Solutions to the Liar Paradox according to which paradoxical sentences are meaningless face all sorts of challenges. For one thing, the partisan of such a solution needs to have a plausible error theory to explain the widespread intuition that such sentences are meaningful. For another thing, they must find a way to defuse familiar "revenge" Liars, like $, below.
$ The sentence marked with a dollar sign is either false or meaningless.
These are major obstacles, and whether or not they can be plausibly overcome is a subject for another time. What I want to focus on is an objection which I find far less initially plausible, but which I hear a surprising amount of the time.
It goes, more or less, something like this:
"Even if Liar sentences are meaningless, they're still not true, right? Meaningless claims aren't true, so that solution doesn't even help with the Strengthened Liar. ('This sentence is not true.')"
Now, in whatever sense in which we are speaking sense when we say "meaningless sentences aren't true," surely it would be exactly equally correct to say that "meaningless sentences aren't false." Meaningfulness is surely a prerequisite for falsity, just as it's a prerequisite for truth.
Forget, for a moment, about the Liar and its kindred semantic paradoxes. Let's just think about a normal case of a sentence whose meaninglessness is much less controversial, like "Green ideas sleep furiously."
Now, given the two claims we just endorsed:
(1) Meaningless sentences aren't true.
(2) Meaningless sentences aren't false.
Given these two claims, Disjunctive Syllogism, Conjunction-Addition and the Principle of Bivalence (for every P, either P is true or P is false), we can easily derive a contradiction about a normal, non-paradoxical meaningless sentence like "Green ideas sleep furiously."*
Let's symbolize "Green ideas sleep furiously" as G. Given Bivalence, we've got our first premise:
1. Tr(G) v F(G)
Symbolizing (1), above, we've got our second premise:
From 1, 2 and Disjunctive Syllogism, we can conclude:
Symbolizing (2), above, we get:
And finally, of course, from 3, 4 and Conjunction-Addition, we conclude:
5. F(G) & ~F(G)
So, given Bivalence and a couple of basic logical rules, the claim that meaningless sentences aren't true or false entails contradictions. Perhaps the very notion of meaninglessness as a separate category from truth and falsity is inconsistent!
But wait. Even if we're willing to give up on the claim that any sentence anywhere is meaningless, what about questions. Surely questions exist. Can questions be true or false? How about bits of burning candle wax? Are they true? No? Are they false? Also no? Well, if G symbolized not a meaningless declarative sentence but a question or a bit of burning candle wax, we could use precisely the same five-step proof to derive an outright contradiction about the semantic status of the question or the bit of burning candle wax.
Clearly, something has gone horribly wrong in our reasoning.
Here's what it is:
When we say "meaningless statements aren't true," we might be making one of two claims:
1-Let M(P) mean "P is meaningful." For every P, if ~M(P), then ~Tr(P).
2. Meaningless sentences aren't the sort of thing to which truth talk meaningfully applies.
If you mean 1, you're confused. (It's significant that no dialetheist has ever used the proof above as an argument for the existence of true contradictions. And if that argument were available with them, why would they bother to swim in the murkier waters of semantic paradoxes?) When we try to symbolize a meaningless statement and perform logical operations on them, we're engaged in a nonsensical category mistake, of exactly the same sort that we'd be engaged in if we tried to symbolize and perform logical operations on a big of burning candle wax.
If you think meaningless sentences aren't true, and when you say that, you actually mean to assert of every meaningless sentence the negation of the claim that that sentence is true, you are necessarily saying something meaningless. After all, given the universal intersubstitutivity of P and Tr(P) for every P, if you say that "Green ideas sleep furiously" is not true, meaning ~Tr(P)--where P is "Green ideas sleep furiously"--then you are, in effect, asserting ~P. As the philosophers of the Vienna Circle were so fond of pointing out, the negation of nonsense is nonsense.
Unless you're willing to accept that green ideas fail to sleep furiously--and that there are true contradictions about the truth-value of every meaningless sentence--when you say that "meaningless statements aren't true", you'd better mean it in sense 2.
Now, like I said before, none of this helps the partisan of the meaninglessness view against revenge paradoxes crafted to fit the details of the view. (For example, given the discussion above, one might wonder about the following sentence, which we could call The Babbler: "This sentence is not the sort of thing to which truth talk meaningfully applies.") And that's fair enough.
