Sentence theories of truth, meaningless sentences and untrue sentences

By sentence theory I just mean a theory of truth carriers that implies that some sentences are true or some are false. Not necessarily a monist sentence theory (=theory that implies that sentences are the only kind of truth carriers) or a theory of sentences as primary truth carriers (=theory that implies that sentences are the primary truth carriers). For more about these terms, see my earlier writings on the subject.

Anyway, I read the newest post on my favorite logic blog (Blog&~Blog). It dealt with the sentences which I have given incredibly clever names (in footnotes):

For all sentences, if it is not the case that it is meaningful, then it is not the case that it is true.

NMNT.1 (∀S)(¬M(S)→T(S))

For all sentences, if it is not the case that it is meaningful, then it is not the case that it is false.

NMNF.2 (∀S)(¬M(S)→F(S))

With the obvious interpretation keys.

This seems like plausible sentences to many when faced with sentences such as the Chomsky:

C. Colorless green ideas sleep furiously.

Which Ben, btw, got wrong as he forgot the first word.

Let’s also agree that:

1. It is not the case that C is meaningful.

¬M(C)

However, this along with some other sentences is inconsistent (=implies a contradiction). First sentence bivalence:

SB.3 For all sentences, it is either true or it is false.

(∀S)(T(S)∨F(S))

The contradiction is easy to derive here:

2. ¬T(C) [from 1, NMNT, MP]

3. ¬F(C) [from 1, NMNF, MP]

4. T(C) [from 3, SB, DS]

5. T(C) ∧¬T(C) [from 2, 4, conj.]

Contradiction! So this doesn’t work. Here I told Ben (author of the blog) that I would drop SB.4 However that apparently doesn’t work either.

Say hi to the T-schema, or the semantic theory of truth:

TS1. For all sentences, iff it is true, then it is the case.

(∀S)(T(S)↔S)

TS2. For all sentences, iff it is false, then it is not the case.

(∀S)(F(S)↔¬S)

Now these are obvious to most people. Not something is that plausible to deny unless the alternatives are really bad. However from these one can get their contra-positional versions:

TS1-CP. For all sentences, iff it is not the case, then it is not the case that it is true.

(∀S)(¬S↔¬T(S))

TS2-CP. For all sentences, iff it is not the case that it is not the case, then it is not the case that it is false.

(∀S)(¬¬S↔¬F(S))

And from these, we can derive their converses (and we can do that because these are bi-conditionals that can be conversed without problems). Do the same for TS1 and TS2:

TS1-CP-C. For all sentences, iff it is not the case that it is true, then it is not the case.

(∀S)(¬T(S)↔¬S)

TS2-CP-C. For all sentences, iff it is not the case that it is false, then it is not the case that it is not the case

(∀S)(¬F(S)↔¬¬S)

TS1-C. For all sentences, iff it is the case, then it is true.

(∀S)(S↔T(S))

TS2-C. For all sentences, iff it is not the case, then it is false.

(∀S)(¬S↔F(S))

And these actually need to be simplified too before I can use them, but I’m too lazy to do that, so I’ll just add a simp. step. No big deal.

Now:

6. ¬C [from 2, TS1-CP-C, simp., MP]

7. F(C) [from 6, TS2-C, simp., MP]

8. F(C)∧¬F(C) [from 3, 7, conj.]

Contradiction. And I didn’t need to use double negation to get it though one could do that too with TS2-CP-C, and of course I didn’t use SB either. It seems to me that this is terrible and the best way out of the contradiction is to deny NMNT and NMNF, and believe instead that sentences like C cannot even meaningfully be said to be true or false, nor can they meaningfully be said to be not true or not false. Any complex sentence with a meaningless part is itself meaningless.5

There is a tendency for people to conflate denial of properties with the denial of the meaningful application of these properties to things. This seems to be the case here too. So instead of saying things like:

Meaningless sentences are not true.

Cars are not true.

We should say things like:

Meaningless sentences cannot meaningfully be said to be true.

Cars cannot meaningfully be said to be not true.

Maybe some people sometimes, confusingly, use the first versions as a shorthand for the second. If they do and really mean what the second ones mean, then they should use them.

In a web of beliefs approach one could set up an inconsistent set of sentences and see which one is the least plausible. I figure that my readers can do that in their heads without I needing to write it out in this case. Maybe the readers will agree with me that NMNT and NMNF are the least plausible ones in the set.

Notes

1Not meaningful not true.

2Not meaningful not false.

3Sentence bivalence.

4Because, seen as a set of inconsistent sentences, this one is the least plausible to me.

5One can formulate clever sentences for this principle. I’ll do that in another essay quickly to follow this one.

