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Wednesday, April 21, 2010

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Perhaps I am being dense, but I don't follow the first point at all.

'In general, p does not entail it is true that p'

If we read 'entail' in the usual way, this is to be rendered as

~ [ p --> it is true that p]

which is the same as

~[ ~ [p and it is not true that p ] ]

which resolves into

p and it is not true that p

I don't see how the last is even consistent with the assumption that there are no truths, on account of the 'p' on the left hand side. (The right hand term is also problematic, as Bill points out: there can be negative truths, not-p, as well as positive ones).

I think using possible worlds scenarios can clarify the argument. If there is a possible world which is void of persons and truths, then we can say that a proposition or state of affairs p (in this world) can be, and be true here as a result, but that in the void possible world it is not true that p. This reduces to 'p and it is possible that not p' which is not a contradiction.

Hi Bill,

Thanks for your reply. I follow your logic, and I agree with you that the existence of truths is (at least) a transcendental necessity. But as for whether it is an absolute necessity, I find your reliance on the p-->Tp principle question-begging.

While the p-->Tp principle is luminously plausible in ordinary cases (e.g., if snow is white then it is true that snow is white), it breaks down in cases where the antecedent describes a null state. It is not true that *If nothing exists then it is true that nothing exists*, for if nothing exists then a fortiori no truth-bearers exist, and thus no truths exist. Applying p-->Tp to the null state is simply not taking the idea of 'nothing' seriously.

You say, "if it is possible that no thinkers and no truths exist, then it is true that it is possible that no truths and no thinkers exist." But in what sense is it supposed to be "possible" that no thinkers and truths exist? The denier of the absolute necessity of truth would only admit this as a negative possibility, such that "possibly" only means "not impossible". But your argument requires taking this in a positive sense, such that "possibly" implies "possibly true", and in doing so it begs the question.

Again, you say, "Now suppose the possibility is actual. Then it will be true both that it is possible and that it is actual." But that doesn't follow. If the possibility in question is 'nothing', then its being actual is, still, 'nothing' - no truthbearers, no truths, no cognizers - nothing.

Dear William,

"p and it is not true that p" is only problematic on the assumption that p-->Tp. Grant that and a clear contradiction follows: "Tp and ~Tp". My argument is that adherents of the view Bill is concerned to refute wouldn't and shouldn't concede that assumption.

Alan,

I feel your pain. This is indeed a delicate and tricky issue and I have been known to waver on it. Underlying question: What is the probative reach of retorsive arguments? Can we use such arguments to establish something as true in itself? (If yes, then this validates classical metaphysics, the attempt to know by reasoning alone something about reality in itself.)

Let's consider your example: *If nothing exists, then it is true that nothing exists.* You claim this proposition is false and is a counterexample to *p -->Tp* You seem to think that to take 'nothing' seriously one has to admit that there might have been nothing at all. But I have argued (on the old blog) that there cannot be nothing at all: necessarily, there is something.

It may be that we cannot avoid mutual question-begging. Surely you are right that *There is nothing* entails *There are no truthbearers and no truths.* But that is consistent with *There is nothing* being necessarily false (as I maintain) or possibly true (as you maintain).

Suppose that *There is nothing* is possibly true. Then there is a possible world W in which it is true. In W, the proposition is both true and false. True, because that is how W is defined. False, because there is something in W, namely the proposition that there is nothing.

I conclude that *There is nothing* is not possibly true. Necessarily there is at least one abstract object.

Now I am not sure I have refuted you, but I am pretty sure that you cannot refute me. Are we engaged in mutual question-begging? Is it a stand-off?

Alan,

William Woking is our old friend 'ocham' who used to show up on your blog when it was active. He uses many pseudonyms to stay one step ahead of the U.K. Thought Police.

Hi Bill,

I think we're making progress.

As for retorsive arguments, I don't think they can establish anything as "true in itself". They can establish performative necessity, but not logical necessity.

As for your modal argument, I'm afraid that it too is question-begging. You begin, "Suppose that *There is nothing* is possibly true." But one who denies p-->Tp will not grant that first step. The "possibility" of nothingness, in whatever sense it might be thought to be possible, cannot along with the denial of p-->Tp be taken to imply the possible *truth* of *There is nothing*. Rather, "possibility" must be understood as nothing more than non-impossibility. The non-impossibility of *There is nothing* doesn't imply that *there is* a maximal, internally consistent proposition containing *There is nothing* as one of its conjuncts. Rather, it merely implies an absence of internal contradiction, which follows because there can't be a contra-diction in the absence of "dicta" (broadly construed).

