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Adam I. Gerard
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Criticisms of Truth as a Sentential Operator

Just a draft - please excuse any errors or typos - there are likely more below than usual.

The idea that truth is [just] a sentential operator (the prosentential view) is an interesting one and a conception of truth that was defended by a colleague of mine in graduate school. (Their view was a stronger thesis: that Truth amounts to Actuality - that "That is actual" is just "that is true" and vice-versa).

Such a conception flies in the face of all the great mathematicians, philosophers, and logicians of the last two centuries (only a few philosophers have ever defended prosententialism and none of them great): Frege, Tarski, Gödel, Lukasiewicz, Wittgenstein, Russell, Feferman, Kripke, etc. all defended variants of opposing views.

I work through some of the weaknesses of that specific, stronger, view below.

Truth Values

The first line of criticism converges from three lines of thinking: (1) formal syntax, (2) the coherence of logic without truth-values and/or truth reduced to a sentential operator, and (3) formal semantics.

  1. If truth is [just] a sentential operator that we add to a logic. What then are truth values?
  2. Isn't there room for logic without a truth-operator or predicate? (E.g. - Classical Zero-Order Sentential Logic, Boolean Algebra, and so on.) This is a standard point of demarcation between Zero, First, and Higher-Order logics. If not, all of mathematics and logic would need to be rebuilt. The absence of such translations is tantamount to a denial of eliminative prosententialism (or the strongest variant).

Logical Connectives

How are logical connectives defined if truth is just a non-logical sentential operator?

Typically, logical connectives are defined by recourse to truth-values. Then a truth-predicate is added to a language and defined, in part, by the set of sentences assigned the appropriate truth-value. In this way, one ascends to truth (metaphorically speaking).

If truth-values are eliminated, then how can one define the logical connectives to define the prosentential conception of truth in the first place?

Modality

Shouldn't the concept of truth be independent of modal notions? (Modal operators and all non-logical sentential operators are dependent on truth - for example, ☐p is true whenever a proposition is true at every world - truth-at-a-world is more fundamental than modal notions.)

Independently, shouldn't we be able to evaluate the truth of modal concepts without first operationalizing/formalizing those modal concepts themselves (e.g. actualism, possibilism, the semantics of modal frames, etc.)?

In other words, shouldn't we be able to debate and consider the truth of modal concepts like necessity without casting them into a formal system? We tend to frame modal conceptions using formal systems but we should be able to reason about those systems prior to formalization. (Formalization represents or captures some of those considerations.)

But, if truth reduces to modal conceptions (formally or otherwise) this would seem to be impossible. We are required to formally define modal conceptions of actuality in order to get to truth formally. This seems viciously circular at best and contradictory, viciously circular, and mistaken at worst.

Formal Definitions

Biconditionals imply a left-hand expression's truth-value be the same as the right-hand expression's. This is true of object languages as well as metalanguages.

If truth-values are eliminated, how are the basic logical connectives themselves defined in the meta-language and hence, the object language?

This is a meta-level critique that parallels problems arising within the object language (by itself) noted above.

Some Formal Considerations

Let's explore the view a bit more and see if we can shore it up some.

A fully eliminative view (some logical validities, equivalences) that starts "from scratch" (attempting to drop any modal prerequisites to improve the defensibility of the view):

@ stands for "It is actual that".

  1. @p ⟷ @@p
  2. @¬¬p ⟷ @p
  3. @(p ∧ q) → @p
  4. @(p ∧ q) → @q
  5. @(p ∧ q) ⟷ @p ∧ @q
  6. @¬@¬p ⟷ @p
  7. @¬@p ⟷ @@¬p
  8. @¬@p → @¬@¬@¬p
  9. @¬p ⟷ @¬@p
  10. Fp := @¬p

Weakening the requirement to allow p to stand alone without @:

  1. @(p ∧ q) ⟷ @p ∧ @q
  2. p ⟷ @p
  3. p ⟷ ¬@¬p
  4. ¬p ⟷ ¬@p
  5. ¬p ⟷ @¬p
  6. @¬p ⟷ ¬@p

Some equivalences between negation connective and @:

  1. ¬@ ≡ ¬
  2. @¬ ≡ ¬@
  3. @ 𠪪
  4. p ⟷ ¬@¬¬@¬p
  5. @¬ ≡ ¬¬¬

Tinkering with a "falsity predicate":

  1. Fp := @¬p
  2. FFp ⟷ @¬@¬p

Distributivity of @ across connectives:

  1. @(p ⟷ q) ⟷ (@p ⟷ @q)
  2. @(p v q) ⟷ (@p v @q)
  3. And so on for the others (where appropriate).

