Carnap’s Problem for Intuitionistic Propositional Logic
Abstract
:1. Motivation and Background
2. Algebraic vs. SetBased Semantics
2.1. Carnap’s Problem for Kripke Semantics
 (1)
 ${\left[\right[p\left]\right]}_{v}=v\left(p\right)$
 ${\left[\right[\perp \left]\right]}_{v}=\varnothing $
 ${\left[\right[\top \left]\right]}_{v}=X$
 ${\left[\right[\phi \ua4e5\psi \left]\right]}_{v}={\left[\right[\phi \left]\right]}_{v}\cap {\left[\right[\psi \left]\right]}_{v}$
 ${\left[\right[\phi \ua4e6\psi \left]\right]}_{v}={\left[\right[\phi \left]\right]}_{v}\cup {\left[\right[\psi \left]\right]}_{v}$
 $\begin{array}{l}{\left[\right[\phi \to \psi \left]\right]}_{v}=\{x\in X:\ua4efy\ge x(y\in {\left[\right[\phi \left]\right]}_{v}\Rightarrow y\in {\left[\right[\psi \left]\right]}_{v}\left)\right\}\\ =\{x\in X:x\uparrow \cap {\left[\right[\phi \left]\right]}_{v}\subseteq {\left[\right[\psi \left]\right]}_{v}\}\end{array}$
 (2)
 ${I}_{\mathrm{st}}^{\mathcal{F}}(\perp )=\varnothing $
 ${I}_{\mathrm{st}}^{\mathcal{F}}(\top )=X$
 ${I}_{\mathrm{st}}^{\mathcal{F}}(\wedge )(U,V)=U\cap V$
 ${I}_{\mathrm{st}}^{\mathcal{F}}(\vee )(U,V)=U\cup V$
 ${I}_{\mathrm{st}}^{\mathcal{F}}(\to )(U,V)=\{x\in X:x\uparrow \cap U\subseteq v\}$
2.2. A Hierarchy of Semantics for $\mathsf{IPC}$
 (3)
 Kripke < Beth < Topological < Dragalin < Algebraic
 (4)
 $a\to a=1$
 $a\wedge (a\to b)=a\wedge b$
 $(a\to b)\wedge b=b$
 $a\to (b\wedge c)=(a\to b)\wedge (a\to c)$
 (5)
 $a{\le}^{\mathcal{A}}b$ iff $a{\wedge}^{\mathcal{A}}b=a$
 (6)
 $\begin{array}{l}\mathbf{Up}\left(\mathcal{F}\right)=(\mathit{Up}\left(\mathcal{F}\right),{I}_{\mathrm{st}}^{\mathcal{F}}(\perp ),{I}_{\mathrm{st}}^{\mathcal{F}}(\top ),{I}_{\mathrm{st}}^{\mathcal{F}}(\wedge ),{I}_{\mathrm{st}}^{\mathcal{F}}(\vee ),{I}_{\mathrm{st}}^{\mathcal{F}}(\to ))\\ =(\mathit{Up}\left(\mathcal{F}\right),\varnothing ,X,\cap ,\cup ,{I}_{\mathrm{st}}^{\mathcal{F}}(\to ))\end{array}$
2.3. SetBased Semantics
 A setbased semantics for $\mathsf{IPC}$ is a class $\mathcal{S}$ of structures of some given type. For $X\ne \varnothing $, let $\mathcal{S}\left(X\right)$ be the class of structures in $\mathcal{S}$ with domain X. Furthermore, $\mathcal{S}$ associates with each structure in $\mathcal{S}\left(X\right)$ a set ${\mathit{SV}}_{\mathcal{S}}\left(X\right)$ of subsets of X (but we drop the subscript ${}_{\mathcal{S}}$ whenever feasible): the set of possible semantic values (over X) of formulas, according to $\mathcal{S}$.
 An $\mathcal{S}$valuation on X is a map $v:\mathit{Prop}\to \mathit{SV}\left(X\right)$.
