B1 Introduction
This is the second part of a two-part series on the logic of hyperlogic, a hyperintensional semantics designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) in the object language. To recap, this regimentation is achieved using:
-
• a multigrade entailment operator
; -
• propositional quantifiers
${\forall }{p}$
and
${\exists }{p}$
; -
• interpretation terms
$\iota $
that double as atomic formulas (“
$\iota $
is correct”); -
• hybrid operators
${\mathop {@}\nolimits }_{\iota }$
(“according to
$\iota $
”) and
${\mathop {\downarrow }\nolimits } i$
(“where i is the current interpretation”).
The semantics of hyperlogic introduces the notion of a “hyperconvention,” i.e., a complete interpretation of the propositional variables, Boolean connectives, and
over some space of possible worlds propositions. Interpretation terms denote “conventions,” modeled as sets of hyperconventions. Propositional quantifiers range over (special kinds of) index propositions, i.e., sets of world-hyperconvention pairs. Models in this semantics determine (i) a set of worlds W; (ii) a domain of admissible conventions
$D_{\mathbb {C}}$
; (iii) a domain of admissible index propositions
$D_{\mathbb {P}}$
; and (iv) a valuation V. Truth-in-a-model is evaluated relative to worlds and hyperconventions. Operators like
${\mathop {@}\nolimits }_{\iota }$
can shift the hyperconvention parameter. This allows formulas to be assessed on alternative interpretations of the connectives and entailment. Hyperintensionality is thus captured through shifting these interpretations.
In Part A [Reference Kocurek34], a complete axiomatization for this semantics was given. The axiomatization in Part A captures consequence over the class of all models. Almost no constraints are placed on either a model’s convention or proposition domain. The resulting logic for hyperlogic is, therefore, fairly minimal. For example, no constraints are placed on the interpretations the entailment operator
can receive. Yet, intuitively, it would be a stretch to say
really represents a notion of “entailment” if, say, it wasn’t factive (i.e., if
did not imply
$\phi $
), or if it wasn’t reflexive or transitive. It would then be natural to inquire into how imposing such constraints affects the underlying logic of hyperlogic.
Furthermore, hyperlogic was initially motivated by concerns with the interaction between metalogical claims and hyperintensional operators such as attitude verbs, counterfactuals, and so on. Yet the language of hyperlogic introduced in Part A does not contain any of such operators.
In Part B of this series, we take initial steps to filling these gaps. We start by studying stronger logics for hyperlogic that can be obtained by adding additional rules and axioms in Section B2. These stronger logics can be shown to be sound and complete over classes of models whose convention and proposition domains satisfy certain natural constraints. In Section B3, we examine how the completeness results from Part A carry over to languages with hyperintensional operators. We conclude in Section B4 with some questions left open by this two-part investigation into the logic of hyperlogic. Section B5 is a technical appendix containing proofs of completeness for various classes of hypermodels.
Note: as this is a continuation of a two-part series, I will freely refer back to the definitions, notation, and results from Part A [Reference Kocurek34], rather than repeat them. Labels for sections, definitions, theorems, and tables are prefixed with the part that they refer to (e.g., “Section A3” refers to Section 3 of Part A).
B2 Restrictions on hypermodels
Let us start by exploring constraints we may impose on the class of hypermodels and how that affects the logic of hyperlogic. In Section B2.1, we look at general constraints on the convention domain and present axiomatizations in the quantifier-free fragment over those hypermodels. In Section B2.2, we extend some of these results to languages with propositional quantifiers. Finally, in Section B2.3, we examine constraints on the proposition domain.
B2.1 Quantifier-free fragment
Table B1 contains a sample of constraints one may want to impose on the convention domain. For the analyticity constraint, we write
$c \approx c'$
to mean c and
$c'$
are exactly alike except possibly in how they interpret propositional variables (i.e.,
for all
). The intersection of each class in Table B1 is denoted by concatenation (e.g., the class of analytic and full hypermodels is
$\textsf {An}\textsf {F}$
). Where
$\textsf {X}$
is a class of hypermodels, we define classical and universal entailment over
, written
$\Gamma \vDash _{\textsf {X}} \phi $
and
respectively, as in Definition A2.16 except restricting to hypermodels in
$\textsf {X}$
.
Table B1 Some constraints on convention domains.
Table B2 contains axiomatizations of consequence over various classes. Some of the axioms make use of the following abbreviations:
Here are their truth conditions (where
):
In addition, we write
$\vec {\phi }$
for
${\phi _{1},\dots ,\phi _{n}}$
, and
$\vec {\phi }^{\kappa } = \vec {\psi }^{\lambda }$
for
. Where
$\textbf {L}$
is a logic and
$\text {A}$
is an axiom,
$\textbf {L} + \text {A}$
is the result of extending
$\textbf {L}$
with
$\text {A}$
(i.e., the rules still apply to formulas derived using
$\text {A}$
). If
$\text {R}$
is a rule,
$\textbf {L} + \text {R}$
is the result of closing
$\textbf {L}$
under
$\text {R}$
along with the other rules. Given this, we have the following:
Table B2 Axiomatizations in
${\mathcal {L}^{\textsf {H}}}$
for various classes from Table B1. Axiomatizations in
$\mathcal {L}^{\textsf {HE}}$
(except those appealing to RAn, which becomes infinitary when add
) are obtained by replacing
${\textbf {H}}$
with
and generalizing the corresponding axioms accordingly.
Theorem B2.1 (Relative completeness in
${\mathcal {L}^{\textsf {H}}}$
and
$\mathcal {L}^{\textsf {HE}}$
).
The axiomatic systems in Table B2 are sound and complete for (consequence over) the relevant class of hypermodels.Footnote 1 (See Section B5.1 for the proof.)
In addition, we can consider imposing restrictions specifically on the interpretation of
. Usual suspects include reflexivity, transitivity, monotonicity, etc. But there are also “unusual” suspects to consider (e.g., factivity) since
is an object language operator. Table B3 contains examples of such constraints, with their corresponding axioms stated in Table B4. Following our earlier convention, we write
$\vec {X}$
for
$X_{1},\dots ,X_{n}$
. (If
$n = 0$
, then
$\vec {X} = \left\langle \right\rangle $
.) We also write
for
.
Table B3 Some constraints on the interpretation of
.
Table B4 Axiomatizations in
$\mathcal {L}^{\textsf {HE}}$
for various classes from Table B3.
Theorem B2.2 (Relative completeness for
).
The axiomatic systems resulting from adding the relevant axioms in Table B4 to
are sound and complete for the relevant class of hypermodels.
Proof (Sketch).
We revise the definition of the proposition space for canonical hyperconventions (Definition A3.30) so that
$\pi _{c_{\kappa }} = {\left\{ X {\left|\ {[X]_{\kappa } \neq \emptyset } \right.} \right\}}$
.Footnote
2
The completeness proof in Section §A3.2 remains in tact. We just need to verify that if we impose an axiom, the canonical model satisfies the corresponding constraint. The proof is more-or-less the same for each case. We illustrate with the transitivity case. Suppose
. Since
, there are some
$\vec {\phi } \in [\vec {X}]_{\kappa }$
and
$\vec {\psi } \in [\vec {Y}]_{\kappa }$
such that
for each i (note: we can let
$\vec {\phi }$
be the same for each
$\psi _i$
by Lemma A3.29 and
). Since
, there is a
$\chi \in [Z]_{\kappa }$
such that
. By Tr,
. Hence,
.
B2.2 Adding quantifiers
Adding propositional quantifiers to the language allows us the ability to distinguish between classes of models that previously generated the same logic. Notably, the consequence relations over
$\textsf {F}$
,
${\textsf {U}_{\textsf {q}}}$
,
$\textsf {At}$
, and
$\textsf {B}$
are now all distinguishable. In addition, we can now present an axiomatization for
$\textsf {An}$
, which was absent from Section B2.1 (see footnote 1).
Axiomatizations for some of those classes are given in Table B5. Where
$\Sigma $
is a set of axioms of the form
$\Vdash \sigma $
, we let
$\textbf {L} \cup \Sigma $
be the proof system defined as follows:
$\Gamma \Vdash _{\textbf {L} \cup \Sigma } \phi $
iff
$\Gamma \cup {\left\{ \sigma {\left|\ (\Vdash \sigma ) \in \Sigma \right.} \right\}} \Vdash _{\textbf {L}} \phi $
(in other words,
$\Sigma $
are treated as premises, not axioms; this means, among other things, that one cannot necessarily derive the universal generalization of members of
$\Sigma $
). The axiomatizations in Table B5 make use of the following abbreviations:
These have the obvious truth conditions assuming
$\left| V(\kappa )\right| = \left| V(\lambda ) \right| = 1$
(which is the only relevant case for the axiomatizations below).
Table B5 Axiomatizations in
${\mathcal {L}^{\textsf {QH}}}$
for various classes from Table B1.
Theorem B2.3 (Relative completeness in
${\mathcal {L}^{\textsf {QH}}}$
).
The proof systems in Table B5 are sound and complete for the relevant class of hypermodels. (See Section B5.2.)
Notice that no axiomatization for
$\textsf {F}$
is stated. This is because consequence over
$\textsf {F}$
is unaxiomatizable.
Theorem B2.4 (Unaxiomatizability of full consequence in
${\mathcal {L}^{\textsf {QH}}}$
).
$\vDash _{\textsf {F}}$
in
${\mathcal {L}^{\textsf {QH}}}$
is unaxiomatizable. Moreover, where
$\textsf {X}$
is the intersection of any of the classes mentioned in Tables B1 and B6, if
$\textsf {F}\textsf {X} \neq \emptyset $
, then
$\vDash _{\textsf {F}\textsf {X}}$
in
${\mathcal {L}^{\textsf {QH}}}$
is unaxiomatizable.
Table B6 Some constraints on proposition domains.
Proof. Let
. It is easy to verify that if c is full, then
$\mathcal {M},w,c \Vdash \textsf {At}(p)$
iff
$\left|{V(p)(c)}\right| = 1$
. Let
$\Delta $
consist of the following formulas:
Since the propositionally quantified modal logic
$\textbf {K}\pi +$
is unaxiomatizable [Reference Fine19], it suffices to show that for any
$\phi \in {\mathcal {L}^{\textsf {Q}}}$
(the language of propositionally quantified modal logic),
$\vDash _{\textbf {K}\pi +} \phi $
iff
$\Delta , \left|{k}\right| {}_1 \vDash _{\textsf {F}} {\mathop {@}\nolimits }_k\phi $
. We do this by constructing, for each
$\textbf {K}\pi +$
-model, an equivalent full hypermodel of
$\Delta $
and vice versa.
First, let
$\mathcal {K} = \left\langle W,R,V \right\rangle $
be a
$\textbf {K}\pi +$
-model. Let
$c_k$
be defined as follows:
Define
$\mathcal {M}^{\mathcal {K}} = \left\langle W,D_{\mathbb {C}},D_{\mathbb {P}},V^{\mathcal {K}} \right\rangle $
so that (i)
$c_k \in D_{\mathbb {H}}$
, (ii) each
$c \in D_{\mathbb {H}}$
is full, (iii)
$V^{\mathcal {K}}(p) = P_p$
, and (iv)
$V^{\mathcal {K}}(k) = {\left\{ c_k \right\}}$
. Clearly,
$\mathcal {M}^{\mathcal {K}},w,c_k \Vdash \Delta \cup {\left\{ {\left| {k} \right| {}_1} \right\}}$
. Moreover, by induction, for all
$\phi \in {\mathcal {L}^{\textsf {Q}}}$
and all
$Q_{1},\dots ,Q_{n}$
where
$Q_i(c_k) = X_i$
, we have
$\mathcal {K}^{q_{1},\dots ,q_{n}}_{{X_{1},\dots ,X_{n}}},w \Vdash \phi $
iff
$(\mathcal {M}^{\mathcal {K}})^{q_{1},\dots ,q_{n}}_{{Q_{1},\dots ,Q_{n}}},w,c_k \Vdash \phi $
. Hence,
$\mathcal {K},w \Vdash \phi $
iff
$\mathcal {M}^{\mathcal {K}},w,c \Vdash {\mathop {@}\nolimits }_k\phi $
.
