TOC

• Hyperintensionality
• Logic
• Comparison
• Outlook

Hyperintensionality

Hyperintensional contexts are simply contexts which do not respect logical equivalence.

A hyperintensional concept draws a distinction between necessarily equivalent contents.

Def.

\(O\) is hyperintensional iff \[ \phi\dashv\vdash\psi\nRightarrow O(\phi)\dashv\vdash O(\psi) \]

• Failure of substitutivity
• Non-congruential operator

Questions

(Ciardelli, Groenendijk, Roelofsen 2018)

• \(?\phi\dashv\vdash \phi\lor\neg \phi\) (Question)
• \(\phi\lor\neg\phi\dashv\vdash\psi\lor\neg\phi\) (Classicality)
• \(?\phi\dashv\vdash ?\psi \)

Because

(Fine 2011, Schnieder 2011)

• \(\phi< Tr(\ulcorner\phi\urcorner)\) (Aristotle)
• \(\phi\dashv\vdash Tr(\ulcorner\phi\urcorner)\) (Tarski)
• \(\phi\dashv\vdash \psi\Rightarrow \theta < \phi \dashv\vdash \theta < \psi\) (Substitutivity)
• \(\phi < \phi \)

FCP

(Hilpinen 1982)

• \(P(\phi_1\lor\phi_2)\vdash P(\phi_i)\) (FCP)
• \(\phi\dashv\vdash\psi\Rightarrow P(\phi)\dashv\vdash P(\psi)\) (Substitutivity)
• \(\phi\dashv\vdash (\phi\land\psi)\lor(\phi\land\neg\psi)\) (Total possibility)
• \(P(\phi) \vdash P(\phi\land\psi) \)

SDA

(Ellis, Jackson, & Pargetter 1977)

• \((\phi_1\lor\phi_2)>\psi\vdash \phi_i>\psi\) (SDA)
• \(\phi\dashv\vdash\psi\Rightarrow \phi>\theta\dashv\vdash \psi>\theta\) (Substitutivity)
• \(\phi\dashv\vdash\phi\lor(\phi\land \psi)\) (Absorption)
• \(\phi>\theta \vdash (\phi\land\psi)>\theta \)

Implications

• Logically & philosophically relevant
• Nolan (2012): Hyperintensional revolution
• Rethink (formal) semantics!

Possible-worlds model

• \(\llbracket \phi\rrbracket=\{w:w\vDash\phi\}\) (Propositions)
• \(\phi \dashv\vdash \psi\Rightarrow\llbracket\phi\rrbracket=\llbracket\psi\rrbracket \)
• \(\llbracket O(\phi)\rrbracket=f_O(\llbracket\phi\rrbracket)\) (Compositionality)
• \(\phi\dashv\vdash \psi\Rightarrow\llbracket O(\phi)\rrbracket= \llbracket O(\psi)\rrbracket\)

The way forward

• Williamson (2021, forthcoming): Wait, I can explain!
• A scientific revolution!
• New frameworks:
• Truthmaker semantics (Fine & Jago forthcoming)
• Inquisitive semantics (Ciardelli, Groenendijk, & Roelofsen 2018)

Logic

Perspectives

• Linguistics, metaphysics, CS, ...
• Logic valid inference

Assumptions

• Methodological pluralism
• Logics are models of valid inference
• Agnostic about propositions

Def.s

 

Logics

\((\mathcal{L},\vdash)\), where \(\vdash~\subseteq \wp(\mathcal{L})\times\mathcal{L}\) is a consequence relation

 

Models

\((\mathcal{A},C)\), where \(C:\wp(A)\to \wp(A)\) is a closure operator with \[\Gamma\vdash\phi\Rightarrow \forall h:L\to A,  h(\phi)\in C(h(\Gamma))\]

Def. (Odintsov & Wansing 2021)

\((\mathcal{L},\vdash)\) is hyperintensional iff \(\exists O\in\mathcal{L}\), \[\phi\dashv\vdash\psi\nRightarrow O(\phi)\dashv\vdash O(\psi)\]

