that grounds social conventions (including conventions of trust and truthfulness) and thereby holds our community together? Is its sacrosanctity the source of the primitive-ness (and, therefore, a-historicity) of truth in language and cognition? What is inside this Kodesh Hakodashim (the Holy of Holies) of rationality? What would be the epistemic relation between the voice from Mount Sinai and its people. Is this religious metaphor (rather than Brandomian social metaphors) appropriate for understanding truth?