Next, let
$\mathcal {M} = \left\langle W,D_{\mathbb {C}},D_{\mathbb {P}},V \right\rangle $
be a full hypermodel satisfying
$\Delta \cup {\left\{ {\left|{k}\right| {}_1} \right\}}$
. Let
$c_k$
be such that
$V(k) = {\left\{ c_k \right\}}$
. Define
$\mathcal {K}^{\mathcal {M}} = \left\langle W,R,V^{\mathcal {M}} \right\rangle $
so that (i)
$wRv$
iff for all
$X \mathrel {\subseteq } W$
, if
, then
$v \in X$
, and (ii)
$V^{\mathcal {M}}(p) = c_k(p)$
. We establish by induction that for all
$\phi \in {\mathcal {L}^{\textsf {Q}}}$
and all
${Q_{1},\dots ,Q_{n}}$
where
$Q_i(c_k) = X_i$
, we have
$\mathcal {M}^{q_{1},\dots ,q_{n}}_{{Q_{1},\dots ,Q_{n}}},w,c_k \Vdash \phi $
iff
$(\mathcal {K}^{\mathcal {M}})^{q_{1},\dots ,q_{n}}_{{X_{1},\dots ,X_{n}}},w \Vdash \phi $
. The only interesting case is the
-clause. Observe that
.Footnote
3
For notational ease, let
$\mathcal {M}^* = \mathcal {M}^{q_{1},\dots ,q_{n}}_{{Q_{1},\dots ,Q_{n}}}$
and
$\mathcal {K}^* = (\mathcal {K}^{\mathcal {M}})^{q_{1},\dots ,q_{n}}_{{X_{1},\dots ,X_{n}}}$
.
(
$\Rightarrow $
) Suppose
. Thus,
. Let
$v \in R[w]$
. Then for all
$X \mathrel {\subseteq } W$
, if
, then
$v \in X$
. Hence,
, which by IH means
. Hence,
.
(
$\Leftarrow $
) Suppose
. Thus,
. Since
$c_k$
is full, by Definition A2.11 (constraint (ii) on
$D_{\mathbb {P}}$
), there exists a P such that
. By the definition of
$\Delta $
,
. Let Q be such that
. Thus,
$Q(c_k) = {\left\{ v \right\}}$
for some
. By IH,
, i.e.,
$\mathcal {K}^*,v \nVdash \phi $
. And since
, that means
$v \in R[w]$
, and so
.
Corollary B2.5 (Unaxiomatizability of classically complete consequence in
${\mathcal {L}^{\textsf {QH}}}$
).
$\vDash _{\textsf {Co}_{{cl}}}$
in
${\mathcal {L}^{\textsf {QH}}}$
is unaxiomatizable, as is
$\vDash _{\textsf {Co}_{{cl}}\textsf {X}}$
for any
$\textsf {X}$
that is the intersection of any of the classes mentioned in Tables B1 and B6 where
$\textsf {Co}_{{cl}}\textsf {X} \neq \emptyset $
.
Proof. Since
$V({cl}) = {\left\{ c \in {\mathbb {H}_{W}} {\left|\ c \text { is classical} \right.} \right\}}$
, there is a
$c \in V({cl})$
such that c is full. So adding
${\mathop {@}\nolimits }_{cl}{\forall }{p}{\exists }{q}(p = {\mathop {@}\nolimits }_k q)$
to
$\Delta $
is enough to ensure that
$c_k$
is full.
B2.3 Constraints on propositions
Let’s now turn to constraints on the proposition domain. A sample of such constraints is given in Table B6. For strong closure, we write
$\mathcal {M} \approx \mathcal {M}'$
to mean
$\mathcal {M}$
and
$\mathcal {M}'$
are based on the same hyperframe (i.e., only differ in valuation). Axiomatizations for consequence over some of these classes are stated in Table B7. Some of the axioms use the following abbreviation:
. Completeness for the intersections of these classes can be gotten from combining the relevant axiomatizations, with the exception of
${\textsf {Cl}_{\Phi }}{\textsf {Df}_{\Phi }}$
and
${\textsf {Cl}_{\Phi }^+}{\textsf {Df}_{\Phi }}$
, which are mentioned explicitly in Table B7.
Table B7 Axiomatizations in
${\mathcal {L}^{\textsf {QH}}}$
for various classes from Table B6.
B3 Hyperintensional operators
In this section, we enrich the language of hyperlogic with hyperintensional operators and explore their logic(s). We start by adding a counterfactual conditional and then show how a similar approach can apply to belief and knowledge operators. In Section B3.1, we expand the syntax and semantics from Section §A2 to include a counterfactual conditional (following Kocurek [Reference Kocurek33]). In Section B3.2, we axiomatize the minimal counterfactual hyperlogic on this semantics. In Section B3.3, we extend this axiomatization to include an entailment operator/propositional quantifiers. In Section B3.4, we explore stronger counterfactual hyperlogics obtained by imposing restrictions on the selection function. Finally, in Section B3.5, we show how a similar approach can address the hyperlogic of belief/knowledge.
B3.1 Selection semantics
For any language
$\mathcal {L}^*$
mentioned in Part A, we can consider the language
that results from extending
$\mathcal {L}^*$
with a counterfactual conditional
. For instance,
is the result of extending
${\mathcal {L}^{\textsf {0}}}$
with
,
the result of extending
${\mathcal {L}^{\textsf {H}}}$
with
, and so on. To extend hyperlogic with a counterfactual conditional, Kocurek [Reference Kocurek33] proposes we allow counterfactuals to shift the hyperconvention parameter of an index. This can be achieved by simply replacing worlds in the standard (intensional) selection semantics for counterfactuals [Reference Lewis39, Reference Stalnaker and Rescher48] with world-hyperconvention pairs. Thus, we revise Definitions A2.11 and A2.12 as follows:
Definition B3.7 (Selection hypermodel).
A selection hyperframe is a tuple
$\mathcal {F} = \left\langle W,D_{\mathbb {C}},D_{\mathbb {P}},f \right\rangle $
where
$\left\langle W,D_{\mathbb {C}},D_{\mathbb {P}} \right\rangle $
is a hyperframe and
${f}\colon {\mathop {\wp }{\mathbb {I}_{D_{\mathbb {H}}}} \mathrel {\times } {\mathbb {I}_{D_{\mathbb {H}}}}} \rightarrow\, {\mathop {\wp }{\mathbb {I}_{D_{\mathbb {H}}}}}$
is a selection function. A selection hypermodel over
$\mathcal {F}$
is a selection hyperframe paired with a valuation for
$\mathcal {F}$
. Satisfaction is defined as in Definition A2.12 with the following addition, where
:
At the outset, we impose no restrictions on the selection function. Some theorists (e.g., Cohen [Reference Cohen15] and Nolan [Reference Nolan43]) argue that if counter(meta)logicals are nonvacuous, then the logic of counterfactuals is trivial. For example, it is nearly universally accepted that
. Yet, here is an alleged counterexample:
-
(1)
-
a. If every instance of conjunction elimination had failed, Alice and Beth would be sad.
-
b.
$\stackrel {?}{\mathbin {\not \Rightarrow }}$
If every instance of conjunction elimination had failed, Alice would be sad.
-
Similar “counterexamples” can be constructed to nearly every principle of counterfactual reasoning.Footnote
4
Even principles such as
have been called into question [Reference Nolan43, p. 555].Footnote
5
Hyperlogic offers refuge to those who find this disheartening. As we’ll see, even though counter(meta)logicals are nonvacuous in hyperlogic, its counterfactual logic is nontrivial: the standard counterfactual principles can be salvaged when the connectives used to state those principles are classically rigidified. This means, among other things, that imposing constraints on the selection function is not incompatible with the nonvacuity of counter(meta)logicals, such as those in (1).
B3.2 Completeness
Let’s turn to the logic of counterfactual hyperlogic. Given that we are not placing any constraints on the selection function, what counterfactual principles, if any, are valid?
Kocurek and Jerzak [Reference Kocurek and Jerzak35, Appendix] show that the logic of classical consequence in
is the same as the logic of the standard “impossible worlds” semantics for counterfactuals, where we can model an impossible world as an arbitrary set of formulas. But this is only because (as Cohen [Reference Cohen15] and Nolan [Reference Nolan43] suggest) there are no valid principles of counterfactual reasoning that aren’t already substitution instances of
$\textbf {S5}$
-theorems. Thus, without further constraints, (1) is invalid in the hyperconvention semantics when regimented so:
Fortunately, counterfactual hyperlogic in
is more interesting, since it has the expressive resources to “hold fixed” the interpretation of a certain connective within the scope of a counterfactual [Reference Kocurek and Jerzak35, p. 21]. If we require a certain formula within a counterfactual to be interpreted according to, say, a classical hyperconvention, then any entailments that formula generates in classical logic must be preserved. For example, the reason (1) seems to invalidate conjunction elimination in the consequent is that the word “and” in the consequent is being interpreted relative to a logic where conjunction elimination fails. If we force that “and” to be interpreted classically, however, then the argument is valid. That is, (1) is valid when regimented so:Footnote 6
This could explain why (1) has a ring of plausibility to it. Even though the counterlogical supposition is asking us to interpret conjunction so that conjunction elimination fails, it’s nevertheless tempting to hold on to our “standard” classical way of interpreting “and” when evaluating the consequent.Footnote 7
We can generalize this observation by axiomatizing consequence in
. The axiomatic system
is given in Table B8. Some notation:
Table B8 Axioms and rules for provability in
(with some derivable rules). The rules for
$\Vdash $
can be converted into rules for
$\vdash $
(given
$\kappa $
isn’t
${cl}$
) by applying C2U, U2C, and Cl.
As before, let
be the expansion of
${\mathcal {L}^{\textsf {H}}}$
with
$\textsf {Prop}^+$
and
$\textsf {INom}^+$
.
Definition B3.8 (Lindenbaum set).
A set
is Lindenbaum if it is a
-maximal consistent set that satisfies constraints
$(i)$
–
$(iii)$
from Definition A3.23 (nominalized, witnesses
s, differentiates terms) as well as the following:
-
iv.
$\Gamma ^+$
differentiates antecedents: if
, then
$\left|{l^+}\right| {}_1 \in \Gamma ^+$
and
$(\phi \neq _{l^+} \phi ') \in \Gamma ^+$
for some fresh
$l^+ \in \textsf {INom}^+$
. -
v.
$\Gamma ^+$
witnesses actual
s: if
, then
$\left|{l^+}\right| {}_1 \in \Gamma ^+$
and
for some fresh
$l^+ \in \textsf {INom}^+$
. -
vi.
$\Gamma ^+$
witnesses possible
s: if
, then
$\left|{l^+}\right| {}_1 \in \Gamma ^+$
and
for some fresh
$l^+ \in \textsf {INom}^+$
.
Lemma B3.9 (Counterfactual Lindenbaum).
If
is consistent, then there is a Lindenbaum set
such that
$\Gamma \mathrel {\subseteq } \Gamma ^+$
.
Proof. The construction is the same as that in Lemma A3.24 except for how we define
$\Gamma _{k+1}$
from
$\Gamma _k'$
(both
$l^+$
and
$p^+$
are unused throughout):
Suppose for reductio that
$\Gamma _{k+1}$
is inconsistent. The only cases we need to check are where
, where
, and where
. Assume throughout the contradiction is derivable from
${\gamma _{1},\dots ,\gamma _{n}} \in \Gamma _k$
.
