Truth as a Contingent Property

Truth is not a property. We cannot characterize true propositions, for there is no common property shared by all true propositions. [Read, p. 31]

The redundancy theory of truth -- that "P is true" ascribes no property to the proposition P and is just a long form for P-- is usually held to be incompatible with the correspondence theory, since in the latter, the truth of P corresponds to the holding of the state of affairs (the sense of P), and that holding seems rather like a property of P.

Some of this is right, within the framework of Tractarian semantics, since only objects (elsewhere in these pages called parts) have properties, contingent on their states. So "P is true" assigns no property to P if P is a proposition because a proposition, as a logical form instantiated by a propositional sign, has no states and is thus not the sort of thing which can have properties. But this does not exclude the possibility of a property which we could identify as "truth", contingently predicable of some object.

We might try taking as the object, not the proposition, but the medium in which the proposition, as a propositional sign, is 'inscribed'. But that obviously won't work, since whether the medium is inscribed with P has, in most cases, no connection with whether the state of affairs sense(P) is the case. We need to combine the medium with a part of the world one of whose states is sense(P). Actually, we only need that part of the medium which has two states, in one of which the medium is inscribed with P and the other in which it is not, combined with that part of the World whose two states are the state of affairs in which sense(P) holds and its complement: the state of affairs in which P is false. But whether sense(P) holds or not is only relevant in states of the World in which P is actually inscribed in the medium. The combination is thus an object m with three states:

m is not inscribed with P,
m is inscribed with P and P is false,
m is inscribed with P and P is true.

We'll call this the truth-object for P = tr_m,P.

By 'inscription', we just mean the state of a part of the world which can be construed as a propositional sign. A spoken proposition and a thought proposition are both, in this very general sense, 'inscriptions'.

The truth-property we are looking for is a property we will call Tr_{L, P}, which holds for an object when its state is construable as an inscription in a language L expressing the proposition P, and P is true. It is therefore the collection of all states of affairs A, such that A is the state of some truth-object tr_m,P, and A = sense(P) .

If we now "sum" Tr_{L, P} over all propositions, we obtain a general 'property-equivalent' of truth as a contingent property Tr_L common to, not true propositions since propositions don't have (contingent) properties, but to truth-objects. A truth-object x has the property Tr_L when its state is the one in which what is inscribed in x can be interpreted in L as a true proposition.

We don't actually need to refer to truth objects to describe Tr_L (they were just defined to make explicit to what the general truth property applies.) Leaving aside the truth-objects gives us a simpler definition: the general truth property for a language L is (extensionally) the collection of senses of contingent propositions expressible in L.

As a result, truth properties are predicable of other objects than truth-objects; e. g., if X is the sense of a proposition expressible in L, then Tr_L is a contingent property of any object one of whose states is X. Notice that we don't have to take into account whether the proposition whose sense is X is inscribed somewhere, since that is necessarily the case if X holds.

It should be clear that there is no circularity in defining the symbol Tr_L to be the name of a property which happens to be describable in terms of the concept of propositional truth, and at the same time, use Tr_L to provide a definition for propositional truth. For if we are realists about properties, then the definition of Tr_L only describes and does not construct the property itself. We can therefore, without impropriety, define truth for propositions in L in terms of Tr_L in the obvious way:

A proposition P is true if the propositional sign which expresses it in the language L is inscribed in a medium m and the truth-object tr_m,P has the property Tr_L property.

(But why should we be realists about contingent properties? Because contingent properties of objects are constituted (extensionally) by the states of affairs in which the objects have those properties, and if we believe in the reality of anything, we should believe in the reality of states of affairs.)

The definition of truth-object brings to light a feature of propositional senses which, to my knowledge, has not previously been explicitly identified, namely that in order for an actual proposition to be true, its sense -- the state of affairs in which it is true -- must entail a state of affairs in which the proposition is inscribed as a propositional sign. For the proposition is what is expressed by a propositional sign, and the propositional sign is a state of affairs.

"3.142 Only facts can express a sense . . ." , Tractatus

If the propositional sign, as a state of affairs, does not hold (the proposition is not inscribed), then in that state of the World, there is no proposition of which it can be said that it is either true or false. (To try to ascribe truth or falsity to a specific proposition only by name won't work; we will also have to define which proposition is named and by so doing, we will inscribe it somewhere.)

The fact that truth as a property in this sense is only applicable to actual, that is, inscribed propositions, has a remarkable coonsequence -- that truth can only be ascribed to an actual elementary proposition (Wittgenstein's Sachverhalten) in the special case that truth and inscription are equivalent properties. For details, see the forthcoming "When are EPs True?".

Constructing a truth-property

An example may help to clarify the definitions given above:

Suppose we have a microworld with states {a, b, c, d, e, f, h} and two objects which we will interpret as "media":

m1 = abc-def-h and
m2 = abch-ef-d.
(As usual, the notation abc-def-h is short for {{a, b, c}, {d, e, f}, {h}}.)

In language L1, the state abc of m1 is construed as the proposition P, whose sense is the state of affairs ac. The state abch of m2 is construed as the proposition R, whose sense is ah.

In language L2, the state def of m1 is construed as the proposition Q, which has the sense ef, and the state abch of m2 now expresses the proposition P. (The same inscription can express different propositions in different languages, and different propositional signs can express the same proposition,)

The truth-objects corresponding to P, Q, and R for each medium are given as follows:

tr_m1,P = ac-b-defh,

(here ac is the state of the medium in which it is inscribed with a propositional sign for P and P is true; b is the state in which P is inscribed but is false, and defh is the state in which P is not inscribed in m1.)

and

tr_m1,Q = abch-de-f,
tr_m2,P = ac-bh-def,
tr_m2,R = ah-bc-def.

The global or "common" truth-properties shared by these truth-objects whenever the contents of their associated media is true are given by the extensions:

Tr_L1 = {ac, ah}, and Tr_L2 = {ac, ef}.

Last Updated: July 8, 2008