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.
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.
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?
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.
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.
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".
@p ⟷ @@p
@¬¬p ⟷ @p
@(p ∧ q) → @p
@(p ∧ q) → @q
@(p ∧ q) ⟷ @p ∧ @q
@¬@¬p ⟷ @p
@¬@p ⟷ @@¬p
@¬@p → @¬@¬@¬p
@¬p ⟷ @¬@p
Fp := @¬p
Weakening the requirement to allow
p to stand alone without
@(p ∧ q) ⟷ @p ∧ @q
p ⟷ @p
p ⟷ ¬@¬p
¬p ⟷ ¬@p
¬p ⟷ @¬p
@¬p ⟷ ¬@p
Some equivalences between the negation connective and
¬@ ≡ ¬
@¬ ≡ ¬@
@ ≡ ¬¬
p ⟷ ¬@¬¬@¬p
@¬ ≡ ¬¬¬
Tinkering with a "falsity predicate":
Fp := @¬p
FFp ⟷ @¬@¬p
@ across connectives:
@(p ⟷ q) ⟷ (@p ⟷ @q)
@(p v q) ⟷ (@p v @q)
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.
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
@ and appended
@ to some sentences and
@¬ to others. (This might count as a replacement truth assignment.)
Wbe a set propositional variables (
Ibe an operation that prepends
@to every sentence of
w+is a subset of
@¬to the rest.
Icomprise the set
wffand so is
@pdetermines whether the sentence
pis true ("It is true [actual] that
p"). How do further negations work here? One could just run the operation
Iagain but then we have no way to determine whether
@pis true (since both
@¬@pwould exist in
@pin a consistent way (if assigned
@as an affix then
@@pis a wff but not
@¬p). Perhaps that's the best route. So, then if
¬@¬pwould be as well.
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):
I(p)=T ≡ @por
@p := I(p)=T
@ 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:
@ ≡ ¬¬
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 of any of the rest plus negation.
p → q ≡ ¬(p ∧ ¬q)
p → q ≡ ¬p v q
¬(p ∧ ¬q) ≡ ¬p v q
At the level of operators (which logical connectives are), these are equivalences. So, truth would just be negated negation:
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):
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).
p := @N(p)
q := @¬@N(q)
Some conceptions of the T-Schema:
@@p ⟷ @p
@p ⟷ p
@N(p) ⟷ @p
The paradox should be derivable using any of those.
@N(@¬@N(q)) ⟷ @¬@N(q)
@N(q) ⟷ @¬@N(q)
@@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))
@¬@N(q) → @(p ∧ ¬p)
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).
Is there a deeper insight here? Maybe.
r := ¬N(r). (This is not itself a paradoxical expression.)
N(r)maps to sentence
r := ¬rcannot be formed (as a wff) due it not being a wff,
¬ris a wff only if
ris a wff (
rcan’t be defined at the same time as
I(¬r) = Tonly 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.
In the philosopher's, logician's, and mathematician's verbiage:
☐☐phas no self-recursion - the
☐symbol is applied to
☐pgenerating a new sentence.
'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