Suppose
. Thus:
Suppose
. Thus:
Suppose
. Thus:
Lemma B3.10 (Counterfactual existence).
Where
$\Delta \in W_{\Gamma }$
:
-
a. If
, then there is a
$\Delta ' \in W_{\Gamma }$
such that
$\phi \notin \Delta '$
. -
b. If
where
$\left|{l}\right| {}_1 \in \Delta $
, then there is a
$\Delta ' \in W_{\Gamma }$
extending
.
Proof. Start with (a). By the proof of Lemma A3.26,
is consistent. Enumerate all formulas of the form
, of the form
, or of the form
as
. We define a sequence of formulas
depending on the form of
$\chi _{n+1}$
. As before,
. If
, then define
$\delta _{n+1}$
as in Lemma A3.26. If
, define
, where
$l^+$
is the first nominal such that
. Suppose for reductio there were no such
$l^+$
. Reasoning as in Lemma A3.26, we can conclude that
and
for all
$l^+$
, and that
. Since
$\Delta $
witnesses possible
s, there is an
$l^+$
such that
,
.
If
, define
, where
$l^+$
is the first such that
. Suppose there is no such
$l^+$
. Then
and
for all
$l^+$
. As before,
, i.e.,
. By S5,
. Since
$\Delta $
witnesses possible
s, there is an
$l^+$
such that
. By S5 again,
,
.
Now for (b). Let
. Then
is consistent. For suppose not. Then for some
, we have
Since
, that means
, contrary to our initial assumption that
,
.
Now, suppose
. Thus,
(by Rigid,
$\text {Intro}_{{\mathop {@}\nolimits }}$
, Red, Bool, and
$\text {Dist}_{{\mathop {@}\nolimits }}^+$
). By Nec
,
. Hence,
. Since
$l_{\Delta },\left|{l_{\Delta }}\right| {}_1 \vdash {\mathop {@}\nolimits }_{l}{\mathop {@}\nolimits }_{l_{\Delta }}\chi \mathbin {\leftrightarrow } \chi $
, the set
is consistent. The proof strategy from here is essentially the same as in part (a), though some changes need to be made to ensure the steps go through. The main change is that we need to replace
and
with
and
. To illustrate, I’ll use the case where
. As before, we define
, where
$l^+$
is the first such that:
. By Bool and the fact that
$l_{\Delta },\left|{l_{\Delta }}\right| {}_1 \in \Sigma $
, this condition is equivalent to
. Suppose, for reductio, there’s no such
$l^+$
. So for all
$l^+$
, there are some
${\gamma _{1},\dots ,\gamma _{n}} \in \Sigma $
such that
. By
,
. Since
,Footnote
8
this means
for all
$l^+$
. As before,
, i.e.,
. By
,
and
. By
and
,
. Since
$\left|{l_{\Delta }}\right| {}_1 \in \Delta $
, that means
. Since
$l_{\Delta } \in \Delta $
, that means
by Bool. Since
$\Delta $
witnesses possible
s, there is an
$l^+$
such that
. By
,
. By
,
,
.
The proofs of the other intermediate lemmas are all as before. To finish the proof, we need to define the selection function for our canonical model.
Definition B3.11 (Defining formula).
Where
$A \mathrel {\subseteq } {\mathbb {I}_{W_{\Gamma }}}$
, we define the set
.
Lemma B3.12 (Replacement of definitions).
For all
$A \mathrel {\subseteq } {\mathbb {I}_{W_{\Gamma }}}$
, all
$c_{\kappa }$
, all
$\phi ,\phi ' \in [A]$
, and all
$\psi $
, we have
.
Proof. Suppose for reductio that
. Since
$\Gamma $
differentiates antecedents, there are some
$l^+$
such that (by
$\text {Dist}_{{\mathop {@}\nolimits }}$
)
$({\mathop {@}\nolimits }_{l^+}\phi \neq {\mathop {@}\nolimits }_{l^+}\phi ') \in \Gamma $
. Since
$\phi ,\phi ' \in [A]$
,
$({\mathop {@}\nolimits }_{l^+}\phi = {\mathop {@}\nolimits }_{l^+}\phi ') \in \Gamma _n$
,
.
Definition B3.13 (Canonical selection function).
We define
$f_{\Gamma }$
, the canonical selection function for
$\Gamma $
, as follows for all
$A \mathrel {\subseteq } {\mathbb {I}_{W_{\Gamma }}}$
, all
$\Delta \in W_{\Gamma }$
, and all
$c_{\kappa }$
. First, if
$[A] = \emptyset $
, then
$f_{\Gamma }(A,\Delta ,c_{\kappa }) = {\left\{ \left\langle \Delta ,c_{\kappa } \right\rangle \right\}} \cap A$
. Second, if
$\phi \in [A]$
, then
$\left\langle \Delta ',c_{\lambda } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
iff for all
, if
, then
${\mathop {@}\nolimits }_{\lambda }\psi \in \Delta '$
.
By Lemma B3.12, if
$\phi ,\phi ' \in [A]$
, then
iff
, so this definition for
$f_{\Gamma }$
is well-defined.
Definition B3.14 (Canonical model).
The canonical selection hypermodel of
$\Gamma $
is the selection hypermodel
$\mathcal {M}_{\Gamma } = \left\langle W_{\Gamma },{D_{\mathbb {C}}}_{\Gamma },{D_{\mathbb {P}}}_{\Gamma },f_{\Gamma },V_{\Gamma } \right\rangle $
where
$\left\langle W_{\Gamma },{D_{\mathbb {C}}}_{\Gamma },{D_{\mathbb {P}}}_{\Gamma },V_{\Gamma } \right\rangle $
is defined as in Definition A3.32 and
$f_{\Gamma }$
is defined as in Definition B3.13.
Lemma B3.15 (Truth).
$\mathcal {M}_{\Gamma },\Delta ,c_{\kappa } \,\Vdash\, \phi $
iff
${\mathop {@}\nolimits }_{\kappa }\phi \in \Delta $
.
Proof. The inductive steps are all the same as before. We just need to check the
step goes through. First,
iff
. By Lemma B3.12 and by IH (
), this holds iff the following holds for all
$\Delta '$
and
$c_{\lambda }$
:
We now show this condition holds for all
$\Delta '$
and
$c_{\lambda }$
iff
.
(
$\Leftarrow $
) Suppose
. Let
$\Delta '$
and
$c_{\lambda }$
be such that for all
, if
, then
${\mathop {@}\nolimits }_{\lambda }\chi \in \Delta '$
. Since
, we have by
that
. Hence,
${\mathop {@}\nolimits }_{\lambda }\psi \in \Delta '$
.
(
$\Rightarrow $
) Suppose
. Thus,
. Since
$\Delta $
witnesses actual
s, there is an
$l^+$
such that
. By Lemma B3.10, there is a
$\Delta ' \in W_{\Gamma }$
such that
. Hence,
$\left\langle \Delta ',l^+ \right\rangle $
is the counterexample we need.
B3.3 Adding
and quantifiers
What changes if we add an entailment operator or propositional quantifiers to
? In the former case, no additional axioms are required apart from those in
and
: all the proofs in Section B3.2 go through in the presence of
. In the latter case, we do need one additional axiom. Observe that the Barcan formula and its converse are universally valid for counterfactuals (where p does not occur free in
$\phi $
):
The converse Barcan formula is easily derived just by combining
${\textbf {QH}}$
and
:
However, the Barcan formula, which is needed to prove the analogue of Lemma A4.41, must be added separately. Other than that, the proofs of completeness for
${\mathcal {L}^{\textsf {QH}}}$
and
can be straightforwardly combined to yield a proof of completeness for
.
Table B9 Axioms and rules for provability in
B3.4 Constraints on selection function
Let’s now look at some constraints on the selection function. Table B10 contains several such constraints. We can extend the completeness result to include such constraints by adding the relevant axioms from Table B11.
Table B10 Some constraints on selection functions.
Table B11 Axiomatizations in
for various classes from Table B10
.
Theorem B3.16 (Relative completeness in
).
The proof systems in Table B11 are sound and complete for the relevant class of selection hypermodels. (See Section B5.4 .)
Let me briefly explain the motivation behind some of these constraints. Vacuism is the view that all counterpossibles (counterfactuals with impossible antecedents) are vacuously true.Footnote
9
Often, vacuists also endorse the necessary consequent and necessary entailment principles, which are all coderivable given success (the labels come from [Reference French, Girard and Ripley20]). Some of these principles have equivalent “hybrid” formulations. In the hyperconvention semantics (with success), counterpossibles are vacuous when we hold fixed the interpretation of the antecedent. This goes back to one of the main motivations for considering hyperlogic as a semantics for metalogical claims, viz., it can formalize “conventionalist” approaches to hyperintensionality, which explain hyperintensionality in terms of convention-shifting (Section §A1). We can regiment this idea of “holding fixed” an interpretation using the hybrid binder
${\mathop {\downarrow }\nolimits }$
, which is what allows alternative axiomatizations for some of these principles.
Second, the “strangeness of impossibility condition” was introduced by Nolan [Reference Nolan43, p. 550]. If we think of selection functions as selecting the “closest” or “most similar” antecedent-worlds, then the condition effectively says that impossible worlds are always “far out” in that they’re less similar than any possible world.Footnote
10
French et al. [Reference French, Girard and Ripley20] present an impossible worlds semantics where this corresponds to the axiom
, which has an analogue in Table B11. Again, this has an equivalent formulation in terms of convention-shifting: counterconventional readings only arise when the antecedent in question is impossible (on its actual interpretation).
Finally, operational rigidity, in effect, states counterlogical vacuism, i.e., the view that all counterlogicals (counterfactuals with logically impossible antecedents) are vacuously true. Some nonvacuists have held that even if counterpossibles are generally nonvacuous, counterlogicals are a special exception, while others have argued there’s no relevant difference between counterlogicals and other counterpossibles.Footnote 11 In hyperlogic, this turns on whether counterfactuals are allowed to shift the interpretation of the connectives. Thus, those who maintain that counterlogicals are a special exception can hold that counterfactuals are only allowed to shift the interpretation of nonlogical vocabulary.
B3.5 Belief and knowledge
Thus far, we have focused on counterfactual hyperlogic. But the selection semantics (or something like it) is also often employed as a semantics for dyadic belief and knowledge, where
${\textsf {B}^{\phi }}\psi $
says the agent believes that
$\psi $
given
$\phi $
and likewise for
${\textsf {K}^{\phi }}\psi $
.Footnote
12
It is standard to define the monadic belief operator as
(here, we can define
). Letting
, we then have the following semantics for monadic belief:
$$ \begin{align*} \mathcal{M},w,c & \,\Vdash\, {\textsf{B}}\phi &\hspace{-36pt} {{\mathbin{\Leftrightarrow}}\quad} & \text{for all } \left\langle v,d \right\rangle \in R(w,c): \mathcal{M},v,d \,\Vdash\, \phi. \end{align*} $$
Thus, we can import the results in Section B3.2 to give a logic of belief and knowledge within hyperlogic. As in Section B3.4, one could consider imposing any of the usual restrictions on R to obtain stronger logics.
One application of doxastic/epistemic hyperlogic is to the problem of logical omniscience. It is well known that on the standard intensional semantics, belief is closed under classical entailment: if
$\phi \vDash \psi $
, then
${\textsf {B}}\phi \vDash {\textsf {B}}\psi $
.Footnote
13
Attempts to avoid this result generally often appeal to limitations or defects in cognitive states, e.g., lack of computational resources, awareness, or informational access. However, another (less discussed) way logical omniscience can fail is via different views on logic. If Inej believes intuitionistic logic is correct, then her beliefs will not generally be closed under classical consequence even if she is a perfect reasoner.