Alan,

It depends on the retorsive argument. We have to go case by case.

Burden-of-proof considerations are relevant to this discussion. The principle *p --> Tp* is so obvious that the onus probandi rests on the one who would deny it. To be perfectly explicit, the principle is this: Necessarily, for any proposition p, p --> Tp, where '-->' symbolizes material implication and 'T' is elliptical for 'it is true that.' We can use the word 'entails' if we understand that to mean the broadly logical necessitation of material implication. I think we are in agreement about this, but I wanted to be sure.

Notice that the principle quantifies over propositions which by definition are truth-valued entities. We can define 'proposition' by saying that propositions are the primary vehicles of the truth-values. It is the very existence of propositions that underwrites the truth of *p -->Tp.*

You say I beg the question by assuming this principle. But you beg the question against me by denying it. Furthermore, the burden of proof is on you to show it false especially since its truth seems bound up with the very meaning of 'proposition.'

For you the following is false: *Possibly nothing exists --> T(Possibly nothing exists).*

Why? To me it is self-evidently true.

Bill, Alan, William, etc.,

[I follow Bill’s conventions: *p* refers to the proposition expressed by the sentence enclosed; ‘p’ refers to the sentence (type? token?) so enclosed.]

First, I suppose the debate is about the principle of bi-valence: i.e., whether every proposition, sentence, etc., is either true or false. Thus, so far as I can discern the contentious (p v ~Tp) is the bivalent principle at issue (or at least a close approximation: the genuine article will be (p)(T~p v Tp)).

Second, does bi-valence applies to sentences, propositions expressed by sentences, or both? Someone may accept bi-valence regarding propositions, while denying it regarding sentences and maintain that gappy sentences fail to express a proposition. Then, every proposition is indeed true or false, but some sentences (the gappy ones) do not express propositions. Therefore, there are no propositions that are gappy. I am unsure which position Alan endorses.

Third, those who accept truth-value gaps think of ‘not-true’ as ambiguous between ‘false’ and ‘neither-true-nor-false’; i.e., gappy. Similarly, ‘not-(im)possible’ is ambiguous between (i) cases where it is a sentence forming operator such that when attached to a sentence, the result is another sentence (and hence a truth-value bearing entity which is subject to bi-valence and expresses a proposition); and (ii) cases where it is not a sentence forming operator. In such cases the phrase ‘not-(im)possible’ when attached to a sentence expresses the denial that the sentence to which it is attached is a truth-value bearing entity and hence expresses a proposition.

Now let us consider the following quotation from Alan’s latest post:

“The non-impossibility of *There is nothing* doesn't imply that *there is* a maximal, internally consistent proposition containing *There is nothing* as one of its conjuncts. Rather, it merely implies an absence of internal contradiction, which follows because there can't be a contra-diction in the absence of "dicta" (broadly construed).”

If we take entailment to be a relation between truth-value bearing entities (sentences or propositions), then the above must be interpreted roughly along the following lines:

(A) The sentence ‘The proposition that results from attaching the operator ‘not-impossible’ to the sentence ‘There is nothing’’ does not entail the proposition *it is possible that there is nothing* (interpreted as there is a maximally consistent proposition containing as a proper conjunct the proposition *There is nothing*); rather it entails the proposition expressed by the sentence ‘The sentence ‘There is nothing’ does not express a proposition’ (i.e., is gappy). The trouble is that the last sentence I have just written itself expresses a proposition and, therefore, Bill wins.

An alternative interpretation and the one I think Alan intends is the following:

(B) The phrase ‘not-impossible’ should not be viewed as a sentence -forming-operator such that when it is attached to a truth-value bearing entity such as the sentence ‘There is nothing’, it yields another truth-value bearing entity and, hence, one which expresses a proposition. Rather the phrase ‘not-impossible’ should be understood as the denial of the claim that the sentence ‘There is nothing’ has a truth-value at all (and, therefore, it fails to express a proposition). But then the term ‘implies’ in the above quotation is misplaced, since as I noted above entailment is a relation between truth-value bearing entities.