 An $\mathcal{S}$interpretation on X is a function I from $\{\perp ,\top ,\wedge ,\vee ,\to \}$ to corresponding operations on $\mathit{SV}\left(X\right)$. I and v recursively associate with each formula $\phi $ a semantic value ${\left[\left[\phi \right]\right]}_{v}^{I}$, as usual:5 ${\left[\left[p\right]\right]}_{v}^{I}=v\left(p\right)$, ${\left[[\perp ]\right]}_{v}^{I}=I(\perp )$, ${\left[[\top ]\right]}_{v}^{I}=I(\top )$, ${\left[[\phi \wedge \psi ]\right]}_{v}^{I}=I(\wedge )({\left[\left[\phi \right]\right]}_{v}^{I},{\left[\left[\psi \right]\right]}_{v}^{I})$, etc.
 I is consistent with a consequence relation ⊢, if ${\psi}_{1},\dots ,{\psi}_{n}\u22a2\phi $ implies that for all $\mathcal{S}$valuations v, ${\bigcap}_{i=1}^{n}{\left[\left[{\psi}_{i}\right]\right]}_{v}^{I}\subseteq {\left[\left[\phi \right]\right]}_{v}^{I}$.
 Each $\mathcal{S}$interpretation I over X produces (and can be identified with) an algebra$$\mathbf{S}{\left(X\right)}_{I}=(\mathit{SV}\left(X\right),I(\perp ),I(\top ),I(\wedge ),I(\vee ),I(\to ))$$
 There is an $\mathcal{S}$interpretation ${I}_{\mathrm{st}}$ such that $\mathbf{S}{\left(X\right)}_{{I}_{\mathrm{st}}}$ is a Heyting algebra and ${I}_{\mathrm{st}}(\wedge )$ is set intersection. ${I}_{\mathrm{st}}$ is called a standard $\mathcal{S}$interpretation.
 (7)
 $a\wedge x\le b$ iff $x\le a\to b$
2.4. Nuclear Interpretations
 (i)
 $a\le ja$ (inflationarity)
 (ii)
 $jja\le ja$ (idempotence)
 (iii)
 $j(a\wedge b)=ja\wedge jb$ (multiplicativity)
 (iv)
 $a\le b$ implies $ja\le jb$ (monotonicity)
 (8)
 ${\mathcal{A}}_{j}=({A}_{j},j{0}^{\mathcal{A}},{1}^{\mathcal{A}},{\wedge}_{j}^{\mathcal{A}},{\vee}_{j}^{\mathcal{A}},{\to}_{j}^{\mathcal{A}})$
 (9)
 ${\left[\left[p\right]\right]}_{v}^{\mathcal{F},j}=v\left(p\right)$
 ${\left[[\perp ]\right]}_{v}^{\mathcal{F},j}=j\varnothing $
 ${\left[[\top ]\right]}_{v}^{\mathcal{F},j}=X$
 ${\left[\right[\phi \wedge \psi \left]\right]}_{v}^{\mathcal{F},j}={\left[\right[\phi \left]\right]}_{v}^{\mathcal{F},j}\cap {\left[\right[\psi \left]\right]}_{v}^{\mathcal{F},j}$
 ${\left[\right[\phi \vee \psi \left]\right]}_{v}^{\mathcal{F},j}=j({\left[\right[\phi \left]\right]}_{v}^{\mathcal{F},j}\cup {\left[\right[\psi \left]\right]}_{v}^{\mathcal{F},j})$
 ${\left[\right[\phi \to \psi \left]\right]}_{v}^{\mathcal{F},j}=\{x\in X:x\uparrow \cap {\left[\right[\phi \left]\right]}_{v}^{\mathcal{F},j}\subseteq {\left[\right[\psi \left]\right]}_{v}^{\mathcal{F},j}\}$
 (10)
 ${j}_{B}U=\{x\in X:\mathrm{every}\mathrm{path}\mathrm{through}x\mathrm{intersects}U\}$
2.5. Topological Interpretations
 (11)
 ${I}_{\mathrm{st}}^{\Omega \left(X\right)}(\perp )=\varnothing $
 ${I}_{\mathrm{st}}^{\Omega \left(X\right)}(\top )=X$
 ${I}_{\mathrm{st}}^{\Omega \left(X\right)}(\wedge )(U,V)=U\cap V$
 ${I}_{\mathrm{st}}^{\Omega \left(X\right)}(\vee )(U,V)=U\cup V$
 ${I}_{\mathrm{st}}^{\Omega \left(X\right)}(\to )(U,V)=\mathit{int}((XU)\cup V)$
3. Carnap Categoricity
3.1. Algebraic Interpretations
 (i)
 $\mathcal{A}$ is consistent with ${\u22a2}_{\mathsf{IPC}}$ iff $\mathcal{A}$is a Heyting algebra.