Still, whether or not they are ultimately viable when we really look into the ins and outs of revenge paradoxes, intuitive difficulties and so on, meaninglessness solutions can't be batted away with the blunt instrument of pointing out that meaningless sentences aren't true.
*At least, that's one that most people take to be meaningless. (E.g. another commonly heard response to claims that Liar sentences are meaningless is "wait, you don't mean meaningless the same way that 'Green ideas sleep furiously' is meaningless, do you?") If, however, you hold semantic views on which 'green ideas sleep furiously' comes out as meaningless, please accept the following as a substitute:
Sentence S1: 'Green swimming red night fun fun fun!'**
**"But wait," I can hear some of you saying, "Sentence S1 isn't even well-formed!"
Well, I'd argue that any invocation of "well-formedness" as a consideration here misses several points at once. "Well-formed" means something fairly specific for symbolic formulas. It's not clear what it's significance is supposed to be when we start throwing it around with reference to natural language sentences. The closest natural language equivalent of the formation rules of formal systems would be the rules of grammar, and conformity to those is clearly neither necessary nor sufficient for meaningfulness. If someone accuses another person of having done something wrong, and the person being accused responds with Sentence S2:
Sentence S2: "Like hell I did!"
....everyone knows what is meant. If the accuser, trying to catch the accused person in an inconsistency, formalized Sentence S2 with a Greek letter, did the same with some of his other statements and and drew out some logical implications, no one would think the accuser was in the grips of any kind of deep conceptual confusion.
Now, someone trying to desperately hold on to some significant role for natural language "well-formedness" could try to say that the difference is that there are grammatically "well-formed" sentences that mean the same thing as Sentence S2, whereas no grammatically well-formed sentence means the same thing as Sentence S1, but, of course, by definition, no grammatically well-formed sentence *ever* means the same thing as any meaningless sentence, because meaningless sentences don't mean anything. That's what we mean when we call them "meaningless."
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It is confusing that you use P to refer to sentences. You ought to use S and reserve P for propositions.
What about simply disbelieving bivalence for sentences? That's what I think. Some sentences are neither true or false, that should be obvious. Meaningless sentences are neither true or false (and by that I don't mean that it is a category error to speak about true or false sentences). Without bivalence about sentences, your argument is unsound. I see no need to deny ¬M(S)→¬Tr(S). I think that ¬M(S)→(¬Tr(S)∧¬F(S)) is true.
Do you accept the T-Schema and the universal intersubstitutivity of equivalents? If so, even without Bivalence, you're going to face the following problem about accepting 'if ¬M(S)then ¬Tr(S)':
The negation of a meaningless statement is, one would think, itself meaningless--if it's true that green ideas *don't* sleep furiously, the claim that they do is just false, not meaningless--but, given the T-Schema and intersubstitivity of logical equivalents, '¬Tr(S) entails ¬S.
I don't know what the universal intersubstitutivity of equivalents is but I can guess. I looked up the T-schema though my guess was that it is just the semantic theory of truth, and I was right.
I don't know if I accept them or not. Could you please derive the contradiction a bit more formally for me? I'm having trouble following your reasoning.
Let C (for Chomsky) =df the sentence "Green ideas sleep furiously.".
For all sentences, if it is not the case that a sentence is meaningful, then it is not the case that that sentence is true.
It is not the case that C is meaningful.
3. ¬Tr(C) (1, 2, MP)
Thus, it is not the case that C is true.
For all sentences, if it is not the case that a sentence is true, then it is not the case that that sentence.
5. ¬C (3, 4, MP)
It is not the case that C.
I don't know if it is something like this you have in mind. In any case, I do not accept (4).
I'm not claiming to derive a contradiction here, just an absurdity.....which is that, at the end of the proof, you derive ~C. Which is, presumably, precisely as meaningless as C itself.
The T-Schema (or Disquotation Schema, etc.) is the universal claim that "'P' is true iff P." "'Snow is white' is true iff snow is white", "'grass is green' iff grass is green," etc., etc., etc.) Pretty rare to find anyone who denies that, though a few people are driven to it by their inability to get around the Liar Paradox in any other way. (E.g. Gil Harman takes that line in "Change of View.")
The universal intersustitutivity of equivalents is just the claim that, if two claims are logically equivalent--"Claim 1 iff Claim 2"--we can substitute them for each other with no loss of truth. Even without that, though, you can't reject (4) unless you're willing to reject either(a) the T-Schema, or (b) Contraposition (which is just Modus Tollens plus Conditionalization....i.e. from "if P, then Q," you can derive "if not-Q, then not-P"). If 'C' is true iff C, and 'C' is not true, then ~C.
