Abstract
We start by noting that the set-theoretic and semantic paradoxes are framed in terms of a definition or series of definitions. In the process of deriving paradoxes, these definitions are logically represented by a logical equivalence. We will firstly examine the role and usage of definitions in the derivation of paradoxes, both set-theoretic and semantic. We will see that this examination is important in determining how the paradoxes were created in the first place and indeed how they are to be solved in a uniform way. There are three features that are special about these definitions used in the derivation of most of the above paradoxes. The first is the use of self-reference between the definiens and the definiendum, the second is the generality of the definiendum, and the third is the under-determination and over-determination of concepts that usually occur as a result of these definitions. We will examine the impact of these three features on the logical representation of definitions and show how this representation then leads to a uniform paradox solution using an appropriate logic that is both paraconsistent and paracomplete. However, it is the paracompleteness, exhibited through the rejection of the Law of Excluded Middle, together with the rejection of contraction principles, that enables the solution of the paradoxes to go through. We characterize definitions as involving syntactic identity and/or meaning identity. We point out that some paradoxes do not have explicit self-reference or circularity and some may not utilize the generality of the definiendum, but the general characterizations of definitions that we give will still apply. We also look beyond all this to paradoxes that rely on illicit definitions between objects that are essentially different to start with.
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Notes
These syntactic identities are what Bob Meyer has called dishonest definitions on p. 223 in his (1974). He calls these definitions dishonest because they involve making a certain meta-linguistic stipulation, as opposed to his honest definitions which enrich the formal system by introducing a new symbol, together (usually) with some defining axioms.
In this regard, consider the traditional Lesniewskian criteria of eliminability and non-creativity for definitions. Here, we are unable to eliminate the definition of L, but we would still regard this definition of the Liar sentence as a definition since this was clearly the intention and the meaning identity would still stand. Regarding non-creativity, on the other hand, one would expect the introduction of the Liar sentence to be non-creative in that one should not be able to derive new sentences from L which do not involve L, because the paradoxes are blocked by using the logic MC (see below) as this logic lacks Contraction principles and the Law of Excluded Middle. Indeed, Brady showed in (2006) that such a system is simply consistent. Further, such a system steers clear of Ogaard’s paths to triviality in his (2016).
Brady and Meinander (2013) explains that distribution in \(\rightarrow \)-form does not follow from the meanings of conjunction and disjunction as determined by their introduction and elimination rules, and so it should not be regarded as expressing an intensional set-theoretic containment, which is based on meaning rather than just truth-preservation. The reason sets are used here comes from their usage in representing logical contents in the content semantics. See Brady (1996, 2006, 2016). Also, the term ‘containment’ applies to sets x and y such that \(\hbox {x} \supseteq \hbox {y}\) and is thereby a transitive relation. The rule-form of distribution, however, expresses an extensional set-theoretic containment, as rules are truth-preserving and the meaning-containment \(\rightarrow \) is absent from the form of this rule.
The term ‘metacomplete’ was introduced by Meyer in (1976) for logics without negation that satisfy standard truth-conditions for conjunction, disjunction, and the two quantifiers, together with those embracing Modus Ponens and Modus Tollens for implication. This definition was expanded to include logics with negation by Slaney in (1984; 1987), who distinguished M1- and M2-logics, according to whether there are no negated entailment theorems or whether negated entailments satisfy the standard truth-condition. One should note however that truth in this context amounts to provability and that the methodology employed here is proof theory and not semantics.
The meaningful sentences must have some meaning so that valid deduction can be made upon them. If meaningless sentences are included, then we need to add a value ‘non-significance’ to the four given below, but such a value would then extend to any compound sentence that includes such a meaningless sentence. This is what Priest calls a ‘garbage-in garbage-out’ value in his work on Buddhist logic. On this point, reference can also be made to the foundational work, Goddard and Routley (1973).
The conceivability of contradictions is a controversial claim, especially in the context of the paradoxes. There is little to prevent formal logical systems from being contradictory especially if there are no constraints on premises and rules that can be employed. So, the Disjunctive Syllogism, and its deductively equivalent (see footnote 7) Explosion Rule (A, \(\sim \hbox {A} \Rightarrow \hbox {B}\)), should fail for such systems in general, especially for the latter rule where the conclusion B can be totally irrelevant from the A of the premises. However, conceivability of concepts is important in framing formal systems so that they can be usefully employed. Moreover, conceivability can be used as a yardstick to keep contradictions in check as it is very hard (or, indeed, impossible) to see how a contradiction can be conceived in practice. In the case of the paradoxes, I contend that inappropriate assumptions in the logic would have been made that enable the paradoxical contradictions to be derived. This will be seen in Sects. 5, 6 and 7.
Deductive equivalence is a general term that applies to deductions in both directions between two theorems or between a theorem and a derived rule or between two derived rules. In the process of such deduction, uniform substitution upon theorems can be used along with the usual usage of theorems and derived rules of the logic. Thus, a two-way derived rule, constituted as derived rules in both directions, is a special case of a deductive equivalence, as one cannot use uniform substitution in performing the derivation of a derived rule.
See foonote 7.
See foonote 7.
See foonote 7.
The disjunctive meta-rule that is needed is the one-place version: if A \(\Rightarrow \) B then \(\hbox {C}\vee \hbox {A} \Rightarrow \hbox {C}\vee \hbox {B}\), derivable from the meta-rule MR1 of MC.
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This paper was presented to the UniLog 5 Congress held during June, 2015, at the University of Istanbul, and to the Logic Seminar of the University of Melbourne during 2015. A big thank you to my audiences in Istanbul and Melbourne, especially to Graham Priest, Hartley Slater and Elia Zardini, who raised pertinent questions, which have been taken into account in the final text. I also wish to thank the editor and two anonymous referees for the considerable effort they have put into this paper and for recommending a number of major and a large number of minor corrections which, when taken into account, have greatly enhanced the overall impact and readability of this paper.
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Brady, R.T. The use of definitions and their logical representation in paradox derivation. Synthese 199 (Suppl 3), 527–546 (2021). https://doi.org/10.1007/s11229-017-1362-7
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DOI: https://doi.org/10.1007/s11229-017-1362-7