Monist sentence theory and modal truths

1. For all things, that it is a truth carrier logically implies that it is a sentence.

2. There exists a thing such that it is a truth carrier and that it is logically necessarily the case.

Thus, 3. There exists a thing such that it is a sentence and that it is logically necessarily the case. [from 1, 2]

4. For all things, that it is a sentence logically implies that (that it is logically necessarily the case logically implies for all possible worlds, that sentence is the case in that possible world).

Thus, 5. For all possible worlds, there exists a sentence such that it is the case in that possible world. [from 3, 4]

6. For all possible worlds and for all things, that a thing is a sentence logically implies (that that a thing is the case in that possible world logically implies that that thing exists in that possible world).
Thus, 7. For all possible worlds, there exists a thing in that possible world such that it is a sentence. [from 5, 6]

8. There exists a possible world, such that it is not the case that there exists a thing such that that thing is a sentence.

Proof of inconsistency

Readers who do not doubt that the above set is inconsistent may skip this section, as it is a technical proof of the inconsistent of the above.

The numbered formulas here are formalization of the above sentences.

Interpretation keys

Domain x ≡ things

Domain w ≡ possible worlds

Tx ≡ is a truth carrier

Sx ≡ is a sentence

Formalization

1. (∀x)(Tx⇒Sx)

2. (∃x)(Tx∧□x)

⊢ 3. (∃x)(Sx∧□x) [from 1, 2, Simp., MP, Conj.]

4. (∀x)(Sx⇒(□x⇒(∀w)(xw)))

⊢ 5. (∀w)(∃x)(xw) [from 3, 4, Simp., Simp., MP,)

6. (∀w)(∀x)(Sx⇒(xw⇒(∃xw)))

⊢ 7. (∀w)(∃xw)(Sx) [from 5, 6, MP]

8. (∃w)¬(∃xw)(Sx)

(7) and (8) are inconsistent. I don’t know if I got the names of the inferences right, I need to read up on that at some point. It should be intuitively clear to anyone that studied predicate logic that the set is inconsistent.

The formalization could have been simplified if I had introduced more domains that were connected to a predicate, such as a domain of sentences. Then I could have avoided the implications inside another implication and could simply have written “for all sentences”.

Discussion

I also think that the above set is minimally inconsistent, by which I mean that if one removed a single sentence, it would no longer be inconsistent. The interesting thing about such minimal inconsistent sets is that the set of all except truth carrier logically implies the negation of the last truth carrier. From such a set the last truth carrier can be constructed a valid argument. Thus, a good deal of arguments can be constructed from the above list.

Suppose a person finds himself believing all the above truth carriers. Which should he stop believing? One might take it as an argument against monistic sentence theory (1), or an argument against a fundamental part of possible worlds semantics, (4), or the additional premise about existence of sentences in the relevant possible world, (6), or as evidence that there are no possible worlds where there isn’t a sentence, negation of (8), or that there are no necessarily the case truth carriers (2). It is very hard to make the decision.

Generally, a rational agent ought to reject the truth carrier that is the least plausible to him. But even that is a tough job. Which one is the least plausible? I think it is (1) given other arguments against monist sentence theory given by Swartz and Bradley. I am not very sure about this and it may be (6) instead which I find the second least plausible. On the other hand, I find (2) the most plausible and (8) the second most plausible. (4) is somewhat plausible I think, even though I have doubts about possible world semantics.

One may construct an ordered set after which truth carrier one has the most reason to reject. To me that would be {1, 6, 4, 8, 2}.1 Though one should bear in mind that these may not be independent. For instance, a web of belief with a belief in (1) would probably result in a more justified web of belief if one also rejected (2), (4) and (6).

1If two truth carriers are tried for plausibility, one may instead have them in an unordered set together inside the other. {1, 2, {3, 4}, 5, 6}

Primary truth bearers, none of them?

Primary truth bearers are the kind of entities that are always true or false. This is in contrast to secondary truth bearers that are only sometimes true or false. It seems possible that there are no primary truth bearers but that there are two or more secondary (= non-primary) truth bearers. All the theories that I am acquainted with are theories that include a primary truth bearers be it sentences, propositions, beliefs or whatever.

Exploring the notion of primary truth bearer

This notion of primary truth bearer seems to me to warrant further investigation and clarification. I would like to find a more rigorous description than the one I gave above. What is meant by “always” in that context?

It seems to me that it is best to think of it in relations to facts. There is something in relation to the fact that the Earth is spherical that is true/false. What is it? To say that there is a primary truth bearer is to say that:

For all facts, there exists a truth bearer of a certain kind in relation to that fact.