Actually, as you stated earlier:
so the biconditional holds
So from 3 and
You can derive a contradiction. First contraposition on the biconditional above
6. (∀S)M(¬S)↔M(S) [from 3, CP, converse(this inference has a name I take it)]
7 is obvious to me. Then
8. M(¬C) [from 4, 7, MP]
9. M(C) [from 6, 8, simp., MP]
which contradicts our original premise
10. 9. ¬M(C)
So somewhere along this path, something is wrong. Or maybe I did some inference wrong. You can hopefully tell me that, you're the expert. :)
I see about the T-schema. I look it up on Wikipedia and get multiple different sentences/propositions/principles. If you just meant the (∀S)Tr(S)↔S then pretty much nobody will disagree I think. Except monist proposition theorists, of course, since they disagree that sentences have truth values at all.
I'm curious. You are defending classical logic but you are always going on about how this and that solution doesn't work in response to the liar paradox, curry etc. Presumably since you are defending classical logic, you have some other solution in mind that actually works (or you think works). Is that so?
From my perspective, what's gone wrong is that we're committing a category mistake by trying to symbolize meaningless strings of words and perform logical operations on them, which is exactly as absurd as trying to apply truth talk to bits of burning candle wax. I think this is brought out sufficiently by the derivation of ~C--which should be meaningless!--but the contradiction helps. Thanks for that.
I argue that "'green ideas sleep furiously' is true" is precisely as meaningless as "green ideas sleep furiously" itself. (Certainly, if you accept Quine's deflationary maxim about truth, that when we say that "'snow is white' is true", we are saying nothing above and beyond attributing whiteness to snow--i.e. that truth attributions don't add any meaning to the sentences to which they are attributed, but are simply a device for asserting the original sentence--it seems to pretty straightforwardly follow that attributions of truth to meaningless sentences are themselves meaningless.) Given that adding a "not" to the mix doesn't convert a meaningless sentence into a meaningful one, it follows that "'green ideas sleep furiously' is not true" is also meaningless.
Of course, there is a sense in which we are clearly talking sense when we say things like "meaningless sentences aren't true", but I think that can adequately be parsed as something like "meaningless sentences aren't the sort of thing that truth talk applies to" (compare: "ideas aren't the sort of thing that color talk applies to").
And yeah, I do have a preferred solution, one I haven't talked about here, mostly because I hate to sound like I'm claiming to 'have an elegant proof that won't fit in the margins of this book', but yeah, I do actually endorse a version of the meaninglessness approach. I think I have a non-question-begging motivation for it, an adequate error theory for the intuition that Liars are meaningful and a workable solution to "this sentence is either false or meaningless" and similar revenge paradoxes as well. But, y'know, doesn't fit in the margins of this book. ;)
Well, you've changed my mind now, sort of. If I ever start accepting a monist sentence theory of truth carriers (I think that is very unlikely), then I will follow this approach or something similar.
I always have this fuzzy feeling when trying to decide if something is meaningless or not; if saying something is a category error or not. Does it make sense to say that sentences are true or false at all? Maybe by proxy (they get their truth value from another truth carrier not from the 'original' truth maker.
At least, when I was considering viewing sentences as truth carriers, it was only as secondary truth carriers that get their truth value by proxy (from propositions). A similar approach can be taken to beliefs.
But then, many people object to ascribing the property of truthness or falsity to another thing beyond a single truth carrier. They invoke Occcam's Razor and indeed, is one not multiplying entities (properties are entities, I take it?) beyond the necessity to explain the things at hand? It seems so. But still, it makes talking so much easier if one doesn't have to write things like "the propositions expressed by the sentence " all the time.
I look forward to reading your dissertation/thesis/paper when you upload it. I hope it's less technical than Priest's In Contradiction because I didn't understand the technical parts of that and I'd say I'm pretty good with formal logic. lol :D
I also did a write up of our discussion here.
I'm actually agnostic about whether the primary (or only!) bearers of truth are sentences, or precissified sentences, or sentences-in-context, or propositions, or thoughts or what. I'm leery about propositions in the same way that my default policy is to be leery about any abstract, non-spatio-temporal objects (given standard epistemic worries, etc.), but that vague sense of leeriness doesn't really add up to a position one way or the other. I generally prefer to talk about sentences mostly for the sake of simplicity, and, in contexts where I'm being a big more careful than I tend to be in blog posts, stick in a footnote to the effect that what I'm saying should be neutral about the issue, and that those who subscribe to some other theory of truth-bearers can mentally substitute in "the proposition expressed by such-and-such sentence" or whatever without losing anything.
