Notes on truth

There are multiple theories of truth,  that is, solutions to the challenge of giving a precise definition of truth, of saying what truth is.  But for a given theory we can object that it is still not precise enough. For instance the 'correspondence theory' of truth.  We make the strict distinction between expressions in a given formal language and what they mean or express.  An expression is  a sentence if it expresses a proposition, which is an extra-linguistic entity like the Fregean Gedanke.  Truth is a property of propositions. I question that we can even speak meaningfully of sentences in isolation being true or true-in-some-language.  That such contradicts linguistic and psychological evidence. For instance:  'Il neige is true in French' can only mean; 'What Il neige means is true and Il neige is a French expression'. Sentences are said to be true only in virtue of, relative to, what they mean or express. Tarski's notion of true-in-L only make sense in the context of a completely artificial set-theoretic construction.

Let us say, for the moment, that there is a realm of proposition-thoughts and a realm of conditions, states-of-affairs. And that there is a correspondence between the two so that a proposition is true if its corresponding condition obtains. We do not see yet how to formalize this.

Technical modal logic is not circular because it is all carried out in set theory and validity for modal sentences is defined purely set-theoretically. But this is not the same as defining necessity or possibility, for instance in terms of 'possible worlds'. A necessary proposition is what correspond to a necessary condition, a condition that must obtain. Thus our burden is to give a non-circular definition of necessary condition.  In our model there is (up to $=_N$) only one necessary condition $\mathfrak{N}$. Thus we define a condition $\phi$ to be necessary iff $[\phi] =_N \mathfrak{N}$.  There is a subset (finite, perhaps) of primitives $P_i$ in each $D_i$, that is $P_i$ is finite and $P_i \subset D_i$. It follows that there is only one $=_N$ equivalence class in $D_0$ containing a $p_0$ which is always true. Bealer wishes to reduce the equivalence classes of primitives to singletons. Thus there are no plural aspects of the primitive phenomenological quality 'being green', $g \in D_1$. 'Being green and big' is a different concept from 'Being big and green' (one may not know the commutative law for conjunction) but nevertheless corresponds to the same quality.  Even at the primitive level we have singletons in general qualities and conditions are to be given a $=_N$ equivalence classes. The problem is that we must pay attention to a primitive being fined or coarse grained.  Suppose 'knows' is a primitive $k$ in $D_1$. Them $comb_{(1)} k a$ may  not define a $=_N$ equivalence class. That is, $comb_{(1)} k b$ can have different truth-values for different $b \approx_N a$. The correspondence (as in the correspondence theory of truth) between thoughts and conditions is thus the taking of equivalence classes !