 (ii)
 $\mathcal{A}$ is consistent with ${\u22a2}_{\mathsf{CPC}}$ iff $(A,{0}^{\mathcal{A}},{1}^{\mathcal{A}},{\wedge}^{\mathcal{A}},{\vee}^{\mathcal{A}},{\prime}^{\mathcal{A}})$ is a Boolean algebra, where ${a}^{\prime \mathcal{A}}:=a{\to}^{\mathcal{A}}{0}^{\mathcal{A}}$
 (a)
 $a{\vee}^{\mathcal{A}}\left(b{\vee}^{\mathcal{A}}c\right)=\left(a{\vee}^{\mathcal{A}}b\right){\vee}^{\mathcal{A}}c$
 (ii)
 The righttoleft direction is well known. Moreover, if $\mathcal{A}$ is consistent with ${\u22a2}_{\mathsf{CPC}}$, the defining equations of Heyting algebras are still valid. It only remains to show that $a\wedge {a}^{\prime}=0$ and $a\vee {a}^{\prime}=1$. The first is a consequence of $a\wedge (a\to b)=a\wedge b$ and the fact that $a\wedge 0=0$. The second identity is the Law of Excluded Middle. □
3.2. Categoricity
 Let $\mathcal{A}$ be an algebraic interpretation. An $\mathcal{A}$interpretation is a map I from the signature to appropriate functions on A. We let$${\mathcal{A}}_{I}=(A,I(\perp ),I(\top ),I(\wedge ),I(\vee ),I(\to ))=(A,{\perp}^{{\mathcal{A}}_{I}},{\top}^{{\mathcal{A}}_{I}},{\wedge}^{{\mathcal{A}}_{I}},{\vee}^{{\mathcal{A}}_{I}},{\to}^{{\mathcal{A}}_{I}})$$
 An Avaluation is a map $v:\mathit{Prop}\to A$. I and v recursively assign a value ${\left[\phi \right]}_{v}^{{\mathcal{A}}_{I}}$ to each formula $\phi $ as usual.
 ${\mathcal{A}}_{I}$ is $\mathcal{A}$consistent with ⊢ if whenever ${\psi}_{1},\dots ,{\psi}_{n}\u22a2\phi $, we have for all Avaluations v and all $c\in A$ that $c{\le}^{\mathcal{A}}{\left[{\psi}_{1}\right]}_{v}^{{\mathcal{A}}_{I}}{\wedge}^{\mathcal{A}}\dots {\wedge}^{\mathcal{A}}{\left[{\psi}_{n}\right]}_{v}^{{\mathcal{A}}_{I}}$ implies $c{\le}^{\mathcal{A}}{\left[\phi \right]}_{v}^{{\mathcal{A}}_{I}}$. Thus, $\mathcal{A}$ is consistent with ⊢ as in Definition 3.3 if $\mathcal{A}$ is $\mathcal{A}$consistent with ⊢.
 (b)
 ${\wedge}^{{\mathcal{A}}_{I}}={\wedge}^{\mathcal{A}}$
 (c)
 ${\perp}^{{\mathcal{A}}_{I}}={0}^{\mathcal{A}}$ and ${\top}^{{\mathcal{A}}_{I}}={1}^{\mathcal{A}}$.
 (d)
 $a{\to}^{{\mathcal{A}}_{I}}b{\le}^{\mathcal{A}}a{\to}^{\mathcal{A}}b$
 (e)
 For every Avaluation v, ${\left[\phi \right]}_{v}^{{\mathcal{A}}_{I}}{\le}^{\mathcal{A}}{\left[\psi \right]}_{v}^{{\mathcal{A}}_{I}}$ iff ${[\phi \to \psi ]}_{v}^{{\mathcal{A}}_{I}}={1}^{\mathcal{A}}$.
