# Truth/actuality as an operator

Frege claimed that "it is true that" adds nothing to the actual meaning of an assertion, and following him along this line are prosentential theories of truth. However, I wonder if this is so. Imagine a joke like:

Why did the utility monster cross the road?

For the greatest good of the greatest number.

Now the punchline can be 'expanded' to, "For the greatest good of the greatest number is the reason the utility monster crossed the road." This already would be a clunky way of delivering the punchline, like a preamble to the kind of 'explanation' that defeats the purpose of telling a joke. But now imagine if someone said, "It is true that for the greatest good of the greatest number is why the utility monster crossed the road." One could read 'is true' here as shorthand for 'the correct punchline to the joke is,' I guess, but even so, and otherwise anyway, there seems to be a little more metaphysical 'oomph' to the last version of the sentence.

Doubtless 'it is true that' does not reduce to 'it is not being said as a joke that,' however. Now, while tinkering with paraepistemic logic, I've reached a juncture where I 'need to decide' between having an explicit actuality operator, and not having one, more or less. There's some 'risk' that a background principle would spawn infinitely many actuality operators, somehow, if even one of these is assumed into this system. So, I'm going to describe some of the subproblems I'm facing and pose my question in those terms. In other words: this is going to take a while to get to the point of...

Existential quantification

I used to be uncertain whether there was some sort of 'trickery' involved in standard existential quantification. Having a special symbol with a special syntax was fine and all, but when explained as a condensation of 'there exists an x such that...', I felt like this notion somehow overlooked the fact that 'there is' sentences, in English, are directly convertible into 'is there' sentences, so that whatever 'is there' means, it is still a predicate and, in turn, a property. However, I have had occasion to reconsider that doubt: for 'there' is an indexical function, more or less, after all. And I am highly tempted by an interleaving of indexicality and actuality, as general forms/concepts in a theory (not in a principally Lewisian way, though, if partly inspired by him nevertheless).

At any rate, I will be working with a logic where quantifier symbols themselves, formally, do have a different enough syntactic relation to other parts of assertoric functions in which they appear, to work like they are 'supposed' to.

So now this opens the door to the basic form of the question: how much difference is there between an actuality operator and an existential quantifier? Or: why have the former, if you have the latter?

Even in zeroth-order logic, you can indicate actuality without using a special operator: you just drop any prefixed "possibly" operators and voila, there's your "actually" operation, apparently.

The plot thickens

One of the most basic aspects of the system is referred to as imperative prelogic. The idea is that, while imperative syntax only mediates genuine logical inference with things like the 'see to it that' or 'make it the case that' operators, which are only able to perform their role as mediators because their operands are assertoric functions, there is still something like "pre-inference" involved in transitions between some imperatives. This belief is most fundamentally cashed out by observing that

The → operator, while in tension with the 'sound' of imperative syntax (at least in (American) English) when A, B in AB are imperatives, can be 'logically tethered' to a different operator that does fully 'comport with' imperative syntax. Let us denote this other operator ⥅. (The reason for this is that in the overview of imperative prelogic, the idea is introduced via having 'do' and 'do not' correspond to two operators on action terms a, with ±a for the first and ¬a for the second (except that "¬" is supposed to be underlined, too; the whole underlining thing is a nod to an attempt at imperative logic I remember seeing once where the semiotics at play was in underlining sentences).) So xy is meant to be shorthand for, "From y, infer x," or, "Infer x from y."

Another crucial ur-element of the system is erotetic (pre?)logic. I incorporated a function which I am for now calling the Åqvist-Hintikka function, into the system, which takes questions for its inputs and outputs corresponding epistemic imperatives. This is not quite 'true to' that Åqvist/Hintikka thesis, here, which I think is strictly taken as a reduction of erotetic to epistemic-imperative structure; but consider a question like, "Fight the utility monster?" Unless taken as always shorthand for, "Do you want me to fight the utility monster?" it is hard for me to take this question as being reducible to, "Let me know whether you want me to fight the utility monster." At any rate, the synthetic apriority, as it were, of the relationship between erotetic functions and epistemic imperatives, grounds them as importantly in the system as if they were analytic reductions (or perhaps even more importantly...).

The reason that this all complicates the issue of an actuality operator is how it 'bleeds into' the power of the paraepistemic level of the system. The axiom I am working with is

Every paraepistemic operator encodes an imperative characterized in terms of which specific operator is doing the encoding.

So, "It is known that..." goes with, "Know that...", "It is axiomatic that..." goes with, "Axiomatize the proposition that...", "It is understood that..." goes with, "Understand that...", and so on and on down the line.