If Alan opts for (B), then the following questions need to be answered:

(a) Under what conditions the phrase not-(im)possible’ should be interpreted as a sentence forming operator yielding another sentence bearing a truth-value (and, hence, expressing a proposition) and under what conditions it merely indicates that the sentence to which it is attached is gappy?

(b) What consideration compel us to view attaching ‘not-impossible’ to the sentence ‘There is nothing’ in the second way, which renders ‘There is nothing’ truth-valueless?


Hi Bill,

You say that *p-->Tp* is "obviously" correct, whereas I say it's "obviously" false when p = *Nothing exists*. In that sort of dialectically situation, appeal to burden of proof considerations does nothing to advance the discussion.

If there is a non-question-begging argument that can resolve our dispute, it cannot be one that assumes either the truth or falsity of *p-->Tp*. I think I've already provided such an argument. Here it is spelled out:

(1) If *Nothing exists*, then no truth-bearers exist.
(2) If no truth-bearers exist, then it is not the case that *Nothing exists* is true.
Therefore,
(3) If *Nothing exists*, then it is not the case that *Nothing exists* is true.
(4) If *p-->Tp* were true, then (3) would be false.
But since (3) is true (by (1) and (2)), it follows that
(5) *p-->Tp* is false.

I don't think this argument is question-begging against you. If you think it is, then I ask you to indicate how. Do you reject premise (1), (2), (4) the inference from (1)-(2) to (3), or the inference from (3)-(4) to (5)? And why?

Hi Peter,

My argument with Bill don't at all depend on bivalence. (In my previous comment I give an argument against *p-->Tp* that makes no appeal to bivalence one way or the other.)

You are right, though, to sense a connection, for there is a dependence in the other direction: One who denies bivalence should reject *p-->Tp* in favor of *p-->~T~p*. In a way, this helps my case, since bivalence is and should be controversial (especially in cases of vagueness), likewise, *p-->Tp* should be controversial, or at least more controversial than Bill takes it to be.

*p-->Tp* is effectively a rationalist metaphysical booting-strapping principle, one that would allow one to get certain items (in this case, truth-bearers and truths) into one's ontology for "free". In that respect, it's akin to the ontological argument for God's existence.

Perhaps this will help, Bill.

I think the intuition that *p-->Tp* is obviously true stems from a conflation of that principle with a close analogue, one which is obviously true.

Let i be a generic index. It could denote a possible world, the actual world, a time, a place, or any well-defined conceptual space.

Let Ti(p) be an indexed truth operator: *It is true at i that p*.

Let pi be the indexed proposition *p-at-i*, or more colloquially, *p is the case at i*.

In those terms, the following principle, I submit, is a truism:

*Ti(p) iff pi*

What the truth operator does is allow one to strip the index off of the proposition. For example, *It is true at world w that p* iff *p is the case at w*.

For truth simpliciter just let the i denote the totality of actuality.

Note that *Ti(p) iff pi* entails *pi-->Tip* but not *p-->Tp*. One gets the latter by ignoring the indices, but that principle isn't an innocuous truism.

Peter writes, >>First, I suppose the debate is about the principle of bi-valence: i.e., whether every proposition, sentence, etc., is either true or false. Thus, so far as I can discern the contentious (p v ~Tp) is the bivalent principle at issue (or at least a close approximation: the genuine article will be (p)(T~p v Tp)).<<

We are not discussing Bivalence or Excluded Middle but a different principle, the principle that for any proposition p, p entails Tp, where 'T' is an operator on propositions, the operator 'it is true that ___.'

Bivalence: there are exactly two truth-values.
Excluded Middle: For every p, p v ~p.

Note that Bivalence and Excluded Middle are not the same. Suppose that Bivalence is false and that there are three truth-values. It can still be the case that every proposition is either true or not true. (In a 3-valued logic, 'not true' is not the same as 'false.') So Excluded Middle does not entail Bivalence. Therefore Excludefd Middle is not the same as Bivalence.

Alan and I are assuming both Excluded Middle and Bivalence, or at least I am.