 (f)
 ${\vee}^{{\mathcal{A}}_{I}}={\vee}^{\mathcal{A}}$
 (g)
 ${\to}^{{\mathcal{A}}_{I}}={\to}^{\mathcal{A}}$
 (12)
 $a{\to}^{j}b{=}_{\mathrm{def}}a\to jb$
3.3. Application to SetBased Semantics
 (13)
 ${\left[[\neg \neg \phi ]\right]}_{v}^{{I}_{\mathrm{st}}^{\Omega \left(X\right)}}=\mathit{int}(\mathit{cl}({\left[\left[\phi \right]\right]}_{v}^{{I}_{\mathrm{st}}^{\Omega \left(X\right)}}))$
 (14)
 ${\left[[\neg \neg \neg \phi ]\right]}_{v}^{{I}_{\mathrm{st}}^{\Omega \left(X\right)}}=\mathit{int}(X\mathit{int}(\mathit{cl}({\left[\left[\phi \right]\right]}_{v}^{{I}_{\mathrm{st}}^{\Omega \left(X\right)}})))$
 (15)
 ${I}^{\Omega \left(X\right)}(\wedge )(U,V)=\mathit{int}\left(\mathit{cl}\left(U\right)\right)\cap \mathit{int}\left(\mathit{cl}\left(V\right)\right)$
 ${I}^{\Omega \left(X\right)}(\neg )\left(U\right)=\mathit{int}(X\mathit{int}\left(\mathit{cl}\left(U\right)\right))$
 ${I}^{\Omega \left(X\right)}(\to )(U,V)={I}^{\Omega \left(X\right)}(\neg )\left({I}^{\Omega \left(X\right)}(\wedge )({I}^{\Omega \left(X\right)}(\neg )\left(U\right),V)\right)$
 ${I}^{\Omega \left(X\right)}(\vee )(U,V)={I}^{\Omega \left(X\right)}(\neg )\left({I}^{\Omega \left(X\right)}(\wedge )({I}^{\Omega \left(X\right)}(\neg )\left(U\right),{I}^{\Omega \left(X\right)}(\neg )\left(V\right))\right)$
 (16)
 $g\left(p\right)=p$
 $g(\phi \wedge \psi )=\neg \neg g\left(\phi \right)\wedge \neg \neg g\left(\psi \right)$
 $g(\neg \phi )=\neg \neg \neg g\left(\psi \right)$
 $g(\phi \to \psi )=\neg \left(g\right(\phi )\wedge \neg g(\psi \left)\right)$
 $g(\phi \vee \psi )=\neg (\neg g(\phi )\wedge \neg g(\psi \left)\right)$
 (17)
 (18)
 ${\left[\left[\phi \right]\right]}_{v}^{{I}^{\Omega \left(X\right)}}={\left[\left[g\left(\phi \right)\right]\right]}_{v}^{{I}_{\mathrm{st}}^{\Omega \left(X\right)}}$
4. Some Fragments of ${\u22a2}_{\mathsf{IPC}}$
4.1. ${\u22a2}_{\wedge}$
 (ii)
 If $\mathcal{A}$ is a meet semilattice, and ${\mathcal{A}}_{I}$ is $\mathcal{A}$consistent with ${\u22a2}_{\wedge}$, then ${\mathcal{A}}_{I}=\mathcal{A}$.
 (iii)
 ${\u22a2}_{\wedge}$ is categorical in every setbased semantics.
 (ii):
 As in the proof of (b) in the proof of Theorem 3.6, using that $p\wedge q{\u22a2}_{\wedge}p$, $p\wedge q{\u22a2}_{\wedge}q$, and $p,q{\u22a2}_{\wedge}p\wedge q$.
 (iii):
 As in the proof of Corollary 3.8. □
4.2. ${\u22a2}_{\top \wedge \to}$
 (ii)
 If $\mathcal{A}$ is a Brouwerian semilattice, and ${\mathcal{A}}_{I}$ is $\mathcal{A}$consistent with ${\u22a2}_{\top \wedge \to}$, then ${\mathcal{A}}_{I}=\mathcal{A}$.
 (iii)
 ${\u22a2}_{\top \wedge \to}$ is categorical in every setbased semantics.
 (ii):
 Again, the corresponding parts of the proof of of Theorem 3.6 work: (b), half of (c), (d) (which depends on (7), which holds in Brouwerian semilattices), (e) (likewise, note that ${1}^{\mathcal{A}}$ is used), and (g). The ${\u22a2}_{\mathsf{IPC}}$consequences used in the proof all hold for ${\u22a2}_{\top \wedge \to}$.
 (iii):
 As in the proof of Corollary 3.8. □
4.3. ${\u22a2}_{\vee}$
 (ii)
 There is a nonstandard Kripke interpretation consistent with ${\u22a2}_{\vee}$.