In any case, in terms of the discussion of meaningfulness and truth, I'm not convinced much of anything hinges on sentences vs. propositions vs. claims vs. whatever. Given that presumably only meaningful sentences can express propositions, I'd think that the same basic arguments should show that, if there are such things as propositions, we're committing a category mistake if we try to apply truth talk to sentences that can't express propositions (because they're meaningless).
And yeah, I know what you mean, re: fuzzy feelings. That's why I like to use "green ideas sleep furiously", since it's a standard example in the literature and I don't have to puzzle out whether people are likely to agree that it's meaningless first.
I'm also agnostic (I do not like to use that word since in my terminology it means has no knowledge, not has no belief) (=without belief) on the matter of primary truth carriers (=truth-bearers) and which things are truth carriers. However I'm pretty sure primary truth carriers are not sentences. See the work of Swartz for my justification.
But I'm more open to sentences as true by proxy (=secondary truth carriers) and beliefs too. I even wrote a bit about their truth conditions, if it is the case.
Well, Priest also talks of sentences and says that people can substitute their favorite truth carriers if they so wish without problems. But it is not actually true always (but often). Sometimes the problem he is talking about applies only to sentences and not propositions or beliefs. For instance, these problems of meaningless sentences do not apply to propositions (because it is gibberish to say that propositions are meaningful/-less) or beliefs (same as with propositions).
You got the Chomsky sentence wrong, btw (as I wrote in my write up). It is Colorless green ideas sleep furiously. You keep forgetting about the colorless part. :) Not that it matters in this context.
"For instance, these problems of meaningless sentences do not apply to propositions (because it is gibberish to say that propositions are meaningful/-less) or beliefs (same as with propositions)."
Surely it's not gibberish to say that only meaningful sentences can express propositions, that we're committing a category mistake if we try to symbolize non-proposition-expressing sentences and perform logical operations on them, etc.?
I agree with what you say. What I meant was that it is nonsense/meaningless to speak of "meaningful propositions".
even without Bivalence... given the T-Schema and intersubstitivity of logical equivalents, '¬Tr(S) entails ¬S.
Without bivalence, Tr(S) and S are not equivalent!
If S is neither true nor false, Tr(S) is false. Thus, if S can be neither true nor false, S and Tr(S) are not equivalent.
Not necessarily. I mean, a Bivalence-denier certainly *could* take that line, but they could (and some actually do) extend whatever their analysis of P is to Tr(P) as well. I guess it depends on one's motivations for denying Bivalence in the first place. E.g. if the point is to get around the paradoxes (in which case, presumably, one is also going to deny Excluded Middle), then it seems like one of the *advantages* of that approach is that it allows you to block the contradictions without having to deny any instances of the T-Schema. If you end up denying T-Schema instances anyway, why not just have *that* be your solution to the paradoxes and avoid having to deny any instances of Bivalence or Excluded Middle?
How is that not necessary? If you deny bivalence, you admit that some S could be neither true nor false. If S is neither true nor false, Tr(S) is false. If S and Tr(S) can have different truth values, they are not logically equivalent.
If you end up denying T-Schema instances anyway
Which T-Schema instance am I denying?
given the T-Schema and intersubstitivity of logical equivalents, '¬Tr(S) entails ¬S.
You also say:
The T-Schema (or Disquotation Schema, etc.) is the universal claim that "'P' is true iff P."
'Tr(P) entails P
'Tr(¬S) entails ¬S.
Which is not the same as:
'¬Tr(S) entails ¬S
Note the different position of the ¬. If one rejects bivalence, moving that ¬ changes the meaning of the sentence. If S and ¬S are neither true nor false, Tr(¬S) is false, but ¬Tr(S) is true.
So were I to reject bivalence, I would still accept all T-Schema instances including "'Tr(¬S) entails ¬S".
But I would deny that "'¬Tr(S) entails ¬S" is a T-Schema instance or logically equivalent to a T-Schema instance.
Incidentally, the above isn't really arguing with your main point. "'Green ideas sleep furiously' is meaningless" does not imply that "Green ideas don't sleep furiously" is true (and meaningful!), fortunately. But that doesn't mean the claim that sentences like the Liar's Paradox are meaningless frees you from paradox.pho
Stupid browser, captcha goes in the captcha box.
