# Modeling the concept of truth using the largest intrinsic fixed point of the strong Kleene three valued semantics (in Croatian language)

Dissertation, University of Zagreb (2004)

**Abstract**

The thesis deals with the concept of truth and the paradoxes of truth.
Philosophical theories usually consider the concept of truth from a wider
perspective. They are concerned with questions such as - Is there any connection between the truth and the world? And, if there is - What is the
nature of the connection? Contrary to these theories, this analysis is of a
logical nature. It deals with the internal semantic structure of language, the
mutual semantic connection of sentences, above all the connection of sentences that speak about the truth of other sentences and sentences whose
truth they speak about. Truth paradoxes show that there is a problem in
our basic understanding of the language meaning and they are a test for any
proposed solution. It is important to make a distinction between the normative and analytical aspect of the solution. The former tries to ensure that
paradoxes will not emerge. The latter tries to explain paradoxes. Of course,
the practical aspect of the solution is also important. It tries to ensure a
good framework for logical foundations of knowledge, for related problems
in Artificial Intelligence and for the analysis of the natural language.
Tarski’s analysis emphasized the T-scheme as the basic intuitive principle for the concept of truth, but it also showed its inconsistency with the
classical logic. Tarski’s solution is to preserve the classical logic and to restrict the scheme: we can talk about the truth of sentences of a language
only inside another essentially richer metalanguage. This solution is in harmony with the idea of reflexivity of thinking and it has become very fertile
for mathematics and science in general. But it has normative nature |
truth paradoxes are avoided in a way that in such frame we cannot even
express paradoxical sentences. It is also too restrictive because, for the same
reason we cannot express a situation in which there is a circular reference
of some sentences to other sentences, no matter how common and harmless
such a situation may be.
Kripke showed that there is no natural restriction to the T-scheme and we have to accept it. But then we must also accept the riskiness of sentences
| the possibility that under some circumstances a sentence does not have
the classical truth value but it is undetermined. This leads to languages
with three-valued semantics. Kripke did not give any definite model, but he
gave a theoretical frame for investigations of various models | each fixed
point in each three-valued semantics can be a model for the concept of truth.
The solutions also have normative nature | we can express the paradoxical
sentences, but we escape a contradiction by declaring them undetermined.
Such a solution could become an analytical solution only if we provide the
analysis that would show in a substantial way that it is the solution that
models the concept of truth.
Kripke took some steps in the direction of finding an analytical solution.
He preferred the strong Kleene three-valued semantics for which he wrote
it was "appropriate" but did not explain why it was appropriate. One
reason for such a choice is probably that Kripke finds paradoxical sentences
meaningful. This eliminates the weak Kleene three valued semantics which
corresponds to the idea that paradoxical sentences are meaningless, and thus
indeterminate. Another reason could be that the strong Kleene three valued
semantics has the so-called investigative interpretation. According to this
interpretation, this semantics corresponds to the classical determination
of truth, whereby all sentences that do not have an already determined
value are temporarily considered indeterminate. When we determine the
truth value of these sentences, then we can also determine the truth value
of the sentences that are composed of them. Kripke supplemented this
investigative interpretation with an intuition about learning the concept
of truth. That intuition deals with how we can teach someone who is a
competent user of an initial language (without the predicate of truth T)
to use sentences that contain the predicate T. That person knows which
sentences of the initial language are true and which are not. We give her
the rule to assign the T attribute to the former and deny that attribute to
the latter. In that way, some new sentences that contain the predicate of
truth, and which were indeterminate until then, become determinate. So
the person gets a new set of true and false sentences with which he continues
the procedure. This intuition leads directly to the smallest fixed point of
strong Kleene semantics as an analytically acceptable model for the logical
notion of truth. However, since this process is usually saturated only on
some transfinite ordinal, this intuition, by climbing on ordinals, increasingly
becomes a metaphor.
This thesis is an attempt to give an analytical solution to truth paradoxes. It gives an analysis of why and how some sentences lack the classical
truth value. The starting point is basic intuition according to which paradoxical sentences are meaningful (because we understand what they are
talking about well, moreover we use it for determining their truth values),
but they witness the failure of the classical procedure of determining their
truth value in some "extreme" circumstances. Paradoxes emerge because
the classical procedure of the truth value determination does not always
give a classically supposed (and expected) answer. The analysis shows that
such an assumption is an unjustified generalization from common situations
to all situations. We can accept the classical procedure of the truth value
determination and consequently the internal semantic structure of the language, but we must reject the universality of the exterior assumption of a
successful ending of the procedure. The consciousness of this transforms
paradoxes to normal situations inherent to the classical procedure. Some sentences, although meaningful, when we evaluate them according to the classical truth conditions, the classical conditions do not assign them a unique
value. We can assign to them the third value, \undetermined", as a sign of
definitive failure of the classical procedure. An analysis of the propagation
of the failure in the structure of sentences gives exactly the strong Kleene
three-valued semantics, not as an investigative procedure, as it occurs in
Kripke, but as the classical truth determination procedure accompanied by
the propagation of its own failure. An analysis of the circularities in the
determination of the classical truth value gives the criterion of when the
classical procedure succeeds and when it fails, when the sentences will have
the classical truth value and when they will not. It turns out that the
truth values of sentences thus obtained give exactly the largest intrinsic
fixed point of the strong Kleene three-valued semantics. In that way, the argumentation is given for that choice among all fixed points of all monotone
three-valued semantics for the model of the logical concept of truth. An
immediate mathematical description of the fixed point is given, too. It has
also been shown how this language can be semantically completed to the
classical language which in many respects appears a natural completion of
the process of thinking about the truth values of the sentences of a given
language. Thus the final model is a language that has one interpretation
and two systems of sentence truth evaluation, primary and final evaluation.
The language through the symbol T speaks of its primary truth valuation,
which is precisely the largest intrinsic fixed point of the strong Kleene three
valued semantics. Its final truth valuation is the semantic completion of
the first in such a way that all sentences that are not true in the primary
valuation are false in the final valuation.

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