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It is interesting to me that in some conventions of logic I have seen (generally, common ones), the form of logical language is designed to make “truth” implicit. For example, merely to write:

P(x)

is taken to mean, “’P(x)’ is true.

It seems like such a system is incapable of expressing an idea such as, “‘P(x) is true’ is false.” You could write not P(x), but this isn’t distinguishable from an interpretation “’P(x)’ is not true.

Why does this matter?

Well, first of all, to me it stimulates some thoughts regarding self-referential paradoxes like “This sentence is false” (more in that in one second). Second, it is interesting to me to what extent the concept of a “predicate” subsumes a large number of syntactic forms in logic. For example, are modal operators just predicates? Is the negation operator just a predicate? And in that case, algebraically, is a predicate just a symbol that can range over any sequence of symbols enclosed in parentheses? Like X(xxxxx) (which could be written in a computer-parsible way without the use of parentheses, Xxxxx.)

When someone says, “This sentence is false,” I think we can use time to avoid possible contradictions (similar to how a computer program must be evaluated and computed in a sequence of discrete moments). First, we hear, “This”. We know this is a syntactic operator that requires a complement. Next, we receive its complement, “sentence”. Now “this” is bound to “sentence”, and assuming ‘this’ (from linguistics) is evaluated with deixis, let’s assume we find it clear that “this sentence” is referring to the sentence in front of you, that you are reading. (To be fair, from the reference point of temporal processing, we haven’t seen the rest of the sentence yet, so again, there is kind of a form waiting for its closure - the human interpreter anticipates the sequence of words they are reading will terminate in a sentence, and when they arrive at its terminus, that sequence of words will become the value of the name or identifier, “this sentence” - but that hasn’t happened yet. (To be contrasted with the case where to one’s surprise it is an unterminating sequence of words we continue reading)). Then we get to is, which is not too meaningful or determining in meaning, but we certainly expect a complement for it as well. Lastly, we get to ‘false’. Like a computer program, we just now convert the arbitrary token ‘false’ into what it ‘compiles’ into (what it means). Let’s say that intuitively, we know that if a sentence is false (or, “not true”), we often have the ability to distribute the ‘not’-ness over some of its components - maybe we could have, “(Not this sentence), is false.” Or, maybe we could have, “This sentence is (not false)”, or “(is not) false”. (I don’t know if you could have “This (not sentence) is false”, but the idea interests me.)

As you “compile”, you keep re-evaluating based on what the present form of the sentence says. It may continue transforming itself. It’s essentially a recursive program (or self-rewriting rule). It gives instructions in how to change itself into new instructions, then operationalizes those instructions, ad infinitum, until it either stabilizes at a constant value (converges), or, (so far as we know), never halts (it diverges). From this perspective, what are commonly called “paradoxes” in a-temporal (or pan-temporal, as it treats all instances of evaluation as occurring at the same time with each other) semantic interpretation, corresponds to non-terminating processes in stepwise evaluation procedures. (We can consider that simultaneity of evaluation is the limit as the temporal duration of evaluation goes to zero).

If you assign the value This sentence is false to the name S1, you must now evaluate it by saying, “not S1”. We could call that sentence S2: S2 = not S1. What happens when we evaluate it now? We could choose to “solidify” “this sentence” so that its value has already been set as S1 (in programming, I believe this is called “lazy” vs. “strict/eager” evaluation, in programming).

Anyway, this is a sketch of some ideas, but getting back to the main topic.

Imagine if truth were treated as a predicate. We would have P(x) means, “x is True”. x on its own is like a subject without a predicate, in English: just “dog”. We could nest this predicate, like P(P(x)). Would this change the properties of logic in some way? Or can it be shown to be equivalent to keeping truth implicit?

Of course, there is a problem. If truth is a predicate, then how do we evaluate the truthhood/validity of instances of that predicate? P(x) says that ‘x’ is true - but we haven’t stated whether or not ‘P(x)’ itself is true or not: truth is not implicit, in our system. (We would need an infinitude of “is true” predicates: …””””P is true” is true” is true” is true”…).

