Quantified Modal Logic and Essentialism

When quantifiers and modal operators interact, logic must confront whether objects have properties necessarily — essentialism — and how to interpret quantification across possible worlds.

Definition

Quantified modal logic combines first-order quantification with modal operators; essentialism is the thesis that objects have some of their properties necessarily (de re), independently of how they are described.

Scope

This topic covers the extension of modal logic with quantifiers and identity, and the metaphysical commitments this generates. It treats the Barcan and converse Barcan formulas, the choice between constant- and variable-domain semantics, the problem of identity and existence across worlds, Quine's objections to de re modality, and Kripke's defence of essentialism and rigid designation.

Core questions

• How should quantifiers and modal operators interact, and is the Barcan formula valid?
• Should the domain of individuals be fixed across worlds or vary?
• Is de re modality intelligible, or does it collapse under Quine's objections?
• Which properties, if any, do objects have essentially?

Key concepts

• Barcan formula and its converse
• constant vs. variable domains
• transworld identity
• rigid designation
• de re vs. de dicto modality
• essential vs. accidental properties

Key theories

Essentialism and rigid designation

Kripke argues that proper names are rigid designators picking out the same object in every world, which makes de re modal claims coherent and supports necessary truths about origin and constitution.

Quine's critique of de re modality

Quine argues that quantifying into modal contexts is unintelligible because substitution of co-referring terms can change truth value, so essentialism would require an objectionable distinction between an object's necessary and accidental traits.

History

Ruth Barcan Marcus initiated quantified modal logic in 1946-1947, proving results about identity and introducing the Barcan formula. Quine attacked the whole enterprise as committed to an unintelligible essentialism, but Kripke's Naming and Necessity (1980) rehabilitated de re modality through rigid designation and the necessity of identity, making essentialism respectable again.

Debates

Is de re modality coherent?

Whether ascribing necessary properties to objects independently of description is intelligible, as Kripke's rigid designation implies, or whether Quine is right that such talk presupposes an indefensible essentialism.

Key figures

• Ruth Barcan Marcus
• Saul Kripke
• W. V. O. Quine
• Alvin Plantinga
• David Kaplan

Seminal works

• barcan1947
• quine1953reference
• kripke1980

Frequently asked questions

What is the Barcan formula?

The Barcan formula says that if it is possible that something is F, then there is something that is possibly F — moving the existential quantifier outside the possibility operator. It and its converse are valid on constant-domain semantics but fail when domains vary across worlds, which is why they bear on whether the same individuals exist in every possible world.