Identity, Haecceity, and the Godzilla Problem

1979. Identity Logics. (Co-author: Steven Ziewacz) Notre Dame Journal of Formal Logic 20, 777–84. MR0545427 (80h: 03017) In this paper we prove the completeness of three logical systems IL.1, IL2, and IL3. IL1 deals solely with identities {a = b), and its deductions are the direct deductions constructed with the three traditional rules: (T) from a = b and b = c infer a = c, (S) from a = b infer b = a and (A) infer a = a (from anything). IL2 deals solely with identities and inidentities {a ≠ b); its deductions include both the direct and the indirect deductions constructed with the three traditional rules. IL3 is a hybrid of IL1 and IL2: its deductions are all direct as in IL1 but it deals with identities and inidentities as in IL2. IL1 and IL2 have a high degree of naturalness. Although the hybrid system IL3 was constructed as an artifact useful in the mathematical study of IL1 and IL2, it nevertheless has some intrinsically interesting aspects. The main motivation for describing and studying such simple systems is pedagogical. In teaching beginning logic one would like to present a system of logic which has the following properties. First, it exemplifies the main ideas of logic: implication, deduction, non-implication, counterargument, logical truth, self-contradiction, consistency, satisfiability, etc. Second, it exemplifies the usual general meta-principles of logic: contraposition and transitivity of implication, cut laws, completeness, soundness, etc. Third, it is simple enough to be thoroughly grasped by beginners. Fourth, it is obvious enough so that its rules do not appear to be arbitrary or purely conventional. Fifth, it does not invite confusions which must be unlearned later. Sixth, it involves a minimum of presuppositions which are no longer accepted in mainstream contemporary logic. These logics are far superior to propositional logic a pedagogical catastrophe. Tarski’s INTRODUCTION TO LOGIC doesn’t start with propositional logic but with an identity logic that is more complicated that these. They are also superior to Aristotelian logic (or syllogistic), which is irrelevant to contemporary applications despite its intrinsic beauty and its undeniable historical importance. *The authors wish to thank Wendy Ebersberger (SUNY/Buffalo) and George Weaver (Bryn Mawr College) for useful criticisms and for encouragement.

Rivista di estetica, 2007

Journal of Philosophical Logic

Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free logics usually reject the claim that names need to denote in (ii), and of the systems considered in this paper, the positive free logic concedes that some atomic formulas containing non-denoting names (namely self-identity) are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject (i), are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired str...

Advances in Logic, 2001

In this paper I present a dialogical formulation of free logic which allows a straightforward combination of paraconsistent and intuitionistic logic-I call this combination Frege's Nightmare. The ideas behind this combination can be expressed in few words: in an argumentation, it sometimes makes sense to restrict the use and introduction of singular terms in the context of quantification to a formal use of those terms. That is, the Proponent is allowed to use a constant for a defence (of an existential quantifier) or an attack (on a universal quantifier) iff this constant has been explicitly conceded by the Opponent. When the Opponent concedes any constant occurring in an atomic formula he concedes tertium non-datur (for this formula) to. This yield a free-logic which combines classical (for propositions with singular terms for realities) with intuitionistic logic (for propositions with singular terms for fictions). This extended free-logic can be also combined with paraconsistent logic in such a way that contradictory objects can be included in the domain of fictions. The idea is here to combine the concept of formal use of constants in free logics and that of the formal use of elementary negations in paraconsistent logics.

1994

There is a long philosophical tradition concerning the problem of non-existing entities, what status do they have? What do they refer to? Can statements involving non-existing entities have truth-values? Are all non-existing entities the same? If so, does this imply that "Pegasus = Sherlock Holmes" is true? If not, how do we distinguish between non-existing entities, if such is possible? In contemporary philosophy, the problem crops up when one is dealing with, e.g., possible worlds. If these are considered to be fictional entities, how many possible worlds are there? According to some philosophers of mathematics, the nominalists in particular, most mathematical concepts are fictions. If so, how do we deal with them? Obviously, any answer(s) to this set of problems must first of all deal with the problem of distinguishing between existent and non-existent entities. To a certain extent-and this is the position defended by Karel Lambert in his introduction to this volume The Nature of Free Logic-the history of logic can be interpreted as a gradual explicitation of the underlying existence assumptions in formal reasoning. In Aristotelian syllogistic reasoning, the inference from "All 5 are P" to "Some 5 are P" was considered correct because it was implicitly assumed that a universal statement carried an existential commitment that there is at least one 5 that is P. Hence the correctness of the conclusion. In Fregean logic this is no longer the case. It must be explicitly added that there is at least one 5 that is P to derive, trivially, the conclusion. But modern logic allows the use of singular terms (or individual constants). If t is such a singular term then it is true that t = t. Apply the rule of existential generalization and one obtains (3x)(x = t). Hence, whatever it is that t refers to, it must exist. Obviously, when talking about Pegasus or Sherlock Holmes, I do not wish to commit myself to their existence. A similar move as in the transition from Aristotelian to modern logic is required vis-à-vis singular terms. Free logic has set itself

In this paper an outline of a class of heterodox logics is presented. These systems, roughly speaking, question the standard notion of identity as given by classical logic and standard mathematics. Due to the apparent fundamentality of this notion, it is necessary to provide a strong motivation for the elaboration of such systems, and perhaps the main one comes from quantum mechanics. This link is mentioned, and some references are made, but no details are presented; the references provide more detailed works. We just describe some non-reflexive logics and mathematics, ending with a general discussion about the necessity of identity.

British Journal for the Philosophy of Science, 2013

We show that finitely axiomatized first-order theories that involve some criterion of identity for entities of a category C can be reformulated as conjunctions of a non-triviality statement and a criterion of identity for entities of category C again. From this we draw two conclusions: Firstly, criteria of identity can be very strong deductively. Secondly, although the criteria of identity that are constructed in the proof of the theorem are not good ones intuitively, it is difficult to say what exactly is wrong with them once the modern metaphysical view of identity criteria is presupposed.

2012

The identity of indiscernibles is the principle stating that two or more entities are identical if they have all their properties in common, and vice versa. In spite of its apparent trivial truthfulness, Max Black put forward a famous argument against its validity. I intend to take his claim into account in order to analyze the matter, following two connected ways. Unlike Peter Geach, who reasoned in favor of a relativistic conception, I uphold that identity, conceived as the primitive relation of numerical identity, is an absolute concept. The universal quantification of the F’s, which, according to Geach, is responsible for difficulties, really concerns only indiscernibility; in fact, the principle is formulated as a double implication, so I shall argue that the " " symbol connects two atomic statements that express the formal definition of numerical identity and the formal definition of indiscernibility respectively. The question, however, still remains as to whether t...

2025

This work proposes a structural-ontological foundation for logic based on the principle of self-contradicted identity, formulated as a relational tension within being itself. Rejecting both pragmatic and formalist approaches, the theory advances a second-order metalogical function, Φ, which operates on the set of relational structures over a given language and generates a class of stable configurations. Logic is not grounded in epistemic utility or inferential practices, but emerges from the iterability of a primary metaphysical contrariety, where identity and difference coexist as a generative tension. Through a formal definition of Φ and its convergence properties, the theory provides an account of logic as a manifestation of the structure of being, prior to any axiomatic system or linguistic articulation. The work engages with contemporary logical theories (e.g., Leitgeb, Da Costa) and classical foundations (e.g., Frege, Hilbert), offering a unified framework capable of encompassing both classical and non-classical logics. Ultimately, logic is redefined not as a tool of thought, but as an ontological-metaphysical necessity.