Doxastic hyperlogic is well-equipped to handle such cases. While it does not require that beliefs are closed under classical consequence, it does validate a more modest closure principle:
. Restricting to hypermodels where
is factive and noncontingent (Table B3), we can simplify this principle:
. In other words, beliefs are closed under whatever logic the agent adopts (if there is one, assuming it’s reasonable). We obtain the “classical” picture only when we assume
${\textsf {B}}{cl}$
holds.Footnote
14
Of course, doxastic hyperlogic is not a complete solution to the problem of logical omniscience. For one, it still assumes agents are perfect reasoners within their own logic, and is thus not a good model of logical error. Moreover, beliefs are still assumed to be closed under universal consequence: if
, then
. The moral, rather, is that there are several different problems of logical omniscience that likely need to be addressed with different tools. Appeals to computation, awareness, fragmentation, etc. are better equipped for modeling logical error, whereas doxastic hyperlogic is better equipped for modeling ideal yet nonclassical agents.
B4 Conclusion
This concludes the two-part series exploring the logic of hyperlogic. In Part A of this series, we axiomatized a minimal logic of hyperlogic. In Part B, we extended these results to stronger logics over a restricted class of models as well as to languages with hyperintensional operators. In this final section, I wish to sketch a few possible directions for future research that would be worth pursuing.
First, it is an open question how best to extend hyperlogic with first-order quantifiers. We could have hyperconventions specify a domain of individuals, but this might bring technical complications with tracking the denotations of variables across shifts in hyperconvention. Another option would be to have hypermodels directly specify a single domain across all hyperconventions. This might be more manageable, though it builds in substantive metaontological assumptions. Second, there are many questions remaining for the model theory of hyperlogic, especially concerning “finite” hypermodels. For example, it is easy to see that any satisfiable
${\mathcal {L}^{\textsf {QH}}}$
-formula is satisfiable in a convention-finite model (i.e., one where
$D_{\mathbb {C}}$
is finite): just reduce the hypermodel to the denotations of the free terms in the formula. Yet, there are satisfiable (quantified)
${\mathcal {L}^{\textsf {QH}}}$
-formulas that not satisfiable in a hyperconvention-finite model (i.e., one where
$D_{\mathbb {H}}$
is finite). What about any satisfiable
${\mathcal {L}^{\textsf {H}}}$
-formula, though? Does
${\textbf {H}}$
satisfy the finite model property?
Third, we made a number of choice points regarding the initial setup of the hyperconvention semantics that could be revised. One is that we required the “classical” hyperconventions to all interpret
and
as universal modals. It would be natural to weaken this requirement so that
and
only obey weaker normal modal logics. Another choice point concerned whether to treat iterated “according to” operators as redundant. I suspect there is more than one way to naturally generalize the semantics for “according to” so that iteration matters.
Finally, the hyperconvention semantics only concerns claims about logics for the propositional modal language. It does not have a way of capturing metalogical claims concerning alternative logics for hyperlogic—specifically, for the propositional quantifiers, hybrid operators, or counterfactuals (except insofar as they also concern alternative logics for the connectives). While Kocurek [Reference Kocurek33, Section 6] sketches a possible extension to such a language, it is unclear what the resulting logic of this proposed solution is or whether there might be more elegant solutions waiting to be explored.
B5 Appendix: Proofs of relative completeness
In this appendix, we give the proofs of various completeness theorems relative to restricted classes of models (Theorems B2.1, B2.3, B2.6, and B3.16). First, we state a helpful lemma, which follows straightforwardly from Corollary A3.27 and Definition A3.30:
Lemma B5.17 (Canonical operations).
Let
$\left|{\kappa }\right| {}_1, \left|{\lambda }\right| {}_1 \in \Gamma $
. Where
$\phi _i \in [X_i]_{\kappa }$
and
$\psi _i \in [Y_i]_{\lambda }$
,
iff
.
In each case, the proof of soundness is straightforward and left to the reader. Completeness requires showing the canonical model is in the relevant class. In some cases, we must revise the canonical model construction and/or the Lindenbaum construction.
B5.1 Theorem B2.1
The proofs of completeness for
$\textsf {F}$
,
${\textsf {U}_{\textsf {q}}}$
,
$\textsf {At}$
, and
$\textsf {B}$
are immediate since the canonical hypermodel (Definition A3.32) is full (and thus, quantification uniform, atomic, and boolean).
${\textsf {U}_{\textsf {o}}}$
We need to make a slight revision to the definition of a canonical hyperconvention. In particular, we need to revise the third clause to say that
$c_{\kappa }$
interprets the connectives classically if the following is in
$\Gamma $
for some
${\iota _{1},\dots ,\iota _{n}}$
and
${\lambda _{1},\dots ,\lambda _{n}}$
:
$$ \begin{align*} (\kappa \in \iota_1) \mathbin{\wedge} (\lambda_1 \in \iota_1) \mathbin{\wedge} (\lambda_1 \in \iota_2) \mathbin{\wedge} (\lambda_2 \in \iota_2) \mathbin{\wedge} \cdots \mathbin{\wedge} (\lambda_n \in \iota_n) \mathbin{\wedge} (\lambda_n \in {cl}) \end{align*} $$
So unlike Definition A3.30,
$c_{\kappa }$
can be classical even if
${\mathop {@}\nolimits }_{\kappa }{cl} \notin \Gamma $
, so long as it satisfies this “zigzag” condition. Now, Lemma A3.33 needs to be restated as the following:
Claim. If
$(\kappa \in \iota ), (\lambda \in \iota ) \in \Gamma $
and
$c_{\kappa }$
is classical, then
$c_{\lambda }$
is classical.
Proof. Suppose first that
$\kappa $
satisfies the zigzag condition. Then the zigzag can be extended to
$\lambda $
via
$\iota $
, and thus
$c_{\lambda }$
is classical. Suppose instead that
$\kappa $
does not satisfy the zigzag condition. Then
. Suppose
$[X]_{\kappa } = \emptyset $
. Then
. But since
$c_{\kappa }$
is classical,
. So
$X = W$
, even though
$(p \mathbin {+} {\mathop {\sim }\nolimits } p) \in [W]_{\kappa }$
,
. Hence, there is no X where
$[X]_{\kappa } = \emptyset $
. This can only happen if
$W_{\Gamma }$
is finite. List the members of
$W_{\Gamma }$
as
${\Delta _{1},\dots ,\Delta _{n}}$
. Since these are all distinct maximal consistent sets, there must be some
${\delta _{1},\dots ,\delta _{n}}$
such that
$\delta _i \in \Delta _j$
iff
$i = j$
. Each
$X \mathrel {\subseteq } W_{\Gamma }$
is then definable by a disjunction of these
$\delta _i$
s (if
$X = \emptyset $
, then it’s definable by
$\bot $
). Now, let
$l_{\Gamma } \in \Gamma $
and for each
$X \mathrel {\subseteq } W_{\Gamma }$
, let
$\delta _X = {\mathop {@}\nolimits }_{l_{\Gamma }}\delta _{i_1} \mathbin {+} \cdots \mathbin {+} {\mathop {@}\nolimits }_{l_{\Gamma }}\delta _{i_k}$
, where the disjunction of
${\delta _{i_{1}},\dots ,\delta _{i_{k}}}$
defines X. By Red and
$\text {Dist}_{{\mathop {@}\nolimits }}$
,
$((\delta _X)^{\kappa } = (\delta _X)^{\lambda }) \in \Gamma $
for every X. By
$\text {Uni}_{\textsf {o}}$
,
. Thus,
by Lemma B5.17, and so
$c_{\lambda }$
is classical.
Using this claim in place of Lemma A3.33 in the inductive step for the connectives in Lemma A3.34, the completeness proof goes through as before. We just need to check that
${D_{\mathbb {C}}}_{\Gamma }$
is operationally uniform. Let
$c_{\kappa },c_{\lambda } \in C_{\iota }$
. By the above claim,
$c_{\kappa }$
is classical iff
$c_{\lambda }$
is classical. If both are classical, then we’re done. So suppose otherwise. I just prove the
-case for illustration. If
$[X]_{\kappa } = \emptyset $
, then
$[X]_{\lambda } = \emptyset $
(otherwise, if
$\phi \in [X]_{\lambda }$
, then
${\mathop {@}\nolimits }_{\lambda }\phi \in [X]_{\kappa }$
). If
$[X]_{\kappa } = [X]_{\lambda } = \emptyset $
, then
. So suppose
$\phi \in [X]_{\kappa }$
and
$\psi \in [X]_{\lambda }$
. Then
${\mathop {@}\nolimits }_{\kappa }\phi \in \Delta $
iff
${\mathop {@}\nolimits }_{\lambda }\psi \in \Delta $
. By Corollary A3.27 and Bool,
${\mathop {@}\nolimits }_{\kappa }\phi = {\mathop {@}\nolimits }_{\lambda }\psi \in \Gamma $
. By
$\text {Uni}_{\textsf {o}}$
,
. Hence,
by Lemma B5.17.
$\textsf {Si}$
Completeness is straightforward. To establish that
${\textbf {H}} + \text {Sing} = {\textbf {H}} + \text {Self-Dual}^+$
, we just need to show that Sing is coderivable with
$\text {Self-Dual}_{{\mathop {@}\nolimits }}$
in
${\textbf {H}}$
.
$\text {Self-Dual}_{{\mathop {@}\nolimits }}$
trivially follows from
$\text {Dist}_{{\mathop {@}\nolimits }}$
and Sing. Here’s the other direction:
$$ \begin{align*} \iota, i & \,\Vdash\, {\mathop{\sim}\nolimits} {\mathop{@}\nolimits}_{\iota}{\mathop{\sim}\nolimits} i & \qquad & \text{Elim}_{{\mathop{@}\nolimits}}, \\ \iota, i & \,\Vdash\, {\mathop{@}\nolimits}_{\iota} i & \qquad & \text{Self-Dual}_{{\mathop{@}\nolimits}}, \\ \iota & \,\Vdash\, {\mathop{\downarrow}\nolimits} i.{\mathop{@}\nolimits}_{\iota} i & \qquad & \text{Gen}_{{\mathop{\downarrow}\nolimits}}, \text{Vac}_{{\mathop{\downarrow}\nolimits}}, \\ & \,\Vdash\, \left|{\iota}\right| {}_1 & \qquad & \text{Gen}_{{\mathop{@}\nolimits}}, \text{Ref}, \text{ def. of } \left|{\iota}\right| {}_1. \end{align*} $$
$\textsf {An}\textsf {F}$
,
$\textsf {An}{\textsf {U}_{\textsf {q}}}$
We revise the Lindenbaum construction, specifically the definition of
$\Gamma _{k+1}$
. Let
$\kappa \nsim \lambda $
abbreviate
, where
$\vec {p^+}$
and
$\vec {q^+}$
are unused at this point in the construction. Then we revise the definition of
$\Gamma _{k+1}$
so that
$\Gamma _{k+1} = \Gamma _k' \cup {\left\{ {\kappa \nsim \lambda }\right\}}$
if
$\phi _k \in \Gamma _k'$
where
. Suppose
$\Gamma _{k+1}$
is inconsistent in this case. Then for some
${\gamma _{1},\dots ,\gamma _{n}} \in \Gamma _k'$
, we have
for each
. By RAn,
$\widehat {\gamma }, \kappa \in \iota , \lambda \notin \iota , \left|{\lambda }\right| {}_1 \,\Vdash\, (\kappa \in \iota ) \mathbin {\equiv } (\lambda \in \iota )$
. Hence,
$\Gamma _k'$
is inconsistent,
.
