Logic talk

Abstract

Sentences about logic are often used to show that certain embedding expressions (attitude verbs, conditionals, etc.) are hyperintensional. Yet it is not clear how to regiment “logic talk” in the object language so that it can be compositionally embedded under such expressions. In this paper, I develop a formal system called hyperlogic that is designed to do just that. I provide a hyperintensional semantics for hyperlogic that doesn’t appeal to logically impossible worlds, as traditionally understood, but instead uses a shiftable parameter that determines the interpretation of the logical connectives. I argue this semantics compares favorably to the more common impossible worlds semantics, which faces difficulties interpreting propositionally quantified logic talk.

Appendix

1.1 A representability theorem

In this appendix, I establish that any finitary single-conclusion logic can be represented in the hyperconvention semantics. In fact, we can represent any logic using just a single hyperconvention. In addition, I show that this improves upon the semantics in Kocurek and Jerzak 2021, which can only represent a smaller set of logics.

First, we need to clarify what we mean by ‘logic’ and ‘represent’.

Definition 11

(Logic) A logic over ${\mathcal {L}}$ is a set $\mathbf{L }\mathrel {\subseteq }{\mathcal {L}}^{<\omega } \mathrel {\times }{\mathcal {L}}$ of pairs of the form . (We allow the lefthand side to be the empty tuple $\diamondsuit$.) We may write in place of . We use $\vec {\phi }$ as shorthand for , where the length is implicit (possibly zero).

Definition 12

(Representability) A logic $\mathbf{L }$ is representable in $\mathcal {M}= \left\langle W,D_{\mathbb {C}},D_{\mathbb {P}},V \right\rangle$ if there’s a $c \in \mathbb {H}_{W}$ such that for all :

Theorem 13

(Representation) Every logic is representable in an infinite hypermodel (i.e., $\left| {W}\right| = \aleph _{0}$).

Proof

Let be an injection. Let . Define c as follows.

Let $\mathcal {M}$ be a hypermodel over W with c in its hyperconvention domain such that $V(p)(c) = [p]$ (e.g., we can let $V(p) = P_p$). By a simple induction, for all $\phi \in {\mathcal {L}}$.

Thus, if , then $([\phi _1],\dots ,[\phi _n] \mathbin {\rhd }_c [\psi ]) = W$, so . Conversely, if , then $([\phi _1],\dots ,[\phi _n] \mathbin {\rhd }_c [\psi ]) = [p_1]$, so .

Theorem 13 contrasts with the situation for the semantics developed in Kocurek and Jerzak 2021. There, they do not introduce an entailment operator $\mathbin {\rhd }$ into the language, so there is no obvious way to represent logics in the usual sense. In a footnote, they suggest the following (rephrased in our notation): c represents $\mathbf{L }$ in $\mathcal {M}$ iff iff for all w, if , then $\mathcal {M},w,c \Vdash \psi$. I will now show that on this understanding of representability, a strictly smaller class of logics are representable by a hyperconvention.

Definition 14

(Properties of logics) A logic $\mathbf{L }$ is intensional if it satisfies the following properties:

• Reflexive: $\phi \mathbin {\Rightarrow }_\mathbf{L }\phi$

• Transitive: if $\vec {\psi } \mathbin {\Rightarrow }_\mathbf{L }\chi$ and $\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\psi _i$ for each $\psi _i$, then $\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\chi$

• Commutative: if $\vec {\phi },\vec {\psi } \mathbin {\Rightarrow }_\mathbf{L }\chi$, then $\vec {\psi },\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\chi$

• Monotonic: if $\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\psi$, then $\vec {\phi },\chi \mathbin {\Rightarrow }_\mathbf{L }\psi$

• Congruential: it obeys replacement of logical equivalents:

  • for : if $\phi \mathbin {\Leftrightarrow }_\mathbf{L }\phi '$, then $\mathop {\bullet }\nolimits \phi \mathbin {\Leftrightarrow }_\mathbf{L }\mathop {\bullet }\nolimits \phi '$

  • for : if $\phi \mathbin {\Leftrightarrow }_\mathbf{L }\phi '$ and $\psi \mathbin {\Leftrightarrow }_\mathbf{L }\psi '$, then $\phi \mathbin {\circ }\psi \mathbin {\Leftrightarrow }_\mathbf{L }\phi ' \mathbin {\circ }\psi '$.

Observe that reflexivity, commutativity, and monotonicity entail (in fact, given transitivity, are equivalent to) the following property:

• Reiterative: $\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\phi _i$ for each $\phi _i$.

