3 Problems
The first problems we discuss relate to the proof theory of Gödel logics.
In this context, we might identify analyticity with exhibiting an adequate subformula property. Recall that \({ G_{\textsc {iso}_0} }\) is recursively enumerable (see above).
A possible way to solve this problem might be to first provide an analytic calculus for intuitionistic logic extended by the schema
\(\textsc {iso}_0\) and then embedding it into an analytic hypersequent calculus deriving
\(\textsc {lin}\) and
\(\textsc {qs}\) (see Baaz et al.
1998). Such an extension of intuitionistic logic could be based on weakened eigenvariable conditions that are sound for classical logic by considering that every negated subformula of a formula is classical in presence of
\(\textsc {iso}_0\).
This approach seems promising, as a hypersequent calculus based on intuitionistic logic has been shown to be an adequate analytic calculus for
\(G_{[0,1]}\) (see Baaz et al.
2003). Indeed, the logic
\(G_{[0,1]}\) is clearly better understood. Nonetheless, several questions about it are yet to be answered. An example of this is the issue of interpolation.
Propositional Gödel logic is one of the seven intermediate logics that admit interpolation (Maksimova
1979). Moreover, it admits strong interpolation, as propositional quantifiers are eliminable in propositional quantifier logic. By a result of Mints et al. (
2013), intuitionistic logic augmented with
\(\textsc {qs}\) does not interpolate. In contrast to this, other extensive fragments of
\({ G_{[0, 1]} }\) do admit interpolation: let
\(G^-_{[0,1]}\) be the fragment of
\(G_{[0,1]}\) consisting of all formulae of the form
$$\begin{aligned} \bigwedge _i A_i \rightarrow \bigvee _i B_i, \end{aligned}$$
(1)
such that each
\(A_i\) and
\(B_i\) are prenex except perhaps for universal quantifiers on the left-hand side or existential quantifiers on the right-hand side. We call those quantifiers
weak quantifiers. If
\(\varphi \) is of the form (
1), we call its antecedent the
negative part of
\(\varphi \) and its consequent the
positive part.
A related problem is the following:
The prenex fragment of
\({ G_{[0, 1]} }\) admits Skolemization (Baaz et al.
2001). Intuitionistic logic with the addition of an existence predicate also admits Skolemization (Baaz and Iemhoff
2006). This Skolemization cannot be extended directly to Gödel logics because the additional axioms
\(\textsc {lin}\) and
\(\textsc {qs}\) used in a proof are in general not reducible to finitely many universal formulae without extending the base language.
The problem arises from the fact that the usual critical formulae of classical logic
\(A(t) \rightarrow A(\varepsilon _x A(x))\) (see Hilbert and Bernays
1970) are not conservative, i.e., it is not the case that only the translations of valid formulae are derivable. For instance:
Our next problem deals with an extension of Gödel logic. Specifically, consider a notion of identity extending interpretations
\(\mathfrak {I}\) such that
$$\begin{aligned} \mathfrak {I}(s=t) = 1 \text { if, and only if, } \mathfrak {I}(s) = \mathfrak {I}(t). \end{aligned}$$
According to the above proof,
\(2^{\aleph _0}\) is an upper bound on the cardinalities of models of (
2). The following result is well known (see, for example, Bell and Slomson
1969).
We remark that without the operator
\(\varDelta \), the Löwenheim–Skolem theorem holds, because the Gödel logics correspond to finite or countable linear Kripke frames with constant domains (Beckmann and Preining
2007).
Problem
6 and Problem
8 below concern recursive enumerability of various Gödel logics.
We note that first-order \({ G^P_{[0, 1]} }\) admits conversion to prenex normal forms.
A consequence of a positive solution to Problem
6 would be that if one could show that first-order
\({ G^P_{[0, 1]} }\) admits Herbrand disjunctions, then one could construct suitable Skolem functionals for
\({ G^P_{[0, 1]} }\).
Propositional quantifiers are eliminable in propositional
\({ G^P_{[0, 1]} }\) (Baaz and Veith
1998). This logic is the only quantified propositional Gödel logic with this property. However, propositional
\({ G_\uparrow }\) (resp.
\({ G_\downarrow }\)) admit propositional quantifier elimination if an additional one-place connective is added (Baaz et al.
2000; Baaz and Preining
2008), namely
\(\circ Y = \dot{\forall }X \big ((X \rightarrow Y) \vee X\big )\) (resp.
\(\dot{\forall }X\big ((X \rightarrow Y)\rightarrow X\big )\)). This connective assigns to
Y the immediate successor (resp. predecessor) of its value.
We can also restrict our attention to monadic fragments of Gödel logic.
Clearly, the monadic fragments of
\({ G_{[0, 1]} }\) and
\({ G_{\textsc {iso}_0} }\) are r.e. On the other hand, the monadic fragment of
\({ G_\uparrow }\) is not r.e. as any nonvalid sentence has a finitely valued countermodel and the monadic fragment of
\({ G_\uparrow }\) is undecidable (see Baaz and Preining
2016). In particular, we might wonder:
It is known that if we restrict ourselves to monadic fragments with one predicate symbol, there are countably many Gödel logics of increasing complexity (see Beckmann and Preining
2015). These fragments can be considered as quantified propositional logics where the actual set of truth values is determined by the interpretation.
Up to now, we have only discussed problems of validity. The representation of Gödel logics with respect to validity can be always reduced to the sets of valid sentences. However, if we speak of validity and satisfiability in parallel, this representation is useless and we therefore have to refer to the truth-value sets. This is unfortunate, as uncountably many different Gödel logics with respect to truth-value sets correspond to countably many sets of valid sentences (Beckmann et al.
2008). The following problem is a particular instance of this phenomenon.