It suffices to show that the canonical hypermodel is analytic. Suppose
$c_{\kappa } \in C_{\iota }$
and
$c_{\kappa } \approx c_{\lambda }$
. So
$(\kappa \in \iota ) \in \Gamma $
and
$\left|{\lambda }\right| {}_1 \in \Gamma $
. Moreover, if
$(\lambda \in \iota ) \notin \Gamma $
, then by the revised Lindenbaum construction,
$\kappa \nsim \lambda \in \Gamma $
, contrary to
$c_{\kappa } \approx c_{\lambda }$
,
. Hence,
$(\lambda \in \iota ) \in \Gamma $
.
$\textsf {S}_{\textsf {5}}$
We revise Definition A3.30 so that
is always defined classically. The only revision needed to the proofs is to verify the connective case in the truth lemma (Lemma A3.34). This follows from the fact that
is
$({\textbf {H}} +\text {Bool}_{\,\Vdash\, })$
-derivable (by
$\text {Bool}_{\,\Vdash\, }$
,
$\text {Gen}_{{\mathop {@}\nolimits }}$
, and
$\text {Dist}_{{\mathop {@}\nolimits }}$
(for
$\,\Vdash\, $
)).
B5.2 Theorem B2.3
For some of these proofs, we use the lemma below, which follows from Definition A4.42 and
${\exists }$
-witnessing.
Lemma B5.18 (Canonical proposition space).
Let
$\left|{\kappa }\right| {}_1,\left|{\lambda }\right| {}_1 \in \Gamma $
. Then
$\pi _{c_{\kappa }} \mathrel {\subseteq } \pi _{c_{\lambda }}$
iff
$(\pi _{\kappa } \mathrel {\subseteq } \pi _{\lambda }) \in \Gamma $
, and
$\left|{\pi _{c_{\kappa }}}\right| = 1$
iff
$\left|{\pi _{\kappa }}\right| {}_1 \in \Gamma $
.
We omit the proofs for
$\textsf {B}$
,
${\textsf {U}_{\textsf {q}}}$
,
${\textsf {U}_{\textsf {o}}}$
,
$\textsf {Si}$
, and
$\textsf {S}_{\textsf {5}}$
, which are routine.
$\textsf {At}$
Let
$c_{\kappa } \in {D_{\mathbb {H}}}_{\Gamma }$
and
$\Delta \in W_{\Gamma }$
. First, observe that
. For by Atom, Bool, and
$\text {Dist}_{{\mathop {@}\nolimits }}$
,
. Since
$l_{\Delta } \in \Delta $
, we have
. By
${\exists }$
-witnessing,
for some
$p^+$
. By
$\text {Elim}_{\forall }$
, ClEx, and
${\exists }$
-witnessing,
. By S5,
.
So suppose
${\mathop {@}\nolimits }_{\kappa } p^+ \in \Delta '$
and suppose
$\phi \in \Delta $
. Thus,
. By Corollary A3.27,
$\phi \in \Delta '$
. Hence,
$\Delta ' = \Delta $
. So
$p^+ \in [{\left\{ \Delta \right\}}]_{\kappa }$
, i.e.,
${\left\{ \Delta \right\}} \in \pi _{c_{\kappa }}$
.
$\textsf {An}$
Since members of
might not be allowed to denote singletons (since conventions must be closed under
$\approx $
), the Henkin construction needs to be revised so that
$\textsf {INom}^+$
is replaced with
(though we don’t allow formulas in
${\mathcal {L}^{\textsf {QH}+}}$
to bind members of
$\textsf {IVar}^+$
). We also need to make the following amendments to the definition of the canonical model:
The proof of the truth lemma (Lemma A3.34) remains intact (the only difference is the
${\mathop {@}\nolimits }_{\iota }$
-case where
$\iota $
is i and
$\left|{i}\right| {}_1 \in \Gamma $
; in that case,
$\Delta ,c_{\kappa } \,\Vdash\, {\mathop {@}\nolimits }_i\phi $
iff
$\Delta ,c_i \,\Vdash\, \phi $
iff
${\mathop {@}\nolimits }_i\phi \in \Delta $
iff
${\mathop {@}\nolimits }_{\kappa }{\mathop {@}\nolimits }_i\phi \in \Delta $
.) Trivially,
${\left\{ c_{\kappa }{\left|\ c \approx c_i \right.} \right\}}$
is analytic. So we need to show that
$C_{\iota }$
is analytic if
, and that
$C_l$
is analytic if
$\left|{l}\right| {}_1 \in \Gamma $
.
First, suppose
. Let
$c_{\kappa } \in C_{\iota }$
and let
$c_{\lambda } \neq c_{\kappa }$
be such that
$c_{\kappa } \approx c_{\lambda }$
. Since
$(\kappa \in \iota ), \left|{\lambda }\right| {}_1 \in \Gamma $
, it suffices to show that
$(\kappa \approx \lambda ) \in \Gamma $
; for then by An,
$(\lambda \in \iota ) \in \Gamma $
, and so
$c_{\lambda } \in C_{\iota }$
. By Lemma B5.18,
$(\pi _{\kappa } = \pi _{\lambda }) \in \Gamma $
since
$\pi _{c_{\kappa }} = \pi _{c_{\lambda }}$
. Moreover, if
$((p^+)^{\kappa } = (q^+)^{\kappa }) \in \Gamma $
, then
${\left\{ \Delta \in W_{\Gamma } {\left|\ {\mathop {@}\nolimits }_{\kappa } p^+ \in \Delta \right.} \right\}} = {\left\{ \Delta \in W_{\Gamma } {\left|\ {\mathop {@}\nolimits }_{\lambda } q^+ \in \Delta \right.} \right\}}$
. Since
$c_{\kappa } \approx c_{\lambda }$
, that means
. So by Lemma B5.17,
. Therefore,
. Since
$\Gamma $
witnesses
${\exists }$
s,
, i.e.,
. Similarly,
. Hence,
$(\kappa \approx \lambda ) \in \Gamma $
.
Next, suppose
$\left|{l}\right| {}_1 \in \Gamma $
. Let
$c_{\kappa } \approx c_l$
. By the reasoning above,
$(\kappa \approx l) \in \Gamma $
. Since
$\left|{\kappa }\right| {}_1,\left|{l}\right| {}_1 \in \Gamma $
, it follows by
$\text {Many}_{\textsf {INom}}$
that
$(\kappa = l) \in \Gamma $
. Hence, by Lemma A4.44,
$c_{\kappa } = c_l$
, and thus
$c_{\kappa } \in C_l$
.
B5.3 Theorem B2.6
We omit the proofs for
$\textsf {Cr}$
and
$\textsf {Di}$
, which are routine.
${\textsf {Cl}_{\Phi }}$
We revise the Henkin construction. Let
be new propositional variables, and let
${\mathcal {L}^{\textsf{QH}+\circ}}$
be the result of expanding
${\mathcal {L}^{\textsf {QH}+}}$
with
$\textsf {Prop}^{\circ }$
(again, not including formulas with quantifiers binding these variables). Enumerate the members of
$\Phi $
as
. Let
$\Delta $
be the set of all formulas of the form
$p_k^{\circ } =_l \chi _k$
, where
$l \in \textsf {INom}^+$
and
$\chi _k \in \Phi $
. The Henkin construction is the same except we redefine
$\Gamma _k'$
so that
$\Gamma _k' = \Gamma _k \cup {\left\{ \phi _k \right\}}$
if
$\Gamma _k, \Delta , \phi _k \nvdash _{{\textbf {QH}} \cup \text {Ex}_{\Phi }} \bot $
(and
$= \Gamma _k$
otherwise). Clearly, if
$\Gamma _k \cup \Delta $
is
$({\textbf {QH}} \cup \text {Ex}_{\Phi })$
-consistent, then so is
$\Gamma _k' \cup \Delta $
. The proof that
$\Gamma _{k+1} \cup \Delta $
is consistent if
$\Gamma _k' \cup \Delta $
is consistent is essentially the same. Thus, we just need to show that
$\Gamma _1 \cup \Delta $
is
$({\textbf {QH}} \cup \text {Ex}_{\Phi })$
-consistent. Suppose it’s not. Since
$l_{\Gamma }$
occurs nowhere in
$\Delta $
, we can eliminate
$l_{\Gamma }$
by the same reasoning as in Lemma A3.24. Thus, there are some
${\alpha _{1},\dots ,\alpha _{k}}$
that are instances of Ex
$_{\Phi }$
, some
${\delta _{1},\dots ,\delta _{n}} \in \Delta $
where
$\delta _i$
is of the form
$q_i^{\circ } =_{k_i} \psi _i$
for some
$\psi _i \in \Phi $
and
$k_i \in \textsf {INom}^+$
, and some
${\gamma _{1},\dots ,\gamma _{m}} \in \Gamma $
such that
(throughout, I’ll use
$\vdash $
for provability in
${\textbf {QH}}$
and
$\vdash _{\text {Ex}_{\Phi }}$
for provability in
${\textbf {QH}} \cup \text {Ex}_{\Phi }$
). Now, it may be that
$q_i^{\circ } = q_j^{\circ }$
for some i and j. So let
$q_i^{\circ } \approx \psi _i$
be the conjunction all
$\delta _j$
s such that
$q_j^{\circ } = q_i^{\circ }$
—that is,
$q_i^{\circ } \approx \psi _i$
has the form
$(q_i^{\circ } =_{k_{i_1}} \psi _i) \mathbin {\wedge } \cdots \mathbin {\wedge } (q_i^{\circ } =_{k_{i_j}} \psi _i)$
. (Given how
$\Delta $
is defined and how
$\Phi $
is enumerated, it is never the case that
$q_i^{\circ } = q_j^{\circ }$
but
$\psi _i \neq \psi _j$
; so this definition is well-defined.) Thus,
. By Lemma A4.38,
where
${r_{1},\dots ,r_{n}} \in \textsf {Prop}$
are fresh. By
$\text {RK}_{\exists }$
,
$\text {Vac}_{\exists }$
, and
$\text {VDist}_{\exists }$
,
. So by
$\text {Ex}_{\Phi }$
,
,
.
The rest of the proof of the Henkin lemma (Lemma A4.40) goes through as before. And since
$\Gamma _k \cup \Delta $
is
$({\textbf {QH}} \cup \text {Ex}_{\Phi })$
-consistent for each k,
$\Gamma ^+ \cup \Delta $
is
$({\textbf {QH}}\ \cup \text {Ex}_{\Phi })$
-consistent, which by maximality means
$\Delta \mathrel {\subseteq } \Gamma ^+$
. Hence,
$\Gamma ^+$
has the following property: for each
$\chi \in \Phi $
, there is a
$p^{\circ }$
such that for all
$\iota \in \textsf {ITerm}^+$
,
$(p^{\circ } =_{\iota } \chi ) \in \Gamma ^+$
.
To complete the proof, we revise the definition of
$\pi _{c_{\kappa }}$
(when
${\mathop {@}\nolimits }_{\kappa }{cl} \notin \Gamma $
) and
${D_{\mathbb {P}}}_{\Gamma }$
:
$$ \begin{align*} \pi_{c_{\kappa}} & = {\left\{ X {\left|\ {\exists}{p \in \textsf{Prop}^+ \cup \textsf{Prop}^{\circ}\colon} p \in {[X]_{\kappa}} \right.} \right\}}, \\ {D_{\mathbb{P}}}_{\Gamma} & = {\left\{ P \in \mathbb{P}_{{D_{\mathbb{H}}}_{\Gamma}} {\left|\ {\exists}{p \in \textsf{Prop}^* \cup \textsf{Prop}^{\circ}}{\forall}{c_{\kappa} \in {D_{\mathbb{H}}}_{\Gamma}\colon} p \in {[P(c_{\kappa})]}_{\kappa} \right.} \right\}}.\\[-24pt] \end{align*} $$
The rest of the proof goes through as before. So by Lemma A4.47,
, so
$P_{p_i^{\circ }}$
can be our witness for
$\chi _i \in \Phi $
. Hence,
${D_{\mathbb {P}}}_{\Gamma }$
is closed under
$\Phi $
.
