Clear Language, Clear Mind

In an earlier essay I mentioned that that meaninglessness is contagious with respect to sentences. One can pretty easily formulate the principle in normal english – if a sentence is meaningless, then so is any more complex sentence of which it is a part of. To get a proper, formal formulation of this we may simply think of the rules in logic systems used to form well-formed formulas (=wff’s) and then formulate some similar principles for the meaninglessness of sentences. Here’s what I have in mind:

Negation. For all sentences, iff it is not the case that a sentence is meaningful, then it is not the case that the negation of that sentence is meaningful.

(∀S)(¬M(S)↔¬M(¬S)

Conjunction part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the conjunction of that sentence with another sentence is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S∧Z)¹

Disjunction part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the disjunction of that sentence with another sentence is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S∨Z)

Implication/conditional part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the implication of the first sentence to the second is meaningful, and it is not the case that the implication of the second sentence to the first is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S→Z)∧¬M(Z→S))

Bi-implication/bi-conditional part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the bi-implication of the first sentence to the second is meaningful, and it is not the case that the bi-implication of the second sentence to the first is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S↔Z)∧¬M(Z↔S))

This should cover propositional logic. It is left to the reader can invent the relevant principles for modal logics and predicate logic.

Notes

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

This is not because I am not actively writing anything. It is because I am working on a very large (for me) project – a reform proposal to the danish orthography. Currently the draft is about 40 pages and it is not done yet. It is written in danish and I do not expect to translate it into english. I will post it on my sister (brother?) blog when it’s done.

I noticed a small dissimilarity between the two words. As I have pointed out numerous times in the past, the phrase “I don’t believe that p” is ambiguous between belief in not-p and lack of belief in p. However the similar phrase for knowledge, “I don’t know that p” is not similarly ambiguous. It is however ambiguous in another way; between lack of belief in p and in not-p, and lack of knowledge that p.

This is a common yet relatively unknown fallacy. The typical situation is this: Someone is defending some view or theory. That someone acknowledges the existence of a number of objections to the view/theory that he is defending. He then defeats these objections to his own satisfaction and concludes that there are no good objections. Presuming that the person is rational, this is where he ought to conclude that there are no good objections known to him. He should not conclude that there are none.

Interestingly, I found logician, Graham Priest, that commits this fallacy (oh well, even logicians commit fallacies but hopefully less or less frequently than other people). Graham Priest defends his dialetheism theory in his book In Contradiction. On pages 238-240 he defends a view about the transmission of obligations. He defends that view against some objections and then concludes:

“[...attempting to refute objections...] The principle of the transmission of obligation is, therefore, perfectly acceptable.” (p. 240)

Such a thing does not follow. It is possible and even probable that there are other good objections which render the view not perfectly acceptable.

http://www.pagef30.com/2008/08/why-norwegian-is-easiest-language-for.html

Actually a very good read.

Self-explanatory title.

“Web(s) of belief” ≡ “web”

“Object(s) of belief” ≡ “oob”

The justification of the web of belief

A web is more or less justified. The justification of a web is a function of its members in many ways. Here are some ways that I speculate may increase the justification of a web. I do not pretend to offer much argumentation for my thoughts or much certainty in the conclusions. It seems to me that it is extremely hard to have any strong evidence the beliefs about these matters. That shall not keep me from examining the matter and giving my intuitions.

The number of beliefs in a web

Imagine a web with only two beliefs whose oob logically implied each other. Think of any two logically equivalent propositions. The interconnectedness of that web is extremely high since logical implication is one of the strongest relations two oob can stand in (see below) and all the members are connected to each other by logical implication. But still it seems to me that such a web is not very justified. I suggest that we explain that by the number of beliefs in the web. If a person with the aforementioned web gave an argument to another person, the other person would (and should) respond that it is circular. It seems to me that we cannot avoid circularity in our justification (because of the infinite regress argument and that epistemic foundationalism and epistemic infinitism is false). However circularity is not much of a problem when the web contains many thousand belief as it does of any grown-up human.

The number of relations of certain kinds between the oob

The oob are truth carriers. (Just substitute your favorite truth carrier be it propositions, sentences, beliefs etc.)

We may distinguish between three kinds of relations between the oob: (1) positive relations, these are the relations that increase the justification of a web as a function of their number, and the justification of a web is partial function of the positive relations between oob, (2) negative relations, which is the opposite of positive relations; they decrease the justification of a web, (3) neutral relations, relations that have no effect on the justification of a web. We may note that this distinction is true regardless of the distribution of relations in the three categories.

Then we ask ourselves: Which relations are positive relations? Deductive relations such as (for all x, and for all y) “x logically implies y”, “x materially implies y” come to mind. Inductive relations such as “x is explained by y”, and “x gives good reason to believe y”, “x is best explained by y” seem to me to increase the justification of a web.

Similarly, which are negative relations? Basically the same of the above just with the added change that it is the negation of y. If you believe two things, and the one logically implies the negation of the other, you have an inconsistent web. It is impossible for all the oob to be true at the same time in a such web.

That a web has at least two beliefs whose oob are inconsistent does not imply that the justification of the web is zero. To see this we should simply recall that all grown-ups have inconsistent oob and that not all web of grown-ups have an equal level of justification. Hence, it is not the case that if a web contains beliefs whose oob are inconsistent, then the justification of that web is zero. Since if it was the case, then web of all grown-ups would be equally justified, all having zero justification. However, it is still the case that such inconsistent oob reduce the justification of a web, which I why we ought to change our mind when we discover that we hold beliefs whose oob are inconsistent.

I can’t think of any neutral relation, but they are not very relevant anyway, so lets disregard an example of a such. There may be no such relation for all I know.

Interconnectedness

I mentioned this in passing above but it deserves elaboration. The justification of a web is also a partial function of the interconnectedness of a web. If a web consisted of a thousand beliefs but that these were divided into 10 groups of beliefs each of whose oob did not have any positive relations with the oob of the other groups of belief, then it seems to me that the justification of that web would be very low. This seems best explainable by justification being a partial function of interconnectedness too.

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)(x_w)))

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

6. (∀w)(∀x)(Sx⇒(x_w⇒(∃x_w)))

⊢ 7. (∀w)(∃x_w)(Sx) [from 5, 6, MP]

8. (∃w)¬(∃x_w)(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).

Thanks to TorrentFreak for enlightening me.

http://en.wikipedia.org/wiki/Streisand_effect