The third principle -- what the hell should we call it? -- is not the same as Bivalence since it could hold even if Bivalence doesn't. It is not the same as Excluded Middle either since all it says is that for every p, p entails Tp.

So, Peter, I would say you have mislocated (dislocated?) the bone of contention. You may also be confusing Bivalence with Excluded Middle.

Alan is denying this: For any p, p entails Tp. He thinks there are counterexamples to it, e.g. *Nothing exists.* He claims that *Nothing exists* does not entail *It is true that nothing exists.*

His reason seems to be: if *Nothing exists* were true, then nothing at all would exist, hence no truthbearers and no truths.

But I say: if, per impossibile, *Nothing exists* were true, then it would be true that nothing exists (by the third principle) and so something would exist, namely, the truth that nothing exists, which implies that necessarily something exists.

I trust you can see the 'logic' of Alan's position, but also the 'logic' of mine. Can this be resolved, or is it a game of mutual question-begging?

>>Dear William, "p and it is not true that p" is only problematic on the assumption that p-->Tp.

I have argued the oppossite. It is problematic only if we assume that ~ p-->Tp. For in that case the chain of inferences outlined above lead to a contradiction.

I am a long way from understanding this.

Hi Bill,

Quick correction. My argument is not, as you put it above to Peter, that if *Nothing exists* were true then it would not be true that nothing exists. It is rather that if *Nothing exists* (period) then it would not be true that nothing exists.

Your adding the words "were true" is what opens me up to your counterargument. Moreover, only someone who already accepts *p-->Tp* would regard it as innocuous.

Gentlemen:

The fundamental question here is whether suppositional reasoning requires hypothetical reification of the supposit. When one asks "If X then ...?" is one ipso facto asking "If X {existed, obtained, were true, were actual, were the case, etc.}, then ... ?" Is the former question merely elliptical for one of the latter sorts of questions? I take it the proponent of *p-->Tp* says "yes" and the denier says "no".

The "yes" proponent adopts a hypothetical "committed" stance toward the supposit and, accordingly, posits it. The "no" proponent, in contrast, thinks it possible to adopt a hypothetical "withholding" stance toward the supposit, one that neither posits it nor its negation.

To get a feel for what a hypothetical "withholding" stance is like, think of pure mathematics. Contemplate, say, some newly invented axiomatic system apart from any concern about whether it is "true", plausible, or applicable, but simply because it seems "interesting". That's how I think about *Nothing exists*. I don't ask myself, "What if it were true?" I simply ask myself what I get if I start with nothing--absolute nothing--and the obvious answer, it seems to me, is "nothing". Ex nihilo nihil fit.

Alan,

I will focus for now on your post, Friday, April 23, 2010 at 11:33 AM. There you maintain that you gave a non-question begging argument. Premise (1) of your argument states:

(1) If *Nothing exists*, then no truth-bearers exist.

But the antecedent of (1) refers to a proposition (given Bill’s convention regarding sentences enclosed in asterisks) and if all propositions are truth-bearers, then there surely is at least one truth-bearer: namely, the proposition *Nothing exists*. You may deny that all propositions are truth-bearers, which then is equivalent to denying bivalence regarding propositions (what else could such a denial mean?). I should note that you cannot reformulate (1) in terms of sentences rather than propositions (at least not in any obvious way), for substituting the sentence ‘Nothing exist’ in (1) for the proposition *Nothing exists* will require a predicate that applies to the sentence, otherwise you do not have a complete sentence in the antecedent. Thus, it seems to me that (1) presupposes the denial of bivalence regarding propositions. But if you deny that every proposition is a truth-bearer, then you beg the question against Bill, who endorses it.

I want to put to you two questions:

(a) Do you deny bivalence regarding propositions?
(b) By ‘not-true’ do you mean ‘false’ or you mean ‘neither true nor false’?

e.g., in the consequent of (2) the following sentence appears: “it is not the case that *Nothing exists* is true.” Shall we understand the phrase ‘it is not the case that *Nothing exists* is true’ as ‘is false’ or as ‘is neither true nor false’? If the former, then *Nothing exists* is false and, therefore, there is at least one truth-bearer, albeit a false one: hence, the antecedent of (2) which says that there are no truth-bearers would be false. Hence, (2) is true vacuously.

