THE LOGIC OF HYPERLOGIC. PART A: FOUNDATIONS

Abstract Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical interpretation of the connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators.

Recently, Kocurek [30] has developed a hyperintensional system that offers answers to both questions called hyperlogic.To regiment metalogical claims, hyperlogic utilizes a combination of several different devices: a multigrade entailment operator ▷; propositional quantifiers ∀p and ∃p [18] to regiment laws of logic; and terms and operators borrowed from hybrid logic [1, 11], such as nominals (l 1 , l 2 , l 3 , ... ) to regiment claims about which logic is correct, and operators @ to regiment "according to" claims.To illustrate, here is how we could regiment (1)- (6) in hyperlogic. 1 (1) Classic logic is correct.

cl
(2) The law of excluded middle holds.
∃p ∀q((p ∧ ¬p) ▷ q) (4) According to intuitionistic logic, the law of excluded middle doesn't hold.@ il ¬ ∀p(▷(p ∨ ¬p)) (5) In strong Kleene logic, nothing is valid.@ k3 ¬ ∃p(▷ p) (6) Everything that is intuitionistically valid is classically valid.∀p(@ il (▷ p) → @ cl (▷ p)) To assign compositional semantic values to metalogical claims, hyperlogic introduces a shiftable convention parameter-a "hyperconvention"-into points of evaluation.This parameter determines the interpretation of the logical connectives (as well as ▷). 2 The semantic value of a formula is a set of world-hyperconvention pairs.While metalogical claims may express a trivial possible worlds proposition relative to a hyperconvention, they can have nontrivial semantic values that hyperintensional environments can exploit.
My aim in this paper is not to defend hyperlogic as a semantic theory for metalogical claims.Rather, my aim is to address the following question: given that hyperlogic is designed to reason about other logics, what, if anything, can we say about logical consequence within hyperlogic itself ?In other words, what is the logic of hyperlogic? 1 As Kocurek [30, fn.9] points out, there are multiple regimentations of (4) depending on how we interpret the "not" in "does not hold" (classically or intuitionistically).Fortunately, hyperlogic can regiment both readings (see Definition A2.5 for expressing classical negation in the scope of "according to" operators). 