 (ii):
 Using the derivability facts of ${\u22a2}_{\vee}$ listed above, it is not hard to verify:
 (19)
 ${\psi}_{1},\dots ,{\psi}_{n}{\u22a2}_{\vee}\phi $ iff there is ${\psi}_{i}$ such that each atom in ${\psi}_{i}$ occurs in $\phi $.
4.4. ${\u22a2}_{\wedge \vee}$ and ${\u22a2}_{\neg \wedge \vee}$
 (ii)
 $\mathcal{A}$ is consistent with ${\u22a2}_{\neg \wedge \vee}$ iff $\mathcal{A}$ is a pseudocomplemented distributive lattice.
 (20)
 Open problem
5. Discussion
5.1. Carnap’s Question and the Meaning of the Connectives
 (21)
 ${\psi}_{1},\dots ,{\psi}_{n}{\vDash}_{\mathcal{A}}\phi $ iff for all valuations v on A and all $c\in A$, $c{\le}^{\mathcal{A}}{\left[{\psi}_{i}\right]}_{v}^{\mathcal{A}}$ for $i=1,\dots ,n$ implies $c{\le}^{\mathcal{A}}{\left[\phi \right]}_{v}^{\mathcal{A}}$.
5.2. Further Questions
 1.
 One can weaken the logics ${\u22a2}_{\Phi}$ defined in Section 4 by constraining the inference rules in various ways. A case in point is [20], which studies a weaker logic in ${L}_{\neg \wedge \vee}$, called ${\u22a2}_{\mathsf{F}}$, with restrictions on the ∨E and ¬I rules. Among other things, distributivity no longer holds.26 Holliday shows how ${\u22a2}_{\mathsf{F}}$ can be seen as a neutral base logic, from which intuitionistic logic, versions of the orthologic studied in [21], and classical logic, can be obtained by suitable additions or changes to the rules or by corresponding constraints in the algebraic or the relational semantics he provides for ${\u22a2}_{\mathsf{F}}$. Finding a nonstandard interpretation consistent with ${\u22a2}_{\mathsf{F}}$—if there is such an interpretation—might be easier than finding one for ${\u22a2}_{\neg \wedge \vee}$.
 2.
 Theorem 3.6 is an existence and uniqueness result: there is an interpretation consistent with ${\u22a2}_{\mathsf{IPC}}$ (namely, the standard interpretation), and it is in fact the only one. In the prooftheoretic tradition, the existence and uniqueness of propositional connectives is a wellestablished topic; see [12], Ch. 4, for a comprehensive overview. Here, the existence and uniqueness is relative to a set of rules. For example, ∧ and ∨ are unique relative to a natural deduction presentation of $\mathsf{IPC}$, in the sense that if we introduce new connectives ${\wedge}^{\prime}$ and ${\vee}^{\prime}$, with the same rules as for ∧ and ∨, respectively, then $\phi \wedge \psi $ is equivalent to $\phi {\wedge}^{\prime}\psi $, and similarly for ∨ and ${\vee}^{\prime}$. Indeed, we only need the introduction and elimination rules for these two connectives. With the notation from Section 4, we have: $\phi \wedge \psi \u22a3{\u22a2}_{\wedge {\wedge}^{\prime}}\phi {\wedge}^{\prime}\psi $ and $\phi \vee \psi \u22a3{\u22a2}_{\vee {\vee}^{\prime}}\phi {\vee}^{\prime}\psi $.As Humberstone noted, and [22] spells out in detail, there is a difference between ∧ and ∨ here: while the former is implicitly definable in a precise sense, the latter, although unique, is not. Došen and SchroederHeister explore connections to Beth’s Definability Theorem.27It is an intriguing question whether, and in that case how, their prooftheoretic notion of implicit definability relates to our semantic notion of categoricity (which exhibits a similar contrast between ∧ and ∨).
 3.
 We have so far followed the lead of [5] and other papers in considering consequence relations ⊢ where the first argument is a (possibly empty) set of formulas and the second is a formula; the setfmla setting in the terminology of [12]. One could look at other settings, such as fmla (no premises), setset, fmlafmla, or seqfmla (with a sequence of premises). As we have seen, the categoricity results would differ significantly. We think setfmla most naturally corresponds to intuitive ideas about ‘what follows from what’, but one would like a better argument for why the others are less suitable. Alternatively, and less contentiously, an overview of what the results would be in those settings.28
 4.
