On Elizabeth Barnes and J. Robert G. Williams’ theory of metaphysical indeterminacy

I. What is metaphysical indeterminacy according to Barnes and Williams?

p is undetermined if and only if, settled all the relevant semantic and epistemological issues, “it’s simply unsettled whether p or rather ¬p obtains”, i.e., “…there is a precise way that things are [but it is] primitively indeterminate which precise way things are…”

Is it circular?

Well, yes not viciously so, because

Indeterminacy is a metaphysical primitive.

That P is indeterminate does not mean that (but entails that)

• it is impossible to know whether P (that is epistemic indeterminacy)
• it is possible that P and it is also possible that not P (that is contingency)

It also does not mean (and it does not entail either):

• it is neither true that P nor is it false (thus bivalence is false)
• it is neither true that P nor is it true that ~P (thus (P v ~P) is not a tautology)

How is that possible?

How I like to understand indeterminacy:

The world is indeterminate if an only if there are fundamental existential truths and disjunctions, i.e. true disjunctions that are not made true by one or more of their disjuncts and true existential propositions that are not made true by any one of their instances.Thus, there are irreducible disjunctive and existential facts.

 

II. What is its logic?

Well, they are not really interested in answering this question in general, since logics of indeterminacy are almost as old as formal logic itself (Lukaciewicz logic dates from 1920) and a dime a dozen nowadays. Instead, they want to answer the question: “[does] the thesis that indeterminacy is primitive … force.. non- classicism upon us [?]”

Their answer: no.

Because their logic of (formal) indeterminacy [aka “undeterminacy” in much current logical parlance] is an extension of classical logic.

The key semantic principles is that:

A formula is indeterminate if it is true in at least some world in the halo and false in at least another one (Thus it is an intensional logic).

But the worlds themselves are classical, bivalente and complete.

“…all precisifications agree that, for example, (p∨¬p) holds, they disagree over which disjunct makes it the case that the disjunction holds (thus the individual disjuncts are themselves still indeterminate).”

One could add an accessibility relation, but they do not do it and it would be pretty straightforward anyways.

But they do a very nice job showing how the determinacy operator interacts with traditional alethic modalities.

All operators have a double life: as binders and as commiters. They bind because they tell you something about which values are assigned to parameters, but they also committee because they tell you what is special about them. The determinacy operator is no exception, it binds the possible world operator and expresses the commitments that the values of the parameter it ranges over correspond only to ways the world may be in the proper sense of being “worlds which are according to the model not determinately unactualized” aka the ‘ontic precisifications’.

But who cares, really?

In intensional logic, and in formal logic in general, there is something arbitrary on choosing which operators to make explicit.

Already in 1995, Raymond Morado had stated that 

“La [lógica] modal no me parece una extension [de la lógica clásica]. [Tradicionalmente] Primero enseñas la [lógica] clásica y [luego] le añades, la extiendes [con la parte modal], pero pudiste haber dado primero la extension, que entonces ya no es extension, y sacar la otra como un caso particular o sucedáneo”. (Morado 1995: 9)

Think of a language for a modal logic where there is no operator for necessity, and instead bare formulas are read as ‘necessary’ yet there is an actuality operator (which, to make the duality more evident we could make the box). This logic would be a dual of traditional alethic modal logic, where P implies box-P.

Just as in the traditional presentation of modal logic, the local logic is classical, but in the extended language, it is not a theorem that (Box-P v Box-not-P); in this dual presentation of modal logic, the local logic is not classical, i.e., it is not a theorem that (P v not-P). Yet, in the extended logic the restriction to formulas que the only boxed formulas are atomic is classical, and thus it is a theorem that (Box-P v Box-not-P). In other words, classical logic can still be recovered in this version of modal logic, which would be as expected since recovery is the dual of extension.

“Roughly, one logic recovers another if it is possible to specify a subsystem of the former system, which exhibits the same patterns of inference as the latter system. Sometimes, this property is also called recapture.” (Barrio & Carnielli 2019:615)

This justifies my skepticism that there is something really substantial to the classical/non-classical intensional logic distinction and therefore to Barnes and Williams’ motivation beside the technical result; on the first formulation, modal logic is an extension to classical logic, on the second formulation, classical logic can be recovered from it.

I suspect the same thing happens with Barnes and Williams’ logic of indeterminacy: it is an extension of classical logic just because she chooses to present it in this way instead of how it is usually do in the LFU tradition.

It is said that “… paraconsistent logicians do not propose a wholesale rejection of Classical Logic. “They usually accept the validity of classical inferences in consistent contexts…” (Barrio & Carnielli 2019:616) Thus even at the conceptual level there is little of substance in the classical/non-classical distinction.

Logics of indeterminacy are just paraconsistent logics in their dualized disguise … and vice versa (Carnielli, Coniglio & Rodrigues 2020). Paraconsistent logics tend to be presented in such a way that the local logic is not classical. But – this is a point I owe to Luis Estrada’s talk a couple of weeks ago –, they could also be presented the other way. In other words, just as Barnes and Williams have shown us that the indeterminacy can be moved outside the logic and let the local logic be classical, paraconsistency can also be moved outside the logic and let the local logic be classical. As a matter of fact, that is how it used to be done in sub-structural logics of indeterminacy (and paraconsistency) like L4, which in the presentation Teresita Mijangos (2003) and me (2015) have proposed, is locally classical but non-classical (multi-valued and both paraconsistent and paracomplete) at the meta level.

Just as Barnes’ logic of indeterminacy is able to express, inside the object language, the notions of determinacy and indeterminacy, as applied to sentences by adding a unary propositional connective d to the language, where dA is informally interpreted as A is determinate. Logics of formal inconsistency, aka LFIs, are able to express, inside the object language, the notions of consistency, or even inconsistency, as applied to sentences. This is done by adding a unary propositional connective ◦ to the language, where ◦A is informally interpreted as A is consistent.” (Barrio & Carnielli 2019:616) Thus, Barnes and Williams’ logic qualifies as a logic of formal undeterminacy, (so-called so that they can be abbreviated to LFUs and avoid ambiguity with the aforementioned logics of inconsistency).

According to Barrio & Carnielli, the debate turns on how normal are the context where we use one logic or another, classic o non-classical. Traditionally, paraconsistent logicians claim that normal everyday context require non-classical logic, but Barnes and Williams make the opposite point that, regarding indeterminacy, in normal everyday context classical logic suffices. This is so because they think there is a fundamental asymmetry between paraconsistency and indeterminacy: paraconsistent logicians believe that there are true contradictions in the language and thus favour a local non-classical logic, but Barnes and Williams claim that paracompletists need not commit themselves to false tautologies (like P or not P) and thus need not commit to a local non-classical logic. I hope by now I have convinced you that this is just notational smoke and mirrors.