${\textsf {Cl}_{\Phi }^+}$
The proof is roughly the same as
${\textsf {Cl}_{\Phi }}$
, but we need to make some revisions. Let
$\Phi ' = {\left\{ \chi [q_1'/q_1,\dots ,q_n'/q_n] {\left|\ {q^{\prime }_{1},\dots ,q^{\prime }_{n}} \in \textsf {Prop}^* \cup \textsf {Prop}^{\circ } \right.} \right\}}$
. Enumerate the members of
$\Phi '$
as
in such a way that
$p_k'$
never occurs in
${\chi _{1},\dots ,\chi _{k}}$
. Proceed with the Henkin construction in the same manner as before, replacing
$\Phi $
throughout with
$\Phi '$
. To establish that
$\Gamma _1 \cup \Delta $
is
$({\textbf {QH}} + \text {Ex}_{\Phi })$
-consistent, we use the same reasoning, except the last step needs further justification, since
$\psi _i$
may not be in
$\Phi $
but rather of the form
$\psi _i = \chi [q_1'/q_1,\dots ,q_n'/q_n]$
for some
$\chi \in \Phi $
. However, since Ex
$_{\Phi }$
is now an axiom, that means if
$\chi \in \Phi $
, then
$\vdash {\exists }{r}(r \approx \chi )$
. So by
$\text {Gen}_{\forall }$
,
$\vdash {\forall _{q_1}\cdots \forall _{q_n}}{\exists }{r}(r \approx \chi )$
. Hence, by
$\text {Elim}_{\forall }$
,
$\vdash {\exists }{r}(r \approx \psi _i)$
.
Making the same revisions as before, the rest of the completeness proof goes through. So we just need to show now that
${D_{\mathbb {P}}}_{\Gamma }$
is strongly closed under
$\Phi $
. Let
$\mathcal {M} = \left\langle W_{\Gamma },{D_{\mathbb {C}}}_{\Gamma },{D_{\mathbb {P}}}_{\Gamma },V \right\rangle $
. Then
$V(q_i) = P_{q_i'}$
for some
$q_i'$
. Hence, by Lemma A4.36,
. By how
$\Gamma $
was constructed, there is a
$p^{\circ }$
such that for all
$\iota $
,
$p^{\circ } =_{\iota } \phi [q_1'/q_1,\dots ,q_n'/q_n] \in \Gamma $
. By Lemma A4.47,
. Hence,
${D_{\mathbb {P}}}_{\Gamma }$
is strongly closed under
$\Phi $
.
To establish that
${\textbf {QH}} + \text {Ex}_{\Phi } = {\textbf {QH}} + \text {Elim}_{\forall \Phi }$
, it suffices to show that
$\text {Ex}_{\Phi }$
is coderivable with
$\text {Elim}_{\forall \Phi }$
. Deriving
$\text {Ex}_{\Phi }$
from
$\text {Elim}_{\forall \Phi }$
is straightforward by S5 and
$\text {Dual}_{\forall }$
. For the other direction, it follows by induction (or completeness over the class of all models) that if
$\chi $
is free for p, and
${\iota _{1},\dots ,\iota _{n}}$
are the free interpretation terms in
$\phi $
, then
$p = \chi , p =_{\iota _1} \chi , \dots , p =_{\iota _n} \chi \,\Vdash\, \phi = \phi [\chi /p]$
. Hence:
$$ \begin{align*} {\forall}{p}\phi, p = \chi, \dots, p =_{\iota_n} \chi & \,\Vdash\, \phi[\chi/p] & \qquad & \text{above}, \text{Elim}_{\forall}, \\ {\mathop{\downarrow}\nolimits} i.{\forall}{p}\phi, {\mathop{\downarrow}\nolimits} i.(p =_i \chi {\mathbin{\&}} \cdots {\mathbin{\&}} p =_{\iota_n} \chi) & \,\Vdash\, {\mathop{\downarrow}\nolimits} i.\phi[\chi/p] & \qquad & \text{Gen}_{{\mathop{\downarrow}\nolimits}}, \text{Idle}_{{\mathop{\downarrow}\nolimits}}, \text{Dist}_{{\mathop{\downarrow}\nolimits}}, \\ {\forall}{p}\phi, {\mathop{\downarrow}\nolimits} i.(p =_i \chi {\mathbin{\&}} \cdots {\mathbin{\&}} p =_{\iota_n} \chi) & \,\Vdash\, \phi[\chi/p] & \qquad & \text{Vac}_{{\mathop{\downarrow}\nolimits}}, \\ {\forall}{p}\phi,{\exists}{p}{\mathop{\downarrow}\nolimits} i.(p =_i \chi {\mathbin{\&}} \cdots {\mathbin{\&}} p =_{\iota_n} \chi) & \,\Vdash\, \phi[\chi/p] & \qquad & \text{RK}_{\exists}, \text{VDist}_{\exists}, \text{Vac}_{\exists}, \\ {\forall}{p}\phi,{\mathop{\downarrow}\nolimits} i.{\exists}{p}(p =_i \chi {\mathbin{\&}} \cdots {\mathbin{\&}} p =_{\iota_n} \chi) & \,\Vdash\, \phi[\chi/p] & \qquad & \text{BF}_{{\mathop{\downarrow}\nolimits}}, \\ {\forall}{p}\phi & \,\Vdash\, \phi[\chi/p] & \qquad & \text{Ex}_{\Phi}, \text{Gen}_{{\mathop{\downarrow}\nolimits}}. \end{align*} $$
${\textsf {Df}_{\Phi }}$
I will only prove weak completeness here; it’s easy to check that if
$\Phi $
is finite, then strong completeness can be established via the same method. Suppose
$\phi $
is
$({\textbf {QH}} + \text {Gen}_{\forall \Phi })$
-consistent. Enumerate the members of
$\Phi $
as
. Parallel to
, we construct a new sequence of sets
. First,
$\Gamma _1 = {\left\{ \phi \right\}} \cup {\left\{ l_1^+,\left|{l_1^+}\right| {}_1 \right\}}$
and
$\Delta _1 = \emptyset $
. Next, define
$\Gamma _{k+1}$
as in the proof of Lemma A4.40. Finally, define
$\Delta _{k+1}$
as follows:
$$ \begin{align*} \Delta_{k+1} & = \begin{cases} \Delta_k \cup \{q^+ =_{l^+} \chi \mid (q^+ =_{l_1^+} \chi) \in \Delta_k \}, & \text{if (*) holds,} \\ \Delta_k \cup {\left\{ p^+ =_{l^+} \chi {\left|\ l^+ \in \textsf{INom}^+ \text{ occurs in } \Gamma_{k+1} \right.} \right\}}, & \text{if (**) holds,} \\ \Delta_k, & \text{otherwise,} \end{cases} \end{align*} $$
-
(*)
and
$l^+$
is the witness introduced in
$\Gamma _{k+1}$
, -
(**)
$\phi _k = {\exists }{p}\psi $
, where
$p^+$
is the witness introduced in
$\Gamma _{k+1}$
and
$\chi $
is the first of
$\Phi $
such that
$\Gamma _{k+1},\Delta _k, {\left\{ p^+ =_{l^+} \chi {\left|\ l^+ \in \textsf {INom}^+ \text { occurs in } \Gamma _{k+1} \right.} \right\}} \nvdash \bot $
.
Finally,
$\Gamma ^+ = {\mathop {\bigcup _{k \geq 1}}} \Gamma _k$
. We first show that for each k,
$\Gamma _k \cup \Delta _k$
is
$({\textbf {QH}} + \text {Gen}_{\forall \Phi })$
-consistent. Clearly this holds for
$k = 1$
. And clearly if
$\Gamma _k \cup \Delta _k$
is
$({\textbf {QH}} + \text {Gen}_{\forall \Phi })$
-consistent, then so is
$\Gamma _k' \cup \Delta _k$
and
$\Gamma _{k+1} \cup \Delta _k$
. So we just need to show that if
$\Gamma _{k+1} \cup \Delta _k$
is
$({\textbf {QH}} + \text {Gen}_{\forall \Phi })$
-consistent, then so is
$\Gamma _{k+1} \cup \Delta _{k+1}$
. If
$\phi _k = {\exists }{p}\psi $
, then
$\Gamma _{k+1} \cup \Delta _{k+1}$
is
$({\textbf {QH}} + \text {Gen}_{\forall \Phi })$
-consistent by construction of
$\Delta _{k+1}$
, assuming it’s defined. Here’s the proof that it is always defined, i.e., there always is such a
$\chi $
in this case. Suppose otherwise. That means for all
$\chi \in \Phi $
, where
,
, and
${l^+_{1},\dots ,l^+_{n}}$
are the nominals occurring in some formula of
$\Gamma _{k+1}$
,
. By
$\text {Gen}_{\forall \Phi }$
,
. By
$\text {Elim}_{\forall }$
,
. Hence, by S5,
$\gamma ,\delta \vdash \bot $
,
.
Now suppose
and suppose for reductio that
$\Gamma _{k+1} \cup \Delta _{k+1}$
is
$({\textbf {QH}} + \text {Gen}_{\forall \Phi })$
-inconsistent. Then for some formula of the form
$q_i^+ =_{l^+} \chi _i$
, we have
. Repeating the reasoning in Lemma A3.24,
$\gamma ,\delta ,{\mathop {\downarrow }\nolimits }\, i.(q_1^+ =_i \chi _1),\dots ,{\mathop {\downarrow }\nolimits }\, i.(q_n^+ =_i \chi _n) \vdash {\mathop {@}\nolimits }_{\iota }\psi $
. Since
$l_1^+,\left|{l_1^+}\right| {}_1 \in \Gamma _k$
:
$\gamma ,\delta ,(q_1^+ =_{l_1^+} \chi _1),\dots ,(q_n^+ =_{l_1^+} \chi _n) \vdash {\mathop {@}\nolimits }_{\iota }\psi $
. But
$(q_1^+ =_{l_1^+} \chi _1),\dots ,(q_n^+ =_{l_1^+} \chi _n) \in \Delta _k$
. Thus,
$\gamma ,\delta \vdash {\mathop {@}\nolimits }_{\iota }\psi $
. So
$\Gamma _k' \cup \Delta _k$
is already
$({\textbf {QH}} + \text {Gen}_{\forall \Phi })$
-inconsistent,
. Hence,
$\Gamma _{k+1} \cup \Delta _{k+1}$
is
$({\textbf {QH}} + \text {Gen}_{\forall \Phi })$
-consistent. Therefore,
$\Gamma ^+ \cup {\mathop {\bigcup _{k}}} \Delta _k$
is
$({\textbf {QH}} + \text {Gen}_{\forall \Phi })$
-consistent, and so by maximality,
$\Delta _k \mathrel {\subseteq } \Gamma ^+$
for all k.
By construction, for each
$p^+ \in \textsf {Prop}^+$
, there is a
$\chi \in \Phi $
such that
$(p^+ =_{l^+} \chi ) \in \Gamma ^+$
for all
$l^+ \in \textsf {INom}^+$
. From here, the completeness proof proceeds as before. To complete the proof, we show
${D_{\mathbb {P}}}_{\Gamma }$
is definable in
$\Phi $
. Where
$P = P_{p^+} \in {D_{\mathbb {P}}}_{\Gamma }$
, let
$\chi \in \Phi $
be such that
$p^+ =_{l^+} \chi \in \Gamma ^+$
for all
$l^+ \in \textsf {INom}^+$
. Then by Lemma A4.47,
. Hence,
.