Though I am still not satisfied. This “of a certain kind” is not formalizable. I want a formalizable version. The below seems to capture the same idea and it is formalizable:

For all facts, there exists a truth bearer such that that truth bearer has the property true/false in relation to that fact and that truth bearer is of a certain kind.

(∀x)(∃y)(Rxy∧Cy)

The fact and truth bearer distinction

Note the distinction between facts and whatever it is that it true/false. “fact” here means things such as the spherical planet referred to by “Earth”. It is not the Earth that is true/false. Since we’re assuming that there is something that is true/false, it has to be something else.

A joint sentence and proposition theory of truth bearers

It seems possible that there is not a single kind of entity that has the property true/false in relation to all facts. Perhaps a theory could be that in all relations to facts where a sentence of the right kind has been uttered, it is that sentence that is true/false and in cases where such a sentence has not been uttered it is a proposition that is true/false.

What are the merits of such a theory? One thing is that it posits less objects than does a pluralistic theory which posits more than one entity being true/false in relation to a single fact. On grounds of simplicity of ontology, we should prefer the above theory. (Maybe. See the next passage.)

The above theory posits exactly the name number of truth bearers as does a monist proposition theory, that is, one for each fact.

Infinities and comparisons

I’m not well versed in infinite math, but I think there is a sense in which a theory that posits more than exactly one truth bearer for each fact posits more truth bearers than a theory that posits exactly one? There is an infinite number of facts in each case. I don’t know the technical answer.

Simplicity and truth bearers kinds

Is there some reason for preferring a theory that posits only a single kind of truth bearers over a theory that posits multiple kinds if the number of truth bearers is the same? Maybe. We seem to favor theories on the same grounds in other fields.

The sentence theory of truth bearers – problems with cognitively meaninglessness and logical implication

In a sentence theory of truth bearers, what it means to say that a sentence is cognitively meaningful is that it is true or false. To say that it is not cognitively meaningful (i.e. cognitively meaningless) means that it is not true or false.

In contrast, in a proposition theory of truth bearers, what it means to say that a sentence is cognitively meaningful is that it expresses a proposition. To say that a sentence is cognitively meaningless means that it does not express a proposition.1

I wonder if there is some problem with cognitively meaningfulness, logical implication and a sentence theory of truth bearers. Consider:

P. Colorless green ideas sleep furiously.

Q. The Earth is spherical.

Now consider a sentence that is about a logical implication from (P) to (Q):

S. That colorless green ideas sleep furiously logically implies that the Earth is spherical.

Now, it seems to me that if (P) is cognitively meaningless, then any sentence of which (P) is an antecedent or consequent, is cognitively meaningless too.

But now recall how a logical implication is defined. P implies Q iff there is no possible world in which Q is false and P true. But this is the case above. (P) is not true in any possible world at all2 and so any logical implication in which it is the antecedent is true. (Also where it is a consequent.) Thus, (S) is true. Contradiction. Something is terribly amiss.

Maybe some other definition of logical implication is needed. Suppose we stop defining it in terms of truth and falsity, and use the “is the case” phrase instead. Logical implication can then be defined as this: P logically implies Q iff there is no possible world in which P is the case and Q is not the case. Presumably all cognitive meaningless sentences are not the case. They are not false either because the semantic truth relations only hold for cognitively meaningful sentences. Now given the definition of logical implication all logical implications with a cognitively meaningless sentence as the antecedent are true. Again contradiction.

Notes

1By “means” I literally mean “means”. Not to be confused with an implication interpretation. I do not want to imply that some pluralistic proposition theory of truth bearers is false.

2Ignore potential problems with sentences meaning something else in a possible world.

Beliefs as secondary truth bearers in a pluralistic proposition theory

It is common to speak of true beliefs. As an example think of the JTB analyses of knowledge. JTB, that is, justified true belief. One could see “true belief” as a shorthand for “a belief in a true proposition”. This seems to be the case. It is common to call the theory for the JTB analysis of knowledge, but when writing down the three necessary and sufficient conditions, one does not write “has a true belief” but “p is true”.

But perhaps it is a good idea to allow for some or all beliefs to be true/false while still maintaining that it is propositions that are the primary truth bearers. A reason not to think so is again parsimony similar to the case of allowed sentences to be true too. Suppose that it is a good idea anyway.

What are the truth-conditions for beliefs?

First we may note that there seems to be no problem with ambiguity as there is with sentences as truth bearers. Perhaps there are ambiguous beliefs. We will suppose that there are none. We may, then, introduce these simple truth-conditions for beliefs:

A belief is true iff the proposition believed in is true.