If the later, then you maintain that there is at least one proposition (namely, *Nothing exists*) which is neither true nor false; therefore, you deny bivalence. In either case Bill should deny (2) either on the grounds that it presupposes the denial of bivalence or on the grounds that if (2) is true, it is true vacuously (it is true because the antecedent must be false when the consequent is true).


Alan,

Thanks for restating your argument, which is:

(1) If *Nothing exists*, then no truth-bearers exist.
(2) If no truth-bearers exist, then it is not the case that *Nothing exists* is true.
Therefore,
(3) If *Nothing exists*, then it is not the case that *Nothing exists* is true.
(4) If *p-->Tp* were true, then (3) would be false.
But since (3) is true (by (1) and (2)), it follows that
(5) *p-->Tp* is false.

I would say that (1) is incoherent. (1) is a compound proposition the components of which are the propositions *Nothing exists* and *No truthbearers exist.* Propositions by definition are truth-bearers. So if the antecedent is true, then the consequent is false. So I reject your argument by rejecting (1).

I was talking with Peter on the phone. We were wondering whether you accept Bivalence.

Alan writes, >>My argument is not, as you put it above to Peter, that if *Nothing exists* were true then it would not be true that nothing exists. It is rather that if *Nothing exists* (period) then it would not be true that nothing exists.<<

I'm sorry, Alan, but I find this incoherent. *Nothing exists* is a proposition, a truth-bearer. Now given Bivalence, this proposition is either true or false. You cannot abstract a proposition from its having a truth value. If you do, you end up with nothing. Only as a truth-valued entity can it stand in inferential relations or enter as a component into compound propositions.

The principle *For any p, p entails Tp* is part and parcel of the very notion of a proposition, which is why I cannot understand how anyone could deny it.

But of course the principle does not allow one to infer the truth of a proposition from its content. *Bush is president* entails *It is true that Bush is president* despite the fact that both propositions are false. So there is nothing like the ontological argument here, and no boot-strapping.

The existent round square no more exists than the round square. Similarly, prefixing 'It is true that' to a false proposition does not make the proposition true. Nevertheless, if I am dead, then it is true that I am dead.

*p entails Tp* holds across the board.

Peter,

Just saw your comment after I posted my last two.

It makes sense to deny Bivalence, but by my lights it makes no sense to think of some propositions as having no TV at all. Alan, is that what you want to say about *Nothing exists* -- that it has no TV at all?

Bill,

"It makes sense to deny Bivalence, but by my lights it makes no sense to think of some propositions as having no TV at all."

I don't see how these two go together. If by TV you mean either 'true' or 'false', then it would make no sense to deny bivalence, yet affirm that every proposition has a TV. On the other hand, if by TV you mean one of three values, one of which is neither the true nor the false, then of course one could deny bivalence yet maintain that every proposition has a TV, some of which have the value 'neither true nor false'.

Hello Gentlemen,

I agree with Bill that Alan's argument is incoherent. Alan asks us to accept the following...

(2) If no truth-bearers exist, then it is not the case that *nothing exists* is true.

What makes the consequent of (2) follow from its antecedent? Consider the following syllogism...

Premise 1: No abstract immaterial entities are truth bearing entities.
Premise 2: *Nothing exists* is an abstract immaterial entity.
Conclusion: *Nothing exists* is not a truth bearing entity.

From this conclusion it follows that it is not the case that *nothing exists* is true. Now, consider Alan's (1)....

(1) If *nothing exists*, then no truth-bearers exist.

Alan has just argued that the antecedent of this conditional is not a truth bearing entity. So, under this situation it is not coherent to speak of (1) being true. So, where does this leave his argument in terms of its soundness?

Brian

Looking at the above, I am a little clearer on what is going on. My original problem was that (B) follows logically given Alan's claim (A) below.

(A) possibly not [ p --> Tp ]
(B) possibly [p and not Tp]

But it turns out Alan does accept (B), in the case that

(1) There are no truths

From which it follows (according to Alan)

(2) For some p, p and not Tp

E.g. "grass is not green". If nothing exists, then grass is not green. But I cannot add (according to him) 'it is true that grass is not green', for this would commit me to the existence of a truthbearer. Thus Alan seems to hold to the view that

(*) For any p if Tp then some truth-bearer exists.