Anyway, in the absence of any lurking direct contradictions from the above, there's still a question about how truth-assignments would be made on a fully eliminative view. Once we have that sorted out, propositions can enter into the above inference rules/axioms/equivalences.

Having Some Fun

Let's attempt to sketch out a way to do resolve that worry (and potentially push the borders of logical knowledge outward to include alternatives to the truth-value approach that presently dominates).

Consider if we just replaced T with @ and appended @ to some sentences and to others. (This might count as a replacement truth assignment.)

  1. Let W be a set propositional variables (p, q, r, ...).
  2. Let I be an operation that prepends @ to every sentence of w+ where w+ is a subset of W and to the rest.
  3. All sentences with affixes from I comprise the set wff.
  4. If p is a wff then @p is a wff and so is @¬p.
  5. Note that this gets a bit tricky. Since @p determines whether the sentence p is true ("It is true [actual] that p"). How do further negations work here? One could just run the operation I again but then we have no way to determine whether @p is true (since both @@p and @¬@p would exist in wff).
  6. We could preselect some sentences to only receive @p in a consistent way (if assigned @ as an affix then @@p is a wff but not @¬@p nor @¬p). Perhaps that's the best route. So, then if @p were assigned, ¬@¬p would be as well.
  7. Then one could establish the logical equivalences above via such a construction.
  8. Like standard interpretation assignments (which can be represented as Markov chains or via Specification Spaces à la Kit Fine), the process above can be mapped for the entire possibility space of subsets of W (e.g. - for every subset that w+ could be and seen at once).

This gets us to a place where some coherent interpretation assignment equivalent might dwell. That might address some of the concerns from a grammar/syntax/semantics standpoint but doesn't address many of the others. (How would this work at a meta-level? Why are we using modal notions to define truth and not the other way around?)

Some further damning considerations:

I think given the other arguments here (and the unresolved issues above) that the prosententialist would probably accept something weaker - e.g. keeping fundamental truth-values (something that truth-predicate theorists do as well):

  1. I(p)=T ≡ @p or @p := I(p)=T

So, @ depends on some more fundamental operation (e.g. the truth-assignment) and mirrors this 1-to-1.

But this just equates the sentential operator with the mapping of the interpretation function (which is essentially an extension) - that is to say, this appears to equate the sentential operator with something like a predicate.

Even if that criticism is refuted, the @ operator appears to reduce (full stop) to double-negation:

  1. Per the above: @ 𠪪

So, either such an operator is really just a predicate (and, that tracks the assignment of the underlying truth-value T), or it's just shorthand for double-negation.

For predicates, this is not such a problem, since predicates enter into inferences of double-negation elimination (for example, the formulations of the T-Schema involve the use Gödel Numbering/name-forming function not just the sentence level expressions).

For logical connectives (sentential operators) it is. Consider the well-known property that one can define any logical connective out any of the rest plus negation.

  1. p → q ≡ ¬(p ∧ ¬q)
  2. p → q ≡ ¬p v q
  3. ¬(p ∧ ¬q) ≡ ¬p v q

At the level of operators (which logical connectives are), these are equivalences. So, truth would just be negated negation:

  1. Bringing us back full circle to the question above about what negation and the logical operators are supposed to be.
  2. This would also imply that any logic for which double-negation elimination is false (or invalid) along with its entailments (such as the Law of the Excluded Middle) would not have any truth concept whatsoever. It would imply that all Intuitionist and Constructive logics would not only be wrong but impossible. (And, that their inventors were not only wrong but technically unskilled at even the most basic notions within logic.)

Liar Paradox

Can we construct sentences that exhibit the Liar Paradox in such a system?

