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).

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}