Aim

Determine a 'natural' relation \(R\) with \[\phi R \psi\Rightarrow O(\phi)\dashv\vdash O(\psi)\]

 

Granularity problem (Schwarz & Bjerring 2017)

Co-hyperintensionality

• Algebra
• No Lindenbaum-Tarski algebra

Def.s

Frege relation

\(\Lambda C=\{(a,b):C(a)=C(b)\}\)

Tarski congruence

\(\tilde{\Omega} C\) the largest\(_\subseteq\) congruence below\(_\subseteq\) \(\Lambda C\)

Reduced model

\(\tilde{\Omega} C=\Delta A\)

Def.s

Basic full models

\((A,C)\preceq (A,C')\) iff $C(X)\subseteq C'(X)$

Smallest\(_\preceq\) models

Full models

\(\Lambda C\)-quotients of basic full modles

Def.s

Associated algebras (Font 2016)

\(\mathfrak{A}=\{A: (A,C)\text{ is a full reduced model}\}\)

Granularity

\(\mathfrak{G}\) is the absolutely free (term-)algebra of type \(\mathfrak{A}\) generated by the propositional variables \(\mathcal{P}\):

\(\forall A\in\mathfrak{A}~\forall f:\mathcal{P}\to A~\exists !h:G\to A, h_{\restriction \mathcal{P}}=f\)

Def

Comparative granularity

\((\mathcal{L},\vdash_1)\preceq (\mathcal{L},\vdash_2)\) iff \(\exists h:\mathfrak{G}_1\hookrightarrow\mathfrak{G}_2\)

• A hyperintensional hierarchy!

Comparison

Aim

Investigate hyperintensional capabilities of frameworks

Inquisitive semantics

• Inquisitive propositions ()
• Information state: \( s\subseteq W\)
• Issue: \(I\neq\emptyset\subseteq\wp(W)\), where \[s\in I~\&~t\subseteq s\Rightarrow t\in I\]
• Proposition: \(P=(s,I)\)
• Semantic clauses: \(\llbracket\phi\rrbracket=(s,I)\)

• Under the semantics, the inquisitive propositions form a Heyting-algebra
• \(\mathfrak{G}_{Inq}\nvDash P\lor\neg P=Q\lor\neg Q\)
• \(\mathfrak{G}_{Inq}\nvDash P=(P\land Q)\lor(P\land \neg Q)\)
• the \((\mathbf{P},\land,\lor)\)-subalgebra is a lattice
• \(\mathfrak{G}_{Inq}\vDash P=P\lor (P\land Q)\)

\(\Vdash\) Truthmaker semantics

• Truthmaker propositions
• State space: \(S,\sqcup\)
• Condition: \(P\subseteq S\), where \[s,t\in P\Rightarrow s\sqcup t\in P\]
• Proposition: \((P^+,P^-)\)
• Semantic clauses: \(\llbracket\phi\rrbracket=(P^+,P^-)\)

• Under the semantics, the truthmaker propositions form a De Morgen bisemilattice (Brzozowski 2000)
• \(\mathfrak{G}_{Tms}\nvDash P\lor\neg P=Q\lor\neg Q\)
• \(\mathfrak{G}_{Tms}\nvDash P=(P\land Q)\lor(P\land \neg Q)\)
• \(\mathfrak{G}_{Tms}\nvDash P=P\lor (P\land Q)\)

Outlook

A 'universal' \(\mathfrak{G}\)?

• Hornischer (2017), \(\exists\phi_{1,2},\psi_{1,2},\mathcal{O}_1,\mathcal{O}_2\)
• \(\mathfrak{G}_1\parallel \mathfrak{G}_2\)
• \(\mathfrak{G}_0=\mathfrak{G}_1\cap \mathfrak{G}_2\)
• \(\mathfrak{G}_\bot=\bigcap\mathfrak{G}=\mathcal{L}\)

Thanks