 In all categoricity results (in our sense) that we are aware of, it is practically immediate that any interpretation consistent with the relevant consequence relation ⊢ must interpret conjunction (∧) standardly. It is the other logical constants that need some work (except for ⊥ if Ex Falso holds). However, one could argue that this is built into our definition of consistency with ⊢ in Definition 2.1: the intersection of the values of the premises must be included in the value of the conclusion. This guarantees that $U\cap V\subseteq I(\wedge )(U,V)$; the converse inclusion is from singlepremise ∧elimination.By contrast, the semantics for $\mathsf{IPC}$ presented in [24] might well be called a setbased semantics, since the semantic values are sets of states, but it does not fit our Definition 2.1.29 The relevant class of structures is the class of complete, residuated partial orders $\mathcal{F}=(X,\le )$, where completeness, as usual, means that least the upper bounds (⊔) and greatest lower bounds (⊓) of all subsets of X exist.30 Fine’s (standard) interpretation of conjunction is as follows:
 (22)
 ${\left[[\phi \wedge \psi ]\right]}_{v}^{\mathcal{F}}=\{x\bigsqcup y:x\in {\left[\left[\phi \right]\right]}_{v}^{\mathcal{F}}\mathrm{and}y\in {\left[\left[\psi \right]\right]}_{v}^{\mathcal{F}}\}$
$x\bigsqcup y$ is thought of as the ‘fusion’ of x and y. Accordingly, $\phi $ is called an exact consequence of ${\psi}_{1},\dots ,{\psi}_{n}$ if (for all $\mathcal{F}$ and v) $\bigsqcup \{{x}_{1},\dots ,{x}_{n}\}\in {\left[\left[\phi \right]\right]}_{v}^{\mathcal{F}}$ whenever ${x}_{i}\in {\left[\left[{\psi}_{i}\right]\right]}_{v}^{\mathcal{F}}$, for $i=1,\dots ,n$. Fine shows that ${\u22a2}_{\mathsf{IPC}}$ is sound and complete for this notion of semantic consequence. Although Definition 2.1 does not apply here (since it intersects the semantic values of the premises, rather than ‘fusing’ their members), one could well raise the issue of categoricity with respect to Fine’s semantics. However, the results in this note say nothing about that.
 5.
 Another natural line of inquiry would be to investigate categoricity for intuitionistic predicate logic. To what extent do the results of classical predicate logic ([5]) carry over?
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Notes
1  Compositionality is by now an accepted requirement on formal semantics, but the idea was not around in 1943. The result as stated presupposes that propositional atoms are treated as variables, as we do here. Otherwise, one would have to exclude by stipulation the nonstandard compositional interpretations making all sentences true, since these would also be consistent with ${\u22a2}_{\mathsf{CPC}}$. 
2  Most but not all: a semantics for $\mathsf{IPC}$, to which our results do not apply, is given in Section 5. 
3  There are several equivalent ways of defining Heyting algebras; see BH19, Section 2.1. 
4  In earlier work we used ‘possible worlds semantics’ rather than ‘setbased semantics’, but this is potentially misleading. Indeed, the elements of the domains in the structures in $\mathcal{S}$ need not have anything to do with worlds, or states, or possibilities. A more accurate label, in view of clause 6 of the definition, would be intersective setbased semantics, but we find this too clumsy. An example of a semantics for $\mathsf{IPC}$ where semantic values are sets but which is not setbased in our sense is mentioned Section 5.2:4. 
5  We use ${[\xb7]}_{v}^{\xb7}$ for semantic values in algebraic semantics, and ${\left[[\xb7]\right]}_{v}^{\xb7}$ for values in setbased semantics. 
6  BH19 lets valuations be functions v from Prop to $\mathit{Up}\left(\mathcal{F}\right)$ (which they write as $\mathit{Up}\left(X\right)$) and then defines ${\left[\left[p\right]\right]}_{v}^{\mathcal{F},j}=jv\left(p\right)$. Since the standard interpretation of the connectives takes pairs of fixed upsets to fixed upsets, all semantic values belong to $\mathit{Up}{\left(\mathcal{F}\right)}_{j}$. So, we obtain the same result by taking valuations to map propositional variables directly to fixed upsets, as done here. 