The obvious absurdity of this suggests we could turn the whole thing around, and construct a clear case against the existence of "truthbearers". I have actually pointed this out in comments to previous posts: the existence of 'negative truths' points to a clear problem for the whole theory of truthmakers. What is the truthmaker of 'grass is not green', given that one possible causa verititatis of this proposition is that no grass, or nothing at all, exists?

Alan,

"To get a feel for what a hypothetical "withholding" stance is like, think of pure mathematics. Contemplate, say, some newly invented axiomatic system apart from any concern about whether it is "true", plausible, or applicable, but simply because it seems "interesting". That's how I think about *Nothing exists*. I don't ask myself, "What if it were true?" I simply ask myself what I get if I start with nothing--absolute nothing--and the obvious answer, it seems to me, is "nothing"."

One certainly can be interested in a certain uninterpreted axiomatic system for the sake of say some proof-theoretic purposes. Here one is interested only in the syntactic elements, not their interpretation.

The trouble is that this example is unlike the case you describe regarding *Nothing exists*. For the minute you ask "what do I get if I start with nothing" you are interpreting the words 'nothing' and 'exists'; you are not treating them as merely uninterpreted symbols. For in this case you are asking: what would the world be like if this proposition were true. For what else could the question "what do I get" here mean?

Compare to the case of the uninterpreted axiomatic system. Suppose you ask the same question (i.e., "what do I get if I start with this") regarding some uninterpreted formula in your axiomatic system. Since the formula are uninterpreted, the obvious answer is that you get more uninterpreted formula. But if you mean by the question "what do I get" how the world would be like if I start with this formula, then you smuggle in an interpretation and that is a game changer: you no longer treat the formula as uninterpreted.

There cannot be a "withholding" stance which aspires to both completely disregard the way things would be like if such-and-such were the case and at the same time inquire how things would be like if such-and-such were the case.

Another absurdity. Suppose just one thing exists, let it be b, and suppose b is not white. Then

b is not white

is true. But what is the truthmaker for that proposition? Is it b? Surely it is, for if b ceases to exist then nothing exists, and according to Alan there are no truthmakers for "nothing exists". That seems absurd. How can b, which is not white, be the truthmaker for 'b is not white'?

William,

The truthmaker for 'b is not white' is .

Strange, only part of my post actually entered. I repeat: the truthmaker for 'b is not white' is the ordered triple {b, the property of being white, and b's lack of exemplifying this property}.


[Peter] >>The truthmaker for 'b is not white' is the ordered triple {b, the property of being white, and b's lack of exemplifying this property}.

So not only b exists, but b's lack of exemplifying this property also exists.

So do the truth conditions of 'b is not white' not include the possibility that nothing exists? Surely that proposition remains true when b vanishes from existence? But if it remains true, how can the ordered triple contain b, which has just ceased to exist? Either b necessarily exists, which seems absurd, or we have to suppose that the proposition that b is not white ceases to exist when b vanishes.

William,

Truth conditions in the traditional sense presuppose that 'b' has a referent and that something exists (Classical, non-free logic). O/w no truth-conditions can be assigned.

The proposition *Nothing exists* is complicated because it involves a quantified phrase 'nothing'. Hence, the first task is to determine whether the domain of the quantification includes absolutely everything, including abstracta such as propositions. If it does, then we got ourselves a tricky situation where the proposition *Nothing exists* applies to itself. As we all know, this sort of self-referentiality is always tricky. Indeed, in this particular case the result is that the proposition is necessarily false.

Now, since *Nothing exists* is necessarily false, we need not worry what happens when 'b is not white' and 'Nothing exists', for there is no possible world in which both are true.

Brian Bosse,

Thanks for your contribution. I think we are converging on the same point.

Regards,

BV

[peter] >>Truth conditions in the traditional sense presuppose that 'b' has a referent and that something exists (Classical, non-free logic). O/w no truth-conditions can be assigned.

I think 'in the formal sense' rather than 'in the traditional sense'. Yes, I am aware of this, but our object is a general theory of truth that applies to the propositions expressed by ordinary language, rather than some simplified formal system whose scope is clearly very limited.

In ordinary language, we are quite comfortable with referring to things that no longer exist because they have disappeared from existence. Any decent theory of truth should reflect this.

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