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  • I seem to remember Quine going to all kinds of lengths to insist that the essence of the liar paradox persists even if you try to label it away with the kind of treatment you have suggested. 'Yields a falsehood when appended to its own quotation yields a falsehood when appended to its own quotation' was the kind of thing he mentioned, which can't be parsed in quite as many ways as 'this statement is false'. Commented Jan 14 at 7:35

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Your question is quite wide-ranging. To some extent, the issue of whether truth is a predicate leads back to the philosophical question, What is truth? And this in turn informs the issue of how to represent truth in a logical framework. One basic problem is that there is no consensus among philosophers over how to understand or analyse the concept of truth, and this leads to disagreement over whether 'true' is a substantial property of propositions or whether it should be understood in some other way.

Consequently, some logicians are happy to use a predicate True(P) to indicate that the proposition P has the property of being true, while others do not. A popular option is a family of related theories called 'deflationary' under which saying that some proposition "is true" is redundant or is eliminable in some way.

Tarski proposed a convention that is widely accepted and that seems to relate to what you mean by saying that truth is implicit.

⌜ϕ⌝ is true if and only if ϕ.

But this is just an adequacy condition and does not amount to a theory of truth. It leaves open a wide range of choices for how to account for truth in a formal system. Tarski showed that if we have an object language that is rich enough to include arithmetic and we try to include a truth predicate within it then this leads to inconsistency. Tarski's own proposal for avoiding this is to form a hierarchy of language, metalanguage, metametalanguage, etc., so that within each level we can say of a sentence in the level below that it is true (or false). This avoids the problem of the liar paradox, because "this sentence is false" violates the hierarchy and so is illegitimate.

But Tarski's approach is not the only option by any means. Kripke proposed a theory of truth based on model theory that does allow for a truth predicate within the object language. The catch is that some sentences fail to have a truth value, including the liar sentence. Other approaches explicitly permit self-reference and avoid inconsistency in other ways, through stability constraints or by axiomatisation. Dialetheists are willing to accept that some contradictions, including the liar sentence, are true.

There is a handy, though slightly dated article on truth predicates by Michael Sheard, "A Guide to Truth Predicates in the Modern Era". Journal of Symbolic Logic, 1994, Vol. 59 (3), 1032-1054.

Kripke's classic paper is "Outline of a theory of truth", Journal of Philosophy, 1975, vol. 72, 690-716.

The articles on truth in the Stanford Encyclopedia and Wikipedia have useful introductions to the subject of truth generally.

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OP: Imagine if truth were treated as a predicate. We would have P(x) means, “x is True”. x on its own is like a subject without a predicate, in English: just “dog”.

The above would be like saying P(door) means 'door' is true, however you need to say something like "the door is closed" which may then be true or false.

Quoting from Heidegger's What is a Thing?, pages 36-38:

Truth is a fitting to things, a correspondence (Übereinstimmung) with the things. But what is now the character of what fits itself? What does the corresponding? What is this about which we say it may be true or false? Just as it is "natural" to understand truth as correspondence to the things, so we naturally determine what is true or false. The truth which we find, establish, disseminate, and defend we express in words. But a single word—such as door, chalk, large, but, and—is neither true nor false. Only combinations of words are true or false: The door is closed; the chalk is white. Such a combination of words is called a simple assertion. Such an assertion is either true or false. The assertion is thus the place and seat of the truth. Therefore, we likewise simply say: This and that assertion are truths. Assertions are truths and falsities.

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Assertion [...] is threefold: a proposition giving information and which, when carried out vis-à-vis others, becomes communication. This communication is correct when the information is right, i.e., if the proposition is true. The assertion as a proposition, as an assertion of "a, b of H," is the seat of truth. In the structure of the proposition, i.e., of a simple truth, we distinguish subject, predicate, and copula—object, assertion, and connective (Satzgegenstand, Satzaussage, und Verbindungswort). Truth consists in the predicate's belonging to the subject and is posited and asserted in the proposition as belonging. The structure and the structural parts of the truth, i.e., of the true proposition (object and assertion), are exactly fitted to that by which truth as such guides itself to the thing as the bearer and to its properties.

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  • Heidegger isn't someone I would quote on such issues, but OK.
    – user71009
    Commented Jan 16 at 15:58

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