${\textsf {Cl}_{\Phi }}{\textsf {Df}_{\Phi }}$
We use the same construction as in
${\textsf {Df}_{\Phi }}$
. We need to show (i) that we can dispense with the
$\text {Gen}_{\forall \Phi }$
rule in the proof above, and (ii)
${D_{\mathbb {P}}}_{\Gamma }$
is closed under
$\Phi $
. (To establish that
${\textbf {QH}} \ \cup \text {Hom}_{\Phi } \ \cup \text {Ex}_{\Phi } = {\textbf {QH}} \ \cup \text {Hom}_{\Phi } \cup \text {Ex}_{\Phi }^-$
, simply observe that
${\left\{ {\forall }{p}((p = \chi ) \mathbin {\supset } (p =_{\iota _i} \chi )) {\left|\ i \leq n \right.} \right\}}, {\mathop {\textsf {E}}}\chi \,\Vdash\, {\exists }{p}{\mathbin {\& _{i=1}^{n}}} (p =_{\iota _i} \chi )$
.)
For (i), note that
$\text {Gen}_{\forall \Phi }$
was only used to establish that in the Henkin construction, if
$\phi _k = {\exists }{p}\psi $
is added to
$\Gamma _k'$
and
$p^+$
is the witness introduced into
$\Gamma _{k+1}$
, then there is a
$\chi \in \Phi $
such that
$\Gamma _{k+1},\Delta _k, {\left\{ p^+ =_{l^+} \chi {\left|\ l^+ \in \textsf {INom}^+ \text { occurs in } \Gamma _{k+1} \right.} \right\}} \nvdash \bot $
. For all
$\chi \in \Phi $
, there is an
$l^+$
such that
$(p^+ =_{l^+} \chi ) \notin \Gamma _{k+1}$
. Then for all
$\chi \in \Phi $
, there exist some
${\gamma _{1},\dots ,\gamma _{n}} \in \Gamma _{k+1}$
, some
${\delta _{1},\dots ,\delta _{m}} \in \Delta _k$
, some
${\alpha _{1},\dots ,\alpha _{k}} \in \text {Hom}_{\Phi }$
, and some
${\beta _{1},\dots ,\beta _{j}} \in \text {Ex}_{\Phi }^-$
such that
$\widehat {\alpha },\widehat {\beta },\widehat {\gamma },\widehat {\delta }, p^+ =_{l_1^+} \chi ,\dots , p^+ =_{l_n^+} \chi \vdash \bot $
. Since
${\forall }{p}(p = \chi \mathbin {\supset } p =_{l_i^+} \chi ) \in \text {Hom}_{\Phi }$
for each
$l_i^+$
, we can assume these are included in
$\widehat {\alpha }$
. Hence, by
$\text {Elim}_{\forall }$
,
$\widehat {\alpha },\widehat {\beta },\widehat {\gamma },\widehat {\delta }, p^+ = \chi \vdash \bot $
. So by Lemma A4.38, where r is fresh,
$\widehat {\alpha },\widehat {\beta },\widehat {\gamma },\widehat {\delta }, r = \chi \vdash \bot $
. By
$\text {Intro}_{\exists }$
,
$\text {VDist}_{\exists }$
and
$\text {Vac}_{\exists }$
,
$\widehat {\alpha },\widehat {\beta },\widehat {\gamma },\widehat {\delta },{\mathop {\textsf {E}}}\chi \vdash \bot $
. So,
$\widehat {\gamma },\widehat {\delta } \vdash _{\text {Hom}_{\Phi } \cup \text {Ex}_{\Phi }^-} \bot $
,
. (Notice we did not rely on
$\Gamma _{k+1}$
being finite, so the same strategy establishes strong completeness.) For (ii), let
$\chi \in \Phi $
. By
$\text {Ex}_{\Phi }$
,
$(p^+ = \chi ) \in \Gamma $
for some
$p^+ \in \textsf {Prop}^+$
. By
$\text {Hom}_{\Phi }$
and
$\text {Elim}_{\forall }$
,
$(p^+ =_{\iota } \chi ) \in \Gamma $
for all
$\iota \in \textsf {ITerm}^+$
. Hence,
for all
$c_{\kappa }$
, i.e.,
.
${\textsf {Cl}_{\Phi }^+}{\textsf {Df}_{\Phi }}$
Similar to
${\textsf {Cl}_{\Phi }}{\textsf {Df}_{\Phi }}$
, except using
$\text {Ex}_{\Phi }$
as an axiom to show that
${D_{\mathbb {P}}}_{\Gamma }$
is strongly closed (as in the proof of completeness over
${\textsf {Cl}_{\Phi }^+}$
).
$\textsf {Cb}$
Let
$X_1 \in \pi _{c_{\kappa _1}},\dots ,X_n \in \pi _{c_{\kappa _n}}$
where
${c_{\kappa _{1}},\dots ,c_{\kappa _{n}}}$
are distinct. Let
${p^+_{1},\dots ,p^+_{n}} \in \textsf {Prop}^+$
be such that
$p_i^+ \in [X_i]_{\kappa _i}$
. Since
${c_{\kappa _{1}},\dots ,c_{\kappa _{n}}}$
are distinct, by the same reasoning as in
$\textsf {Di}$
,
$(\kappa _i \neq \kappa _j) \in \Gamma $
for
$i \neq j$
. By Split and Bool,
${\exists }{p}{\mathbin {\bigwedge _{i=1}^{n}}} (p =_{\kappa _i} p_i^+) \in \Gamma $
. By witnessing
${\exists }$
s, there is a
$p^+ \in \textsf {Prop}^+$
such that
$p^+ =_{\kappa _i} p_i^+ \in \Gamma $
for
$1 \leq i \leq n$
. Hence,
$P_{p^+}(c_{\kappa _i}) = X_i$
.
To establish that
${\textbf {QH}} + \text {PII}^+ + \text {Split} = {\textbf {QH}} + \text {PII}_1^+ + \text {Split}$
, we just need to show that
$\text {PII}^+$
is
$({\textbf {QH}} + \text {PII}_1^+ + \text {Split})$
-derivable. By
$\text {PII}_1^+$
, it suffices to show that
${\forall }{p}(p^{\iota } = p^{\kappa }), \left|{\iota }\right| {}_1, \left|{\kappa }\right| {}_1, (\iota \neq \kappa ) \,\Vdash\, \left|{\pi _{\iota }}\right| {}_1$
is derivable using Split:
$$ \begin{align*} {\forall}{p}(p^{\iota} = p^{\kappa}) & \,\Vdash\, p^{\iota} = p^{\kappa} & \qquad & \text{Elim}_{\forall}, \\ {\forall}{p}(p^{\iota} = p^{\kappa}) & \,\Vdash\, r^{\iota} = r^{\kappa} & \qquad & \text{Elim}_{\forall}, \\ {\forall}{p}(p^{\iota} = p^{\kappa}), p^{\iota} = q^{\iota}, p^{\kappa} = r^{\kappa} & \,\Vdash\, q^{\iota} = r^{\iota} & \qquad & \text{S5}, \\ {\forall}{p}(p^{\iota} = p^{\kappa}), {\exists}{p}(p^{\iota} = q^{\iota} {\mathbin{\&}} p^{\kappa} = r^{\kappa}) & \,\Vdash\, q^{\iota} = r^{\iota} & \qquad & \text{RK}_{\exists}, \text{VDist}_{\exists}, \text{Vac}_{\exists}, \\ {\forall}{p}(p^{\iota} = p^{\kappa}), \left|{\iota}\right| {}_1, \left|{\kappa}\right| {}_1, (\iota \neq \kappa) & \,\Vdash\, q^{\iota} = r^{\iota} & \qquad & \text{Split}, \\ {\forall}{p}(p^{\iota} = p^{\kappa}), \left|{\iota}\right| {}_1, \left|{\kappa}\right| {}_1, (\iota \neq \kappa) & \,\Vdash\, \left|{\pi_{\iota}}\right| {}_1 & \qquad & \text{RK}_{\forall}, \text{Vac}_{\forall}, \text{Dist}_{{\mathop{@}\nolimits}}. \end{align*} $$
$\textsf {Cp}\textsf {Si}$
We must revise the definition of the canonical model so that
${D_{\mathbb {P}}}_{\Gamma } = \mathbb {P}_{{D_{\mathbb {H}}}_{\Gamma }}$
. The only thing that needs to be redone is the
${\forall }$
inductive step of Lemma A4.47. The argument that if
$\mathcal {M}_{\Gamma },\Delta ,c_{\kappa } \,\Vdash\, {\forall }{p}\phi $
, then
${\mathop {@}\nolimits }_{\kappa }{\forall }{p}\phi \in \Delta $
is the same. For the other direction, we first need the following intermediate result:
Claim. For all
$\phi $
and all
$\lambda \in \textsf {ITerm}^+$
such that
$\left|{\lambda }\right| {}_1 \in \Gamma $
, there is a formula
$\phi ^{\uparrow }$
such that where
$\vec {p}$
are the free propositional variables in
$\phi $
:
-
i.
$\phi ^{\uparrow }$
contains no interpretation binders
${\mathop {\downarrow }\nolimits } i$
. -
ii. If
$\iota $
and
$\kappa $
occur in
$\phi ^{\uparrow }$
and
$\iota $
isn’t
$\kappa $
, then
$(\iota \neq \kappa ) \in \Gamma $
. -
iii. For all
$\Delta \in W_{\Gamma }$
,
$\mathcal {M}_{\Gamma },\Delta ,c_{\lambda } \,\Vdash\, {\forall }{\vec {p}}(\phi = \phi ^{\uparrow })$
. -
iv. For all
$\Delta \in W_{\Gamma }$
,
${\forall }{\vec {p}}(\phi = \phi ^{\uparrow }) \in \Delta $
.
Proof. First, since
$\Gamma $
witnesses
s, for each free
$\iota $
, there is an
$l_{\iota }^+ \in \textsf {INom}^+$
such that
$(l_{\iota }^+ \in \iota ) \in \Gamma $
. By Sing and
$\text {Intro}_{=}$
,
$(l_{\iota }^+ = \iota ) \in \Gamma $
. Let
$l_{\iota }^+$
be the first in
$\textsf {INom}^+$
with this property. By SubId, we can replace each
$\iota $
that occurs free in
$\phi $
with
$l_{\iota }^+$
. Call the result
$\phi '$
. Now proceed as follows:
-
a. If
${\mathop {\downarrow }\nolimits } i$
does not occur in the scope of any
${\mathop {@}\nolimits }_{\kappa }$
or any
${\mathop {\downarrow }\nolimits } j$
, replace each free i in its scope with
$l_{\lambda }^+$
. Then delete this
${\mathop {\downarrow }\nolimits } i$
. -
b. Repeat (a) on the result until there are no more
${\mathop {\downarrow }\nolimits } i$
s that do not occur in the scope of any
${\mathop {@}\nolimits }_{\kappa }$
or any
${\mathop {\downarrow }\nolimits } j$
. -
c. For each subformula of the form
${\mathop {@}\nolimits }_{l^+}\psi $
that does not occur in the scope of any
${\mathop {@}\nolimits }_{\kappa }$
operator, repeat (a) and (b) on
$\psi $
except with
$l^+$
in place of
$l_{\lambda }^+$
. Continue until there are no more binders
${\mathop {\downarrow }\nolimits } i$
left. Call the result
$\phi ^{\uparrow }$
.
It is now easy to verify that
$\phi ^{\uparrow }$
satisfies (i)–(iv).