7  A Heyting algebra $\mathcal{A}$ is complete if ${\bigvee}^{\mathcal{A}}B$ exists for every $B\subseteq A$ (and hence ${\bigwedge}^{\mathcal{A}}B$ also exists). The Heyting algebras exemplified in this note are in fact complete, but this will not play a role for our purposes here. 
8  So $\Omega \left(X\right)\subseteq \mathcal{P}\left(X\right)$, $\varnothing ,X\in \Omega \left(X\right)$, and $\Omega \left(X\right)$ is closed under finite intersections and arbitrary unions. 
9  We are now in the framework that Humberstone [12] calls ≤based algebraic semantics. The definition is suggested by his Remark 2.14.7(i). 
10  Theorem 3.6 generalizes our original formulation of Carnap categoricity for $\mathsf{IPC}$, and was suggested to us by an anonymous referee, to whom we are most grateful. 
11  It is easy to find examples of algebras ${\mathcal{A}}_{1}$, ${\mathcal{A}}_{2}$ of signature $\{\perp ,\top ,\wedge ,\vee \}$ such that ${\mathcal{A}}_{1}$ is a bounded lattice, ${A}_{1}={A}_{2}$, ${\wedge}^{{\mathcal{A}}_{1}}={\wedge}^{{\mathcal{A}}_{2}}$, and ${\le}^{{\mathcal{A}}_{1}}={\le}^{{\mathcal{A}}_{2}}$, but ${\vee}^{{\mathcal{A}}_{2}}$ is not the least upper bound operator. 
12  Dummett algebras have an interesting connection to a proposal in [13] to explain the significance of Beth semantics in terms of a distinction between a formula being verified and assertible at a stage x (BH19, end of Section 3.2). 
13  BH19 gives a simple example of a Dummett algebra that is not a Heyting algebra (just before Theorem 3.25), using the Beth nucleus ${j}_{B}$ on $\mathit{Up}\left(\mathcal{F}\right)$, where $\mathcal{F}$ is the poset $\{0,1\}$ with the standard order, so $\mathit{Up}\left(\mathcal{F}\right)$ is $\{\varnothing ,\{1\},\{0,1\left\}\right\}$ with the inclusion order. However, this does not yield a nonstandard Beth interpretation, since the nonstandard interpretation of → as in (12) does not carry over to $\mathit{Up}{\left(\mathcal{F}\right)}_{{j}_{B}}=\{\varnothing ,\{0,1\}\}$. 
14  The example was originally intended to show that the stronger notion of consistency is needed for the result in [5] about $\mathsf{CPC}$; see Remark 3.9. Here, we have adapted it to $\mathsf{IPC}$. 
15  g is similar to the Kolmogorov translation, say, G, which puts a double negation in front of all subformulas. One can show that if $\phi $ is not an atom, ${\u22a2}_{\mathsf{IPC}}g\left(\phi \right)\leftrightarrow G\left(\phi \right)$. It is well known that ${\u22a2}_{\mathsf{CPC}}\phi \iff {\u22a2}_{\mathsf{IPC}}G\left(\phi \right)$, and since no atom is a theorem, (17) follows. [14] is a survey of various negative translations. 
16  Indeed, it validates all $\mathsf{CPC}$ theorems, by (17), and since the above argument in fact works for all valuations $v:\mathit{Prop}\to \mathcal{P}\left(X\right)$. As Wesley Holliday pointed out, ${I}^{\Omega \left(X\right)}$ evaluates complex formulas using the double negation nucleus, i.e., in the Heyting algebra of regular open sets (sets U such that $U=\mathit{int}\left(\mathit{cl}\right(U\left)\right))$, which is a Boolean algebra. 
17  In most (all?) cases, ${\u22a2}_{\Phi}$ turns out to be the restriction of ${\u22a2}_{\mathsf{IPC}}$ to ${L}_{\Phi}$. 
18  For example, let $\mathcal{A}=(\{a,b\},{\wedge}^{\mathcal{A}})$, where (dropping superscripts) $a\wedge a=a$, $b\wedge b=b$, $a\wedge b=b$, $b\wedge a=a$. Then $x\le y$, i.e., $x\wedge y=x$, holds iff $x=y$, so ≤ is a partial order, but ∧ is not commutative in $\mathcal{A}$. 