"But I would deny that "'¬Tr(S) entails ¬S" is a T-Schema instance or logically equivalent to a T-Schema instance."
Well, the T-Schema gives you S implying Tr(S). (The T-Schema just *is* that, plus vice versa.) Given Modus Tollens, that gets you "'¬Tr(S) entails ¬S." So I don't see how you think you can deny that w/o denying the T-Schema (or Modus Tollens).
Of course, you're right that if P and Tr(P) have different truth-values, they aren't equivalent, but then (given Modus Tollens), if you claim that they have different truth-values, you are of course rejecting the relevant T-Schema instances. "P iff Q" simply isn't true if P and Q have different truth-values.
As far as how someone can deny Bivalence without accepting that "Tr(P)" as false, you might want to look at Hartry Field's "paracomplete" view, which he argues for in his book "Saving Truth From Paradox" and elsewhere. You're right that if one expresses one's rejection of Bivalence by asserting that some sentences are neither true nor false, then one has to reject the equivalence of P and Tr(P), but there are some larger problems with expressing it this way....e.g., if you accept (as you seem to) that ¬Tr(P) is equivalent to "Tr(P) is false", then saying that some sentences are neither true nor false adds up to saying that some sentences are both true and false, which is a significantly bigger counter-intuitive bullet to bite. Plus, even if you *don't* accept the equivalence between "P is false" and "¬P", if you express your denial of Bivalence by saying that some sentences are neither true nor false, you're going to be forced into a contradiction about the Strengthened Liar ("this sentence is not true", which is presumably true if it's neither true nor false). Thus, Field opts not to say of some sentences that they're neither true nor false, but rather to talk about both rejecting the claim that they're true and rejecting the claim that they're false. Whether that's a viable, plausible position for the Bivalence-denier to take is a larger issue--I don't think it ultimately works, but then I'm not a Bivalence-denier anyway--but this is the sort of thing I was talking about when I said before that denying Bivalence doesn't *necessarily* mean rejecting the equivalence of P and Tr(P). It depends on which particular Bivalence-denying view you subscribe to.
that gets you "'¬Tr(S) entails ¬S." So I don't see how you think you can deny that w/o denying the T-Schema (or Modus Tollens).
I'd bite that on Modus Tollens, there's still no need to deny that "S is true" is true iff S is true. You're right that it follows, by Modus Tollens, but unless bivalence holds, "S is not true entails S is false" makes no sense. For Modus Tollens to work without bivalence, you'd need to express it as "if P implies Q, then ~Q implies ~Tr(P)" because indeed if P were true, Q would have to be true, but in a world without bivalence saying that P is not true tells you nothing about whether or not ~P is true. (Isn't Modus Tollens a theorem following from axioms including bivalence?)
if you accept (as you seem to) that ¬Tr(P) is equivalent to "Tr(P) is false", then saying that some sentences are neither true nor false adds up to saying that some sentences are both true and false
It's not obvious to me that would follow, have a proof? Using only rules of inference that don't depend on bivalence in their derivation?
Field opts not to say of some sentences that they're neither true nor false, but rather to talk about both rejecting the claim that they're true and rejecting the claim that they're false
That's not solving the paradox, that's just carefully avoiding talking about the paradox.
I'm not a Bivalence-denier
Oh? Is "green ideas sleep furiously" true, then? Is it false? Are you sure it's one or the other, but you'd need to make some empirical observations of green ideas before you could say either way?
When you say that you're not a bivalence-denier, are you stating a preference for bivalence as an axiom for a formal system of logic, or are you stating a fact about sentences (or just well-formed sentences) in general?
I'd say that to apply truth talk to "colorless green ideas sleep furiously" is to commit a nonsensical category mistake, just as applying it to questions is to commit one. (See above in the comments, and the original post for that matter, for why I think this.) So, strictly speaking, my stance would be that "either 'colorless green ideas sleep furiously' is true or 'colorless green ideas sleep furiously'" is meaningless. I certainly don't assert its negation.
....and, yeah, I'm pretty sympathetic to the gist of your criticism of Field's view. (And, in any case, I don't think it leaves him with anything plausible to say about what I've previously called the Rejection-Liar, which is "This sentence would be rejected by someone who accepted all true sentences and rejected all sentences that they did not reject.") My point was just that there are formulations of Bivalence-denying views on which one says precisely the same things about "P" and "'P' is true", and that there are interesting reasons why Bivalence-deniers might want to say things like this.
