This paper provides a procedure which, from any Boolean system of sentences, outputs another Boolean system called the ‘_m_-cycle unwinding’ of the original Boolean system for any positive integer _m_. We prove that for all \(m>1\), this procedure eliminates the direct self-reference in that the _m_-cycle unwinding of any Boolean system must be indirectly self-referential. More importantly, this procedure can preserve the primary periods of Boolean paradoxes: whenever _m_ is relatively prime to all primary periods of a Boolean paradox, this (...) paradox and its _m_-cycle unwinding have the same primary periods. In this way, we can produce an indirectly self-referential Boolean paradox with the same periodic characteristics as a known Boolean paradox. (shrink)

In Hannes Leitgeb’s article What truth depends on the author provides a formally correct and materially adequate truth definition for the set of all grounded sentences, defined as the least fixed point of a monotone operator of semantic dependence. In this paper we will focus on the mathematical aspects of Leitgeb’s notions of dependence, grounding and truth, recasting Leitgeb’s construction in a functional setting in which we establish some new facts about these notions.

The unwinding that Cook, 767–774 2004) proposed is a simple but powerful method of generating new paradoxes from known ones. This paper extends Cook’s unwinding to a larger class of paradoxes and studies further the basic properties of the unwinding. The unwinding we study is a procedure, by which when inputting a Boolean modal net together with a definable digraph, we get a set of sentences in which we have a ‘counterpart’ for each sentence of the Boolean modal net and (...) each point of the digraph. What is more, whenever a sentence of the Boolean modal net says another sentence is necessary, then the counterpart of the first sentence at a point correspondingly says the counterparts of the second one at all accessible points of that point are all true. The output of the procedure is called ‘the unwinding of a Boolean modal net on a definable digraph’. We prove that the unwinding procedure preserves paradoxicality: a Boolean modal net is paradoxical on a definable digraph, iff the unwinding of it on this digraph is also paradoxical. Besides, the dependence digraph for the unwinding of a Boolean modal net on a definable digraph is proved to be isomorphic to the unwinding of the dependence digraph for the Boolean modal net on the previous definable digraph. So the unwinding of a Boolean modal net on a digraph is self-referential, iff the Boolean modal net is self-referential and the digraph is cyclic. Thus, on the one hand, the unwinding of any Boolean modal net on an acyclic digraph is non-self-referential. In particular, the unwinding of any Boolean modal net on \ is non-self-referential. On the other hand, if a Boolean modal net is paradoxical on a locally finite digraph, the unwinding of it on that digraph must be self-referential. Hence, starting from a Boolean modal paradox, the unwinding can output a non-self-referential paradox only if the digraph is not locally finite. (shrink)

This paper aims to explore the relationship between the necessity predicate and the truth predicate by comparing two possible-world interpretations. The first interpretation, proposed by Halbach et al. (J Philos Log 32(2):179–223, 2003), is for the necessity predicate, and the second, proposed by Hsiung (Stud Log 91(2):239–271, 2009), is for the truth predicate. To achieve this goal, we examine the connections and differences between paradoxical sentences that involve either the necessity predicate or the truth predicate. A primary connection is established (...) through two translations that change only one of the predicates to the other while keeping everything else unchanged. We prove that in bijective frames, a set of sentences that contains one of the two semantic predicates has the same paradoxicality as the corresponding set of sentences that contains the other predicate obtained through translation. However, there are substantial differences as well. First, the necessity predicate and the truth predicate, under the two interpretations, cannot be defined by each other. Moreover, for sentences that involve only the truth predicate, their paradoxicality is preserved under the homomorphisms of frames. For sentences containing the necessity predicate, their paradoxicality is preserved under bounded morphisms, but none of these sentences can have their paradoxicality preserved under the extension of frames. Finally, we also show that paradoxical sentences involving the necessity predicate and those involving the truth predicate differ significantly in terms of mirror symmetry, circularity dependence, and frame compactness. (shrink)

In various theories of truth, people have set forth many definitions to clarify in what sense a set of sentences is paradoxical. But what, exactly, is _a_ paradox per se? It has not yet been realized that there is a gap between ‘being paradoxical’ and ‘being a paradox’. This paper proposes that a paradox is a minimally paradoxical set meeting some closure property. Along this line of thought, we give five tentative definitions based upon the folk notion of paradoxicality implied (...) in Tarski’s undefinability theorem of truth and the dependence relation proposed by Leitgeb (J Philos Log 34(2):155–192, 2005). A definition of ‘being a paradox’ is acceptable if its extension must be sufficiently comprehensive to include all known paradoxes on the one hand, and its standard must also be high enough to exclude those that are evidently not paradoxes on the other hand. It turns out that only the last attempt is acceptable: a set of sentences is a paradox if it is paradoxical and self-dependent, and at the same time, it and any of its paradoxical and self-dependent subsets bisimulate each other. In this way, we articulate in what sense a set of sentences is a paradox, and so establish a connection between the notion of a paradox and the notion of paradoxicality. (shrink)