Yes, in a direct manner (but slightly more roundabout than against naive truth-predicate approaches):

  1. Introduce a name-forming function N(). We do not require that truth be a property of the name of a sentence here (or even sentences). We simply introduce this (e.g. - by itself). Why would we do this? Because we need to talk about sentences (whether or not Truth is just a sentential operator or not).
  2. Sentence p := @N(p)
  3. Sentence q := @¬@N(q)
  4. It trivially introduces the Liar Paradox.

Some conceptions of the T-Schema:

  1. @@p ⟷ @p
  2. @p ⟷ p
  3. @N(p) ⟷p
  4. @N(p) ⟷ @p

The paradox should be derivable using any of those.

Proof:

  1. @N(@¬@N(q)) ⟷ @¬@N(q)
  2. @N(q) ⟷ @¬@N(q)
  3. @@N(q) ⟷ @¬@N(q)

i. We've already arrived at a direct contradiction. But we can make that a bit more explicit.

ii. → @@N(q) ∧ @¬@N(q)

iii. → @@N(q)

iv. → @¬@N(q)

v. @¬@N(q) → @(@N(q) ∧ ¬@N(q))

  1. @¬@N(q) → @(p ∧ ¬p)
  2. q → @(p ∧ ¬p)

One purported advantage of prosententialism is that it can dispel the Liar Paradox from the claim that diagonalization only applies to predicates.

But diagonalization applies to the name-forming function not just the truth-predicate (belying the lack of technical insight in such approaches). And, can therefore still be trivially introduced.

Does this problem also affect the opposing predicate theories? Yes, unless they employ my solution (which treats truth as a predicate that accepts the name of a sentence resolving this same issue via its treatment of Alethic Paradox).

More About Diagonalization

Is there a deeper insight here? Maybe.

  1. Sentence r := ¬N(r). (This is not itself a paradoxical expression.)
  2. N(r) maps to sentence r.
  3. Note that sentence r := ¬r cannot be formed due it not being a wff, ¬r is a wff only if r is a wff (r can’t be defined at the same time as ¬r) and I(¬r) = T only if I(r) = F.

My previous proposal addresses alethic paradox (I'm the first person to formally define what a truth paradox is - a contradiction whose shortest proof involves the T-Schema or F-Schema). We see that there are still problems resulting from the name-forming function.

I've mentioned that restricting diagonalization may be a better result. This would be relevant to Second Incompleteness, the whole gambit of Semantic Paradoxes, and the Liar Paradox.

We can use two predicates: the standard notion above and a restricted operator. The T-Schema would then apply only to the restricted operator (and we might be able to drop the conditionalization of the T-Schema). Other relevant notions can also be treated in such a fashion.

Appendix

In the philosopher's, logician's, and mathematician's verbiage:

Predicates:

  1. Predicates apply to names (constants, variables) and not to objects directly.
  2. Predicates are to properties as names (referrers) are to objects (references).

Metalinguistic predicates:

  1. Metalinguistic predicates apply to the names of expressions.
  2. This is typically handled using Gödel numbering (following Tarksi).
  3. E.g. - "Talking about (parts of) language(s)."

Sentential operators:

  1. Do not take arguments to compute a value. They do have arities (places).
  2. They are infixed, suffixed, or affixed to a wff to generate another wff.
  3. The computed value of an expression is always handled through a function - e.g. an interpretation assignment. For example, this is the role that the [conceptually independent] truth-table plays.
  4. Modal Operators are treated as such.
  5. One cannot diagonalize into a Sentential Operator. For example, consider the Modal Operator: ☐☐p has no self-recursion - the symbol is applied to ☐p generating a new sentence.

Predicates:

  1. A predicate takes arguments and computes a truth-value based on whether the arguments are contained within its extension.
  2. A function - as in "truth-functional".

Conventions:

  1. Logical Connectives are usually treated as Sentential Operators (the arguments above should apply regardless of which kind of sentential operator they are or if they are at all).
  2. The more general word 'operator' has been used in lots of ways throughout mathematics. Many philosophers use the term 'name-forming operator' when a more accurate definition would be 'name-forming function'.

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