As far as the link between "P is neither true nor false" and "P is both true and false", you need:
(1) Double negation
(2) The equivalence between "P" and "'P' is true"
(3) The equivalence between "~P" and "'P' is false"
I'm not sure how you feel about (1). You've already indicated that you think Bivalence-deniers should deny (2), although you've also admitted that the stance you recommend for them requires them to either reject certain the T-Schema, Modus Tollens or both. Your original point in this comment thread--that Bivalence-deniers should reject (2) on the basis of the fact that, if some things are neither true nor false, then there are some P's for which "P" and "'P' is true" have different truth values--assumes (3). In any case, right off the top of my head, its not obvious to me that we can get "P is both true or false" from "P is neither true nor false" without (2), but we can certainly get the very weird result that "P & ~Tr(P)" with just (1) and (3). That's certainly intuitively not too different from an outright contradiction, although it's technically not one.
1. ~Tr(P) & ~F(P)
Use (3) to get:
2. ~Tr(P) & ~~(P)
Finally, use (1) to get:
3. ~Tr(P) & P
I'd say that to apply truth talk to "colorless green ideas sleep furiously" is to commit a nonsensical category mistake, just as applying it to questions is to commit one.
Fair enough, though I'd argue that's basically answering "the former" in response to my question. It's fine to say that a formal system of logic should assume bivalence, note that this system doesn't apply to something as broad as "all well-formed English sentences", and even argue that it's not a drawback that we can't reason about "green ideas sleep furiously" using such a system.
Of course, such an argument might also apply to paradoxes. It might not indicate a flaw in Classical Logic that the Liar's Paradox exists, so long as you can't derive such a paradox from the axioms and rules of inference of Classical Logic.
I'm pretty sympathetic to the gist of your criticism of Field's view
Mind, I'm getting ahead of myself, I still need to read what Field has to say.
(1) Double negation
I think that one follows from the definition of the "not" operator, even if bivalence is left out of the axioms.
(2) The equivalence between "P" and "'P' is true"
(3) The equivalence between "~P" and "'P' is false"
that Bivalence-deniers should reject (2)... assumes (3)
Far from the rejection of 2 assuming 3, wouldn't I deny 3 for the same reason as 2, absent bivalence? If P and ~P are neither true nor false, "~P" would be neither true nor false, but "P is false" would just be false.
Well, I'd deny that I'm just answering "the former." See the end of my post for why I think talk of "well-formed" sentences in natural language is a total non-starter....I wouldn't put it in terms of denying Bivalence for meaningless sentences any more than I'd say that everyone denies Bivalence for questions or for bits of melting candle wax. In all three cases, I'd say that we're just not talking about the sort of entities which can be meaningfully logically symbolized, to have truth-talk applied to them and so on. Whatever one's view about truth-bearers, presumably when we do symbolic logic, what we're symbolizing are truth-bearers or at least (allowing for truth-value gaps) the sorts-of-things-that-can-bear-truth, whether one takes those to be meaningful declarative sentences, the propositions expressed by them, or what. In other words, I'd deny that its simply a matter of formal preferences....but I'm not sure how much rides on the distinction.
In any case, when it comes to your stance on (3), you're certainly right that one could tell a Bivalence-denying story to motivate denying both (2) and (3), but your particular argument for denying (2) did assume (3). What you said in your first comment on this thread was....
"If S is neither true nor false, Tr(S) is false."
That assumes (3), or at least half of (3). As an equivalence claim, (3) goes in both directions, with ~P implying F(P) and F(P) implying ~P. What you said, quoted above, assumes the first half. You're assuming that "~Tr(S)" implies "F(Tr(S))."
but I'm not sure how much rides on the distinction.
Tomayto, tomahto... but I appreciate the clarification.
That assumes (3), or at least half of (3).
I'd actually admit both "halves" for a bivalence denying system: "~P→F(P)" and "F(P)→~P". I'd just deny that those "halves" add up to the (3) above! (Because of precisely the counter-example I mentioned.)
If ~P is neither true nor false, F(P) is false, and neither "~P→F(P)" nor "F(P)→~P" is false (neither evaluates to "[something true] implies [something not true]"). Since ~P and F(P) can have different values even if "~P→F(P) & F(P)→~P" is true, "~P→F(P) & F(P)→~P" does not imply "~P == F(P)".
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