In the last section, the language L we considered is without a truth predicate. This language is interpreted by the base modelsM = hD,Ii. Now we consider an expanded languageL+ that allows for self-reference and contains a predicate
T which is intended to play the role of the truth predicate.
To give the interpretations ofL+, we extend the base models to do so. Specifically,
predicateT as truth. One important constraint onT is that the interpretation of the predicateT should obey the transparent truth principle (TT).
• The Transparent Truth Principle (TT):ThAi = ||=A
for anyA∈ L+. TT ensures that for anyA∈ L+,ThAiandAare intersubstitutable
in any extensional context.
Moreover,Thλi ∨ ¬Thλi, an instance of the LEM, should fail in the logic. A logic where the LEM fails is not hard to come by. In fact, we have2K3 A∨¬A. However,
it is not a trivial matter to construct such a logic in whichThλi ∨ ¬Thλifails, if we require the intersubstitutability ofThAiandA.
Since the target logic is based onK3 and closed under TT, we call the logicK3T T.
In summary, we require thatK3T T has the following desiderata:
i. K3T T interprets the predicateT as truth. Specifically, we should have: any
A ∈ L+,A = ||=K3T T ThAi.
ii. 2K3T T ThAi ∨ ¬ThAi
In what follows, we will discuss how Kripke constructs a logic with these desider- ata.
Firstly, Kripke proposes seeing the predicateT as partially defined. Rather than a simple extension, the predicateT is assigned a pair of sets: the extensionT+and
the nti-extensionT−. The truth predicate is true of the things inT+, and false of
the things inT−
.
The extensionT+and the anti-extensionT−
are mutually exclusive:T+∩T−=∅
. So the predicateT cannot be true and false of the same thing. However, some objects (i.e., (codes of) sentences ofL+) are allowed to fall neither in the extension
T+nor the anti-extensionT−
. In other words, there could be something that the predicateT is neither true of nor false of.
Secondly, to interpret the predicateT as truth, we need to ensure that
• Identity of Truth (IT):ThAiandAalways have the same value.
Notice that IT and TT are different. IT only concerns how models assign value; it says nothing about entailment relation betweenThAiandA. Thus, one should not conflate IT with TT. That being said, IT is a crucial step for obtaining TT. Gen- erally speaking, a countermodel to an argument is defined in such a way that it must assign different values to the premises and the conclusions. Accordingly, if ThAi and A always have the same value, there is no countermodel to an ar- gument fromThAi toAand vice versa. Thus, given that logical consequence is defined as the absence of countermodels, then IT ensures TT.
Imposing IT on models amounts to ensuringT+ to be the set of things of which
it is true, andT−
to be the set of thing of which it is not true. LetT+
M+ be the set
of (codes of) true sentences of M+, and let T−
M+ be the set of objects in D such
that either the objects are not (codes of) sentences ofM+, or are (codes of) false
sentences ofM+. To ensure IT, we need to ensure thatT+ =T+
M+ andT
−=T−
M+.
predicateT is true of (false of)AiffAtakes the value1(the value0). In summary, the target model is as follows.
Definition 6(K3+model). AK3+ model is a structureM+ =hD,I,T isuch that
(i) Dis the domain.
(ii) I is the same asI in theK3 models.
(iii) T is a pairhT+,T−i, such that
– T+assigns a set of objects to the extension of the truth predicateT+⊆
D; – T−
assigns a set of objects to the anti-extension of the truth predicate
T− ⊆ D . – T+∩ T− =∅. – For anyT+,T− ∈ D ,hT+,T−i=hT+ M+,T − M+i, where TM++ is the set of (codes of) true sentences ofM+, and
TM−+ is the set of objects inDsuch that either the objects are not (codes
of) sentences ofL+, or are (codes of) false sentence ofM+.
Logical consequence is construed as the absence of countermodels. In many- valued logics, it is customary to use the notion of designated value to define countermodels. Since we can think of the designated values as the values for good sentences, it is natural to define countermodels as follow:
• A countermodel to an argument from the premisesΓto the conclusions∆is a model that assigns a designated value to every member ofΓ and assigns a non-designated value to every member of∆.
Call this theprinciple of designated value preservation. Since the value1is the only designated value, we have:
• A K3+ model is a countermodel to an argument from the premisesΓto the conclusions∆iff it assigns the value1to every premise ofΓand the value
1
2 or the value0to every conclusion of∆.
Thus,K3T T consequence amounts to:
Definition 7(K3T T Consequence). Γ |=K3T T ∆ iff ifvM+(A) = 1for allA ∈Γ,
thenvM+(B) = 1for someB ∈∆.