A belief is false iff the proposition believed in is false.

Sentences as secondary truth bearers in a pluralistic proposition theory #2

“You said, “a sentence is true [if and only if] it expresses exactly one proposition and that proposition is true. I don’t understand the reasoning behind the “exactly one” condition as you have worded it. An implication of what you said is that a sentence that expresses more than one proposition (hence, not exactly one proposition) is not true because you said, “if and ONLY if”, but I don’t see why you would think that.

[...]

Is it because if one of the propositions is false, then the sentence is both true and false and that’s a contradiction?”

First, yes, it is to avoid conflicts with bivalence about sentences, that is, for all sentences, a sentence is either true or false but not both. But then I realized that maybe one could drop bivalence about sentences but not drop it about propositions. Supposing that one drops bivalence about sentences, then one can adopt much broader truth-conditions of sentences:

A sentence is true iff it expresses a true proposition.

A sentence is false iff it expresses a false proposition.

However it is also possible to accept broader truth-conditions even keeping bivalence about sentences. One could just specify that all the propositions expressed by a sentence has to have the particular truth value. It doesn’t matter if it is one or more:

A sentence is true iff it expresses only true propositions.

A sentence is false iff it expresses only false propositions.

The sentence theory of truth bearers – the problem of ambiguity #2

Ben Burgis over at (Blog&~Blog) has commented on my essay about the monist sentence theory of truth bearers. I have some comments on his comments. Aha! Let the comment wars begin.

Ben makes three somewhat related points. I have comments only for the two first.

The first point

Here’s what he had to say:

(1) The indexical phrasing might make things a bit confusing in this specific case. On one level, it’s surely contingent that Ben Burgis exists, but one might argue that it’s logically impossible that any instance of “I exist” tokened by anyone could ever be false. What one thinks about what to ultimately make of this might depend on what one thinks about the widely alleged essentialness of indexical claims–if “I exist” really *means* Ben Burgis exists, that’s one thing, but given that I could forget that I’m Ben Burgis but still be quite sure that I exist, there are tricky issues at play here.

I certainly did not try to get into problems by using indexicals (such as pronouns). It seems that I can avoid this issue by simply choosing another example (more about this in the second comment) or avoiding indexicals at all. I suppose I could just change it to:

S. It is logically possible that Emil Kirkegaard exists and that Emil Kirkegaard does not exist.1

(Though as for the problems with being wrong about “I exist” (the proposition!), see this discussion over at Philosophyforum.com. There is something curious about the phrase “cannot be wrong” when applied to truth bearers. It is not clear how to properly understand it. I made two quick analyses of the concept in this essay.)

The second point

Ben’s second point:

(2) Another complicating factor about the example is that existence is being treated as a predicate, which seems to assume “noneism,” the view that there are objects that have some properties (like being referred to) but which don’t exist. Anyone who agrees with Quine’s claim in “On What There Is?” that the answer to the question of ontology (“what exists?”) is “everything” would, while agreeing that it’s possible for there to be no object that Ben-Burgisizes, strong object to ◊¬Ei.

I do not believe in “noneism” (never heard of it). I only write it like that because it is simpler and not confusing in most cases. Here are two other ways to formalize the same sentence (original (S)):

1. ◊(∃x)(Ux)∧◊¬(∃x)(Ux)

2. ◊[(∃x)(Ux)∧¬(∃x)(Ux)]

(Where “Ux” is some unique description of me. I will just translate it to “is Emil Kirkegaard”, alternatively it could be “fits the unique description of Emil Kirkegaard”.)

So, in predicate logic english-ish:

1*. It is logically possible that there exists at least one person such that that person is Emil Kirkegaard and it is logically possible that it is not the case that there exists at least one person such that that person is Emil Kirkegaard.

2*. It logically possible that (there exists at least one person such that that person is Emil Kirkegaard and that it is not the case that there exists at least one person such that that person is Emil Kirkegaard).

The sheer length of this is why I usually use ‘simplified predication’ when formalizing.

More ambiguity?

3. 1. ◊(∃x)(Ux)∧¬(∃x)(Ux)

3*. It is logically possible that there exists at least one person such that that person is Emil Kirkegaard and it is not the case that there exists at least one person such that that person is Emil Kirkegaard.

That’s just applying the predicate “it is logically possible” to the first part and not the second.

Notes

1Philosophers have some weird history for using their own names in examples. I shall follow their example. Just for kicks.

2Is there any convention about what to do when both asking a question and mentioning things that require a colon (:)?