Now try assimilating 'it is true that...' to the above matter of the paraepistemic constellation, as a modal operator equal to 'it is actually true that...' (or whatever). But what imperatives could modal operators correspond to? I.e., it seems to me as if they don't, and that this is a serious distinction between modal and paraepistemic operators. On 'the other end of the spectrum,' so to speak, consider also deontic logic. OBA could be taken to correspond to several imperatives: "Make it true that A," "Obligate A'ing," or maybe, "Discharge the obligation to A" (this one itself is not far from the first option, though). But again, there seems to be enough difference between the localized Åqvist-Hintikka maps of the paraepistemic and the deontic operators to say that deontic logic is another 'dimension' of logic just like modal logic is vs. the paraepistemic sphere.

There is, though, an even more intricate and for me puzzling option. The imperative of inference is not the only prelogical imperative in the system. For example, there are erotetic imperatives like, "Ask question X," and assertoric imperatives like, "Assert that T." It is the latter that I am tempted to identify with the Åqvist-Hintikka map of the wished-for truth operator, viz. "It is true that A," goes with, "Assert that A."

Unfortunately, when other erotetic principles are brought into play, we end up with a more complex operation 'to the same effect': "When asked question X, reply with answer Y." This is guaranteed to be a legitimate function of the system, and otherwise 'grounds' the prelogical "assert that..." interlude in the system (i.e. the erotetic prelogical operation yields the imperative prelogical one, it seems, at least as a 'justifiable' directive, so to say). So I have no reason to doubt an "assert that" operator of some form, in the system. However, notwithstanding the relationship between assertoric functions and truth-aptitude (those functions are the truthbearers), a direct truth operator, i.e. as a direct actuality operator, otherwise seems to be of a piece with the other modal operators as far as the Åqvist-Hintikka function is concerned, wherefore it would lack a 'useful' relevant directive, and hence would be as 'uninformative' as Kant thought the modal categories were altogether. So my question is: Do the premises of this system, as stated, favor or disfavor a truth/actuality operator? {Worst-case scenario: the system is easily shown to be inconsistent, and since at present it is not 'strictly' paraconsistent in intent, such inconsistency would trivialize the system.}

• There's a subtle difference between truth and actuality, the latter is of modality nature which is handled by modal logic while the former is a more universal perennial conceptual topic, it's logically though not necessarily true pig possibly can fly, even actually in this world it cannot. So if you're not a deflationist, surely you can be motivated to favor truth predicate (not as a modal operator as reasoned above) such as the axiomatized Kripke–Feferman theory. If you adopt deflationism, you'd better only have truth values/gaps of abstract truth objects like Frege to be fully consistent... Jan 25, 2022 at 6:52
• This reminds me Kant's quip:"we do not add a thing to the concept of a thing when we further state that the thing is". Those who agree, like Frege, fully vest existence into the quantifier, those who disagree, like Meinong, add the existence predicate and deflate the quantifier. What you do depends on your semantics, if you fictionalize the domain of discourse you need the predicate to tell which things are real, if not, you must have Russell's paraphrase for fiction, see What are the counterexamples to existence is not a predicate? Jan 25, 2022 at 7:32
• @DoubleKnot, unfortunately or not, some of my other premises seem to lead straight to a truth predicate/property and a truth operator and a truth value (something similar seems to happen in my deontic logic, where we have an obligation operator on propositions, an obligation state for objects ("is obligated to..."), and a nominalized obligation (i.e. "is an obligation")), and none of these (or the many other such things in the system besides) strikes me as having to be more fundamental than any of the others. Jan 25, 2022 at 13:33
• @Conifold, I felt like there was something oddly similar/"symmetrical"(?) about Frege's analysis of existence-as-a-predicate and truth-as-a-predicate. And I agree with Frege/Russell in coidentifying existence of objects with the reference of terms. But moreover, there's still so much more about quantifiers that I need to consider. Like, I (personally "historically") have been attracted to a plural-quantification solution to Russell's paradox, yet there are also motives for using general quantifiers in my system, and so on and on. Jan 25, 2022 at 16:02
• Over time, I have come more and more to rely on the insight that mathematical logic (and formalisms in general) representations idealizations of the subject matter, rather than mere accounts of it. For example, including LEM in your logic is basically tantamount to excluding propositions like "this sentence is false" and "roses are red" from your domain of inquiry, because True and False are not enough to deal with paradoxical or fuzzy propositions. Likewise, a logic without the proper modal operators can't handle the associated modal propositions. The point of this is that statements like Jan 25, 2022 at 22:49