So suppose
$\mathcal {M}_{\Gamma },\Delta ,c_{\kappa } \nVdash {\forall }{p}\phi $
. By the claim above,
$\mathcal {M}_{\Gamma },\Delta ,c_{\kappa } \nVdash {\forall }{p}\phi ^{\uparrow }$
. Thus, there is a
$P \in \mathbb {P}_{{D_{\mathbb {H}}}_{\Gamma }}$
such that
$(\mathcal {M}_{\Gamma })^p_P,\Delta ,c_{\kappa } \nVdash \phi ^{\uparrow }$
. Let
${l^+_{1},\dots ,l^+_{n}}$
be the interpretation terms in
$\phi ^{\uparrow }$
. By ClEx and
$\,\Vdash\, {\mathop {\textsf {E}}} p$
(by
$\text {Intro}_{\exists }$
), for each
$l_i^+$
, there is a
$p_i^+$
such that
$p_i^+ \in [P(c_{l_i^+})]_{l_i^+}$
, i.e.,
$\Delta ' \in P(c_{l_i^+})$
iff
${\mathop {@}\nolimits }_i p_i^+ \in \Delta '$
. Since
$(l_i^+ \neq l_j^+) \in \Gamma $
when
$i \neq j$
, it follows by Split that
. By witnessing
${\exists }$
s, there is a
$p^+$
such that
. Thus, for each i and
$\Delta '$
:
${\mathop {@}\nolimits }_{l_i^+} p^+ \in \Delta '$
iff
${\mathop {@}\nolimits }_{l_i^+} p_i^+ \in \Delta '$
. Hence,
$\Delta ' \in P(c_{l_i^+})$
iff
${\mathop {@}\nolimits }_i p^+ \in \Delta '$
. By Lemma A4.36,
$(\mathcal {M}_{\Gamma })^p_P,\Delta ,c_{\kappa } \,\Vdash\, \phi ^{\uparrow }$
iff
$\mathcal {M}_{\Gamma },\Delta ,c_{\kappa } \,\Vdash\, \phi ^{\uparrow }[p^+/p]$
. Hence,
$\mathcal {M}_{\Gamma },\Delta ,c_{\kappa } \nVdash \phi ^{\uparrow }[p^+/p]$
. By IH,
${\mathop {@}\nolimits }_{\kappa }\phi ^{\uparrow }[p^+/p] \notin \Delta $
. By
$\text {Elim}_{\forall }$
,
${\forall }{p}{\mathop {@}\nolimits }_{\kappa }\phi ^{\uparrow } \notin \Delta $
. By the claim above,
${\forall }{p}{\mathop {@}\nolimits }_{\kappa }\phi \notin \Delta $
. By
$\text {CBF}_{{\mathop {@}\nolimits }}$
,
${\mathop {@}\nolimits }_{\kappa }{\forall }{p}\phi \notin \Delta $
.
B5.4 Theorem B3.16
In each case, it suffices to show that
$f_{\Gamma }$
satisfies the corresponding constraint given the axiom. Moreover,
$f_{\Gamma }$
is already defined to satisfy the relevant constraint when
$[A] = \emptyset $
. So assume throughout that
$[A] \neq \emptyset $
.
$\textsf {Suc}$
Suppose
$\left\langle \Delta ',c_{\lambda } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
. Let
$\phi \in [A]$
. By Lemma B3.12 and Definition B3.13, if
where
, then
${\mathop {@}\nolimits }_{\lambda }\psi \in \Delta '$
. By
and
,
. So
${\mathop {@}\nolimits }_{\lambda }\phi \in \Delta '$
. By Definition B3.11,
$\left\langle \Delta ',c_{\lambda } \right\rangle \in A$
.
$\textsf {W}$
Let
$\left\langle \Delta ,c_{\kappa } \right\rangle \in A$
and
$\phi \in [A]$
(so
${\mathop {@}\nolimits }_{\kappa }\phi \in \Delta $
). Suppose
. By
and Ded,
. By
$\text {Gen}_{{\mathop {@}\nolimits }}$
and Ref,
. Hence,
${\mathop {@}\nolimits }_{\kappa }\psi \in \Delta $
. By Definition B3.13,
$\left\langle \Delta ,c_{\kappa } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
.
$\textsf {C}$
Suppose
$\left\langle \Delta ,c_{\kappa } \right\rangle \in A$
. Let
$\phi \in [A]$
. Thus,
${\mathop {@}\nolimits }_{\kappa }\phi \in \Delta $
. Reasoning as above, we have
. So if
, then
${\mathop {@}\nolimits }_{\kappa }\psi \in \Delta $
, meaning
$\left\langle \Delta ,c_{\kappa } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
. Moreover, let
$\left\langle \Delta ',c_{\lambda } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
. So for all
, if
, then
${\mathop {@}\nolimits }_{\lambda }\psi \in \Delta '$
. Now, by Cen,
. By
$\text {Gen}_{{\mathop {\downarrow }\nolimits }}$
and
$\text {Vac}_{{\mathop {\downarrow }\nolimits }}$
,
. By
$\text {Gen}_{{\mathop {@}\nolimits }}$
and
$\text {DA}_{{\mathop {@}\nolimits }}$
,
. Since
$\left|{\kappa }\right| {}_1 \in \Delta $
, that means
. By
,
. So
${\mathop {@}\nolimits }_{\lambda } \kappa \in \Delta '$
. By Rigid and Corollary A3.27,
$\left|{\kappa }\right| {}_1 \in \Delta '$
. By
$\text {Intro}_{=}$
,
$(\kappa = \lambda ) \in \Delta '$
. By Lemma A3.31,
$c_{\kappa } = c_{\lambda }$
. We now show
$\Delta ' = \Delta $
. We’ll just show
$\Delta \mathrel {\subseteq } \Delta '$
to illustrate. Let
$\psi \in \Delta $
. By
$\text {Intro}_{{\mathop {@}\nolimits }}$
,
$\text {Elim}_{{\mathop {@}\nolimits }}$
, and Red,
${\mathop {@}\nolimits }_{\kappa }{\mathop {@}\nolimits }_{l_{\Delta }}\psi \in \Delta $
. Thus,
. So
${\mathop {@}\nolimits }_{\kappa }{\mathop {@}\nolimits }_{l_{\Delta }}\psi \in \Delta '$
since
$\left\langle \Delta ',c_{\kappa } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
. So by Red, Rigid,
$\text {Intro}_{{\mathop {@}\nolimits }}$
, and
$\text {Elim}_{{\mathop {@}\nolimits }}$
,
$\psi \in \Delta '$
.
$\textsf {Stal}$
Suppose
$\left\langle \Delta _1,c_{\lambda } \right\rangle , \left\langle \Delta _2,c_{\mu } \right\rangle \in f(A,\Delta ,c_{\kappa })$
. Let
$\phi \in [A]$
. Thus, for all
:
Suppose
. Thus,
by
, and so,
${\mathop {@}\nolimits }_{\lambda }{\mathop {\sim }\nolimits }\lambda \in \Delta _1$
,
. Hence,
. By CEM,
. By
,
, and so,
${\mathop {@}\nolimits }_{\mu }\lambda \in \Delta _2$
. By Rigid and
$\text {Intro}_{=}$
,
$(\lambda = \mu ) \in \Gamma $
since
$\left|{\lambda }\right| {}_1, \left|{\mu }\right| {}_1 \in \Gamma $
. By Lemma A3.31 then,
$c_{\lambda } = c_{\mu }$
. Further,
$(\lambda = \mu ) \in \Delta _1 \cap \Delta \cap \Delta _2$
by Rigid.
We now show that
$\Delta _1 \mathrel {\subseteq } \Delta _2$
(the proof that
$\Delta _2 \mathrel {\subseteq } \Delta _1$
is symmetric). Suppose
$\psi \in \Delta _1$
. By
$\text {Intro}_{{\mathop {@}\nolimits }}$
and
$\text {Elim}_{{\mathop {@}\nolimits }}$
,
. By Red,
. Since
$\left\langle \Delta _1,c_{\lambda } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
,
. By SubId, since
$(\lambda = \mu ) \in \Delta $
,
. By CEM,
. Since
, we have
by
. Since
$\left\langle \Delta _2,c_{\mu } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
, we have
by Bool. So by
$\text {Dist}_{{\mathop {@}\nolimits }}$
,
$\text {Intro}_{{\mathop {@}\nolimits }}$
, and
$\text {Elim}_{{\mathop {@}\nolimits }}$
,
$\psi \in \Delta _2$
.
$\textsf {Vac}$
Let
$A(c_{\kappa }) = \emptyset $
. Suppose for reductio
$f_{\Gamma }(A,\Delta ,c_{\kappa }) \neq \emptyset $
. Let
$\left\langle \Delta ',c_{\lambda } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
and let
$\phi \in [A]$
. By Corollary A3.27 and
$\text {Dist}_{{\mathop {@}\nolimits }}$
,
. By Vac and
$\text {Gen}_{{\mathop {@}\nolimits }}$
,
. By Definition B3.13,
${\mathop {@}\nolimits }_{\lambda }\bot \in \Delta '$
,
.
$\textsf {NC}$
,
$\textsf {NEC}$
,
$\textsf {SIC}$
We just do
$\textsf {NC}$
, since
$\textsf {NEC}$
and
$\textsf {SIC}$
are similar. It’s left as an exercise to the reader to show that the two versions of the relevant axiom are coderivable. Let
$\phi \in [A]$
and let
$\left\langle \Delta ',c_{\lambda } \right\rangle \in f_{\Gamma }(A,\Delta ,c_{\kappa })$
. So for all
$\psi $
, if
, then
${\mathop {@}\nolimits }_{\lambda }\psi \in \Delta '$
. By Rigid and Bool,
. By NC and
,
. Hence,
${\mathop {@}\nolimits }_{\lambda }\kappa \in \Delta '$
. By Rigid,
$\text {Intro}_{{\mathop {@}\nolimits }}$
, and
$\text {Elim}_{{\mathop {@}\nolimits }}$
,
$(\kappa = \lambda ) \in \Delta '$
. Thus,
$c_{\lambda } = c_{\kappa }$
.
$\textsf {R}_{\textsf {o}}$
We revise the definition of a canonical hyperconvention as we did for Theorem B2.2 so that
$\pi _{c_{\kappa }} = {\left\{ X {\left|\ {[X]_{\kappa } \neq \emptyset } \right.} \right\}}$
. Given this, let
$\left\langle \Delta ',c_{\lambda } \right\rangle \in f(A,\Delta ,c_{\kappa })$
and let
$\alpha \in [A]$
. Thus, for all
$\chi $
, if
, then
${\mathop {@}\nolimits }_{\lambda }\chi \in \Delta '$
. We will just show the
-case, i.e., that
, since the others are similar.
First, observe that
$\pi _{c_{\kappa }} = \pi _{c_{\lambda }}$
, since, e.g., if
$\phi \in [X]_{\lambda }$
, then
${\mathop {@}\nolimits }_{\lambda }\phi \in [X]_{\kappa }$
by Red. So let
$X \in \pi _{c_{\lambda }}$
and let
$\phi \in [X]_{\lambda }$
. By
$\text {R}_{\text {o}}$
and
$\text {Gen}_{{\mathop {@}\nolimits }}$
,
. By
$\text {DA}_{{\mathop {@}\nolimits }}$
,
. Hence,
. By
$\text {DA}_{{\mathop {@}\nolimits }}$
,
. By
$\text {Dist}_{{\mathop {@}\nolimits }}$
and Red,
. By Lemma B5.17 (since
${\mathop {@}\nolimits }_{\lambda }\phi \in [X]_{\kappa }$
),
.
Acknowledgement
I am grateful to two anonymous referees for their helpful feedback.
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