CONCLUSIONES

Recall that the paracomplete approaches have problems concerning expressive limitations and revenge paradoxes.

i. The claim that the Liar sentence is neither true nor false, if formalized as

¬(Thλi ∨Th¬λi), does not come out true in paracomplete logics;¬(Thλi ∨

Th¬λi)must take the value1 2.

ii. Adding some extra connectives to increase expressive power gives rise to revenge paradoxes.

It is often argued that the dialetheic approaches a dual problem (e.g., Littmann & Simmons 2004 and Shapiro 2004). Specifically, dialetheic theories have prob- lems of using the notion of ‘just true’ (‘just false’) to charactize non-dialetheic true (false) sentences:

i*. The claim that a sentenceAis just true, if formalized asThAi ∧ ¬Th¬Ai, is a contradiction in dialetheic logics;ThAi ∧ ¬Th¬Aican still have the value 1

2.

(Similar problems arise out of the notion of just false.)

ii*. Increasing expressive power gives rise to revenge paradoxes, trivializing the logics (i.e., the extension ofLP T T andST T T).

In what follows, we discuss whether or not (i*) and (ii*) are tenable.

Falsity.For convenience, let us introduce a falsity predicate. Falsity is defined as truth of negation:

• FhAi iff Th¬Ai

Taking ‘iff’ to be the material biconditional, we have the following truth-table:

A ¬A ThAi FhAi

1 0 1 0 1 2 1 2 1 2 1 2 0 1 0 1

Just True. It is often argued that dialetheic approaches fail to characterize sen- tences as being just true. An obvious way to say a sentenceAis just true would be to say thatAis true but not false:

ThAi ∧ ¬Th¬Ai

Alternatively, we can write the above as:

ThAi ∧ ¬FhAi

However, both are merely equivalent toThAi. Notice that ¬FhAi(or¬Th¬Ai) is equivalent toThAi. To see that, we can show that¬FhAiand ThAialways have the same value:

» Suppose thatAtakes the value 1. ThAitakes the value1. According to the semantics ofF,FhAitakes the value0. Then, the semantics of¬tells us that

¬FhAitakes the value1.

» Suppose thatAtakes the value 1₂. ThAitakes the value 1₂. According to the semantics of F, FhAi takes the value 1

2. Then, the semantics of ¬ tells us

that¬FhAitakes the value 1₂.

» Suppose thatAtakes the value 0. ThAitakes the value0. According to the semantics ofF,FhAitakes the value1. Then, the semantics of¬tells us that

¬FhAitakes the value0.

Hence,ThAi ∧ ¬FhAiamounts toThAi. This is undesirable; because, intuitively, there is a difference between truth and just truth: if a sentence is just true, it is not a dialetheia.

If there were no difference between truth and just truth, there would be some sentences which are dialetheic but just true. Consider the Liar sentenceλ. Recall thatλ must take the value 1

2. By the identity of truth,Thλi takes the value 1 2 as

well. But sinceThAi ∧ ¬FhAiis equivalent toThAi,Thλi ∧ ¬Fhλitakes the value

1

2 as well.

Just False. Analogously, dialetheic theories has the problem of using the notion of just false to characterize non-contradictory false sentences. An apparent way to say that a sentenceAis just false would be to say thatAis false but not true:

FhAi ∧ ¬ThAi

This simply amounts toFhAi. Recall that¬FhAiandThAialways have the same value. It can easily be checked thatFhAiand¬ThAialways have the same value as well.

Moreover, consider a sentence which says of itself that it is just false: (18) (18) is just false.

Using our current expressive resource, we can formalize (18) as:

κ = ||=Fhκi ∧ ¬Thκi

Using familiar reasoning, we can show thatκis both true and just false, that is,

1 Thκi ∨Fhκi LEM 2 Thκi Assumption 3 κ 2: Release 4 Fhκi ∧ ¬Thκi 3: Def ofκ 5 Thκi ∧Fhκi ∧ ¬Thκi 2, 4:∧-Intro 6 Fhκi Assumption 7 ¬Thκi 6: FhAi ≡ ¬ThAi 8 Fhκi ∧ ¬Thκi 6, 7:∧-Intro 9 κ 9: Def ofκ 10 Thκi 10: Capture 11 Thκi ∧Fhκi ∧ ¬Thκi 11, 9:∧-Intro 12 Thκi ∧Fhκi ∧ ¬Thκi 1, 5, 12: Reasoning By Cases

Priest’s Reply. Priest suggests that the problem of ‘just false’ poses no threat to dialetheic theories. Specifically,Thκi ∧Fhκi ∧ ¬Thκiis compatible with dialethe- ism.

This is a contradiction of the kind that will sink any consistent solu- tion, but it obviously does not sink a dialetheic solution. The contra- diction is exactly what one should expect to get in the context. (Priest, 2006, p. 287)

Priest rightly points out that inconsistency is compatible with the dialetheic ap- proaches. However, it is one thing to say that a contradiction is compatible with dialetheism; it is another thing to accept that the notion of just false (just true) is inconsistent. After all, it seems that it is part of the meaning of ‘just false’ (‘just true’) that it behaves consistently (Young, 2015b). In addition, on our current ap- proach, the notion of just false (just true) cannot get the desired interpretation, since it is no different from the notion of falsity (truth).