Given a theory – e.g., in mathematics, theology, physics, sociology, political theory – what is it for the objects of that theory to be identical? Not all true theories give the same answer: it can depend on the objects of the theory. The logic of identity, as current research stands, is largely confined to the standard framework of classical logic. Non-classical logics have proven to be important in philosophy, and they are promising for related disciplines. What we lack, as current research stands, are natural, simple, and widely applicable accounts of identity for the family of non-classical frameworks. This one-day Arché workshop will explore the prospects for non-classical logical approaches to identity and their possible applications.

Venue: G03, Edgecliffe
Time: From 9am till 5pm on Monday April 18th, 2022 (BST)
Format: In person
Registration:  Please register your interest in attending by emailing our workshop organisers at mn85@st-andrews.ac.uk

Schedule

• 9.00-9.30 Coffee and registration
• 9.30-10.45 Speaker: Matteo Nizzardo (University of St Andrews)
  Title: The Identity of Indiscernibles’ Weakest Interesting Interpretation
• 10.45-11.15 Coffee
• 11.15-12.30 Speaker: JC Beall (University of Notre Dame)
  Title: The leibnizian recipe, difference makers, and relative identity
• 12.30-14.00 Lunch break
• 14.00-15.15 Speaker: Greg Restall (University of St Andrews)
  Title: Exploring three-valued models for identity
• 15:15-15.45 Coffee
• 15:45-17:00 Franz Berto (University of St Andrews)
  Title: Counting the Particles: Permutations, Identity, and Indiscernibility

Titles and Abstracts

Matteo Nizzardo — The Identity of Indiscernibles’ Weakest Interesting Interpretation

According to Strawson (1959) the principle of the Identity of Indiscernibles (PII) holds that no two individuals can share all their qualitative properties. This has been challenged by Rodriguez-Pereyra (2006), who suggests a weaker non-trivial interpretation of PII according to which no two individuals can share all their nontrivializing properties. I argue that Rodriguez-Pereyra’s reading of PII entails an infinite regress, which can be avoided only by committing to individuals that differ only numerically. I generalise the result to any interpretation of PII that is weaker than Strawson’s, and conclude that the weakest interesting version of PII is the one restricting the domain of quantification to qualitative properties only.

JC Beall — The leibnizian recipe, difference makers, and relative identity

I think of identity relations as the product of the leibnizian recipe, which basically defines an identity relation R via a biconditional scheme:

aRb is true iff all relevant instances of PHI(a)<—>PHI(b) are true.

aRb is false iff some relevant instance of PHI(a)<—>PHI(b) is false.

Clearly, the entailment behavior of the identity relation R will turn not only on the entailment behavior of the biconditional in the scheme but also on the range of the schematic variable. Example: use classical-logic ingredients (ergo, classical material conditional) and R is an equivalence relation. Use subclassical ingredients (e.g., FDE) and, well, R certainly needn’t be an equivalence relation (though, in the confines of some theories, can be an equivalence relation, depending on the language of the given theory and especially the range of PHI).

All of this is familiar; it’s basically the standard recipe for identity relations. (And if you shop at the standard store for your ingredients, you’ll get the standard identity relation sometimes called ‘classical identity’.) Let’s assume — as I in fact believe — that this standard recipe is correct (though standard ingredients aren’t necessary).

Against the backdrop of (by my lights) the somewhat perplexing debate over “relative identity” (e.g., Geach, Martin, van Inwagen, Rea), I show that relative identity relations simply drop out of the standard leibnizian recipe, at least if associated sets of “difference makers” are made explicit.

Against some (so-called “pure”) relative-identity theorists, I think that we have not only a gazillion relative-identity relations (as all relative-identity theorists believe); we also have many (if not exactly a gazillion) non-relative identity relations — all easily traced back to the leibnizian recipe.

The result should be good news for so-called relative-identity theorists. The reason is that, to my knowledge, such theorists never actually explicitly define relative-identity relations; they just wave at features of “is-the-same-PHI-as” that’s supposed to somehow define the relations. (But they don’t, so far as I’ve seen.) If I’m right, relative-identity theorists are just leibnizian-recipe theorists with a few small twists.  

Greg Restall — Exploring three-valued models for identity

There is an very natural way to interpret the propositional connectives and quantifiers, relative to the algebra of three semantic values, {0, i, 1} where 0 and 1 are understood as the traditional values of falsity and truth, and the third value is some intermediate value. The evaluation clauses do not, by themselves, determine the logic, because for that, you need to determine how models are used to provide a counterexample to a sequent. If a counterexample is given by a model that assigns every premise the value 1 and assigns every conclusion a value other than 1, the resulting logic is Kleene’s strong three-valued logic, K3. If a counterexample is a model assigning every premise the value 1 or i and every conclusion the value 0, the resulting logic is Priest’s logic of paradox, LP. If a counterexample is a model assigning every premise the value 1 and every conclusion the value 0, you get the logic ST of Strict-Tolerant validity. ST is distinctive, in that it is, in some sense, classical logic—every classically valid sequent in this language is ST-valid—but since it has strictly non-classical models, there are ST theories which are not classical theories.

What does this mean for the logic of identity in a three-valued context?

In this talk, I will explain how the classical logic of identity, when interpreted in ST models, gives us a well-behaved class of three-valued models for identity—much larger than the traditional models for identity, used by afficianados of LP or K3—which can be used to model the distinctive non-classical behaviour of identity statements, with a greater degree of freedom than we might have thought.

Franz Berto — Counting the Particles: Permutations, Identity, and Indiscernibility 

I’d like to attack a certain view: the view that the concept of identity can fail to apply to some things although, for some positive integer n, we have n of them. The idea of entities without self-identity is seriously entertained in the philosophy of quantum mechanics. It is so pervasive that it has been labelled the Received View. In this talk, I introduce the Received View; I explain what I mean by “entity” (synonymously, by “object” and “thing”), and I argue that supporters of the Received View should agree with my characterization of the corresponding notion of entity (object, thing). I also explain what I mean by “identity”, and I show that supporters of the Received View agree with my characterization of that notion. Then, I argue that the concept of identity, so characterized, is one with the concept of oneness. Thus, it cannot but apply to what belongs to a collection with n elements, n being a positive integer. I add some considerations on the primitiveness of identity or unity and the status of the Identity of Indiscernibles. Finally, I say something on the problem of how reference to indiscernible objects with identity can be achieved.