Sentences as secondary truth bearers in a pluralistic proposition theory

It seems to me that monist sentence theories are too implausible, but might it not nonetheless be the case that some sentences are true/false? In this essay I will discuss sentences as secondary truth bearers.

Pragmatic value

I can see that it has some pragmatic value to say that sentences are also sometimes true/false in addition to propositions. The pragmatic value is that it makes it easier to talk about certain things without having to use complex phrases like “the proposition expressed by (the sentence) is true (or false)”. Perhaps this is a good enough reason to posit that sentences also in some cases have the properties true/false.

An alternative solution is to invent some shorthands for talking about propositions expressed. See (N. Swartz, R. Bradley, 1979).

Parsimony

The problem I see with it is that of parsimony. “Entities must not be multiplied beyond necessity” (Wiki). Is that not exactly what we are doing? At least if properties are entities. I think they are since entity is the most inclusive set (similar to “thing”)1. But perhaps it is not as problematic to multiply properties as it is to multiply other kinds of entities in an explanation. I don’t know.

What are the conditions for a sentence being true/false?

This is how I see understand the position:

A sentence is true iff it expresses exactly one proposition and that proposition is true.
A sentence is false iff it expresses exactly one proposition and that proposition is false.

The phrase “ expresses exactly one proposition” seems to avoid the ambiguity problem that I wrote about earlier.

Notes

1Yes, I am aware of Russell’s paradox that may arise when defining sets like this. I’m working on a ‘solution’.

Truth bearers

The truth bearers are the kind of entities that have the property true. It is thought that it is the same kind of entities that have the property false too. They are sometimes referred to as the bearers of truth/falsity. I shall just refer to them as “truth bearers”.

Theories of truth bearers

There are multiple theories for what kind of entities truth bearers are. Some think it is sentences that are true/false but I think that there are too many problems with these theories. I shall call such theories for sentence theories of truth bearers.

Others, like me, think that it is propositions that are true/false, proposition theories of truth bearers.

Some presumably think something else, perhaps that it is beliefs that are truth bearers, belief theories of truth bearers.

Multiple truth bearers

It is often written “the bearers of truth/falsity”. Note the definite article “the”, it seems to imply (implicature (SEP), not implication) that there is only one kind of entities that are true/false. But could it not be that there were multiple kinds of entities that are true/false? I can see no good reason not to think this possible. The only objection that I can think of is parsimony/Occam’s Razor (Wiki); “entities must not be multiplied beyond necessity” and this presumably applies to properties too. One should not multiply properties if unnecessary. “Necessary for what?”, one might ask. “Necessary to explain truth and falsity”, I answer.

Proposed terminology

We may call theories that restrict the properties of truth/falsity to a single kind of entities for monist theories of truth bearers. Theories that allow for multiple kinds of entities may be called pluralistic theories of truth bearers.

If there entities that are always true/false, they may be called the primary truth bearers. Other entities that only in some cases bear truth/falsity may be called secondary truth bearers.

The sentence theory of truth bearers – the problem of ambiguity

I think there are numerous problems with the sentence theory of truth bearers. Here I will touch on one problem, that is, the problem of ambiguity. I start by assuming the sentence theory of truth bearers.

The problem

Consider the sentence:

S. It is logically possible that I exist and that I do not exist.

Is (S) true or false? I can’t tell because it is ambiguous. If you don’t see how it is ambiguous try deciding whether the predicate “It is logically possible” applies to only “I exist” or to both “I exist” and to “I do not exist”. Which is it? Logic helps us see the difference. We may formalize the two interpretations like this:

1. ◊Ei∧◊¬Ei
2. ◊(Ei∧¬Ei)

(Where “Ex” means x exists, “i” means I.)

We can translate these into english-ish:

1*. It is logically possible that I exist and it is logically possible that I do not exist.

2*. It is logically possible that (I exist and that I do not exist).

The first is true since my existence, anyone’s existence is a contingent matter (except contradictory entities). The second is false since it is not logically possible that I both exist and not exist (at the same time). That’s a contradiction. The problem is with deciding whether or not (S) is true or not. It can mean either (1) or (2), but which? It seems that there is no way in principle to tell whether (S) is true or not.

Both true and false

Another idea is to accept that it means both and simply say that (S) is both true and false. That doesn’t strike me as a good solution. It is basically giving up classical logic and accepting dialetheism.1

Neither true or false

One more plausible solution is to say that (S) is neither true or false; adopting this principle: All ambiguous sentences are neither true or false. The problem with this is that lots of sentences that we normally use are ambiguous, but maybe not in the context that they are used in. This is the best solution that I know of to the problem of ambiguity. Though it runs into methodological problems. When is a sentence ambiguous and when is it not?