19  We skip the brackets and commas when writing ${\u22a2}_{\Phi}$. 
20  If we add ⊥ to the signature as the smallest element, we have negation, and contradictions, in the language, so e.g., $(a\to a)\to \perp =\perp $. The results corresponding to Fact 4.2 below still hold, by an easy addition to the proof. 
21  This simple nonstandard interpretation was suggested by Wesley Holliday, and replaces a more ad hoc construction than we originally gave. Using another ad hoc construction, one can give a nonstandard Kripke interpretation of ¬, which is consistent with ${\u22a2}_{\neg}$, but we omit the proof of that here. Here, ${\u22a2}_{\neg}$ is defined with the two natural deduction rules $\begin{array}{ccc}& & \\ \frac{\psi \neg \psi}{\phi}& & \frac{\begin{array}{ccc}\left[\phi \right]& & \left[\phi \right]\\ :& & :\\ \psi & & \neg \psi \end{array}}{\neg \phi}\\ & & \end{array}$ 
22  The rows 111, 101, 011, are fixed by $p{\u22a2}_{\vee}p\vee q$ and $q{\u22a2}_{\vee}p\vee q$. The row 000 is fixed by $p\vee p{\u22a2}_{\vee}p$, which is an instance of ∨elimination. 
23  Here 0 is identified with $a\wedge \neg a$. The variety of pseudocomplemented lattices can also be defined by adding the following equations to those for distributive lattices:

24  This was observed by the second author and is stated and generalized in [7]. The set of valid sentences in $\mathcal{L}\left({\mathcal{Q}}_{0}\right)$ is not recursively enumerable. 
25  Questions and comments from several people inspired the remarks in this subsection, in particular from Wes Holliday for part 1 and from Johan van Benthem for part 2. 
26  Holliday presents ${\u22a2}_{\mathsf{F}}$ with a Fitchstyle natural deduction system, where the added constraint becomes a requirement on the Reiteration rule. ${\u22a2}_{\mathsf{F}}$ is also extended to a firstorder language with the quantifiers ∀ and ∃ (but without → and =). Failure of distributivity is a characteristic of quantum logic, but Holliday argues that it also accords with certain facts about natural language semantics, in particular facts about epistemic modals. 
27  
28  In fact, the solution proposed in Carnap [1] to the existence of nonnormal interpretations was in effect to use the setset framework instead. This makes good sense for classical logic. However, for intuitionistic logic there is, as far we know, no obviously correct candidate for the setset version of $\mathsf{IPC}$. For example, $p\vee q\u22a2p,q$ is not valid in standard Beth semantics (since we can have ${j}_{b}(U\cup V)\u2288U\cup V$), although it holds in Kripke semantics. 
29  We thank an anonymous referee for directing our attention to Fine’s semantics, which is based on partial orders, but where semantic values are not upsets. 
30  Fine defines, for $x,y\in X$, $x\to y=\sqcap \{z\in X:y\le x\bigsqcup z\}$ and calls $\mathcal{F}$ residuated if it always holds that $y\le x\bigsqcup (x\to y)$. As pointed out in BH19 (Section 2.3), Fine’s frames are not complete Heyting algebras but complete coHeyting algebras. Fine argues that his semantics comes closer to the BHK interpretation of the connectives than other formal semantics for $\mathsf{IPC}$, in view of the clause (22) for conjunction, as well as the clause for implication: ${\left[[\phi \to \psi ]\right]}_{v}^{\mathcal{F}}=\{\bigsqcup \{x\to f\left(x\right):x\in {\left[\left[\phi \right]\right]}_{v}^{\mathcal{F}}\}:f\mathrm{is}\mathrm{a}\mathrm{function}\mathrm{from}{\left[\left[\phi \right]\right]}_{v}^{\mathcal{F}}\mathrm{to}{\left[\left[\psi \right]\right]}_{v}^{\mathcal{F}}\}$ 
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Tong, H.; Westerståhl, D. Carnap’s Problem for Intuitionistic Propositional Logic. Logics 2023, 1, 163181. https://doi.org/10.3390/logics1040009
Tong H, Westerståhl D. Carnap’s Problem for Intuitionistic Propositional Logic. Logics. 2023; 1(4):163181. https://doi.org/10.3390/logics1040009
Chicago/Turabian StyleTong, Haotian, and Dag Westerståhl. 2023. "Carnap’s Problem for Intuitionistic Propositional Logic" Logics 1, no. 4: 163181. https://doi.org/10.3390/logics1040009