Definition 15

(Intensional representability) A logic $\mathbf{L }$ is intensionally representable in $\mathcal {M}= \left\langle W,D_{\mathbb {C}},D_{\mathbb {P}},V \right\rangle$ if there’s a $c \in \mathbb {H}_{W}$ such that for all :

Theorem 16

(Intensional Representation). A logic $\mathbf{L }$ is intensionally representable in an infinite hypermodel $\mathcal {M}$ iff $\mathbf{L }$ is intensional. Regardless of $\mathcal {M}$’s size, $\mathbf{L }$ is intensionally representable in $\mathcal {M}$ iff $\mathbf{L }$ is intensional and the number of nonequivalent finite sequences of formulas is no greater than $\left| {W}\right|$.

Proof

The left-to-right direction is straightforward: one simply checks that necessary implication has all the features of an intensional logic.

For the right-to-left direction, we begin by defining the finite theory space of $\mathbf{L }$, which is a generalization of the Lindenbaum-Tarski algebra. Where , define:

The finite theory space of $\mathbf{L }$ is the order $\mathbb {F}_\mathbf{L }= \left\langle F_\mathbf{L },\le _\mathbf{L } \right\rangle$ where:

• 

• $[\vec {\phi }]_\mathbf{L }\le _\mathbf{L }[\vec {\psi }]_\mathbf{L }$ iff $\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\psi _i$ for each $\psi _i \in \vec {\psi }$.

From now on, we’ll leave the $\mathbf{L }$-subscripts implicit. Recall that a meet-semilattice is a partial order where the meet (i.e., greatest lower bound) of any two elements exists.

Claim

$\mathbb {F}$ is a meet-semilattice with a top element.

Proof

Since $\mathbf{L }$ is reiterative, for $1 \le i \le n$. So $\le$ is reflexive. The transitivity and antisymmetry of $\le$ follow from the transitivity of $\mathbf{L }$. The top element is simply $[\left\langle \right\rangle ]$. As for the existence of meets, define the meet $\sqcap$ operation as follows:

That $[\vec {\phi },\vec {\psi }] \le [\vec {\phi }]$ and $[\vec {\phi },\vec {\psi }] \le [\vec {\psi }]$ follows from reiterativity. Moreover, if $[\vec {\chi }] \le [\vec {\phi }]$ and $[\vec {\chi }] \le [\vec {\psi }]$, then $[\vec {\chi }] \le [\vec {\phi },\vec {\psi }]$ by definition of $\le$. So $\sqcap$ really is a meet operation.

It is a well-known fact of order-theory that any partial order can be order-embedded into its powerset algebra in a meet-preserving manner. Since $F_\mathbf{L }$ is at most the size of W, that means there’s an injective map such that:

• (i) $[\vec {\phi }] \le [\vec {\psi }]$ iff $f([\vec {\phi }]) \mathrel {\subseteq }f([\vec {\psi }])$

• (ii) $f([\vec {\phi },\vec {\psi }]) = f([\vec {\phi }]) \cap f([\vec {\psi }])$.

What’s more, we can take $f([\left\langle \right\rangle ]) = W$. For if $f([\left\langle \right\rangle ]) \ne W$, define $f^*$ to be exactly like f except $f^*([\left\langle \right\rangle ]) = W$. It is easy to then verify that $f^*$ is still an order-embedding that preserves finite meets.

Define c as follows (where and ):

This is well-defined. For if $f([\phi ]) = f([\psi ])$, then $[\phi ] = [\psi ]$ by injectivity. Thus, $\phi \mathbin {\Leftrightarrow }_\mathbf{L }\psi$. So by congruentiality, $\mathop {\bullet }\nolimits \phi \mathbin {\Leftrightarrow }_\mathbf{L }\mathop {\bullet }\nolimits \psi$, which means $[\mathop {\bullet }\nolimits \phi ] = [\mathop {\bullet }\nolimits \psi ]$. Hence, $f([\mathop {\bullet }\nolimits \phi ]) = f([\mathop {\bullet }\nolimits \psi ])$. (And similarly for binary connectives.)

Throughout, let $\mathcal {M}$ be a hypermodel of the relevant size with c in its hyperconvention domain where $V(p)(c) = c(p)$. By induction:

Claim

For any $\phi \in {\mathcal {L}}$, .

Finally, we show that c intensionally represents $\mathbf{L }$ in $\mathcal {M}$.

$\mathbin {\Leftarrow }$:

Suppose . Thus, . Since f is order-reflecting, . So there is a . Now observe:

Hence, but $\mathcal {M},w,c \nVdash \psi$.

$\mathbin {\Rightarrow }$:

Suppose . Then . Since f is order-embedding:

Therefore, c intensionally represents $\mathbf{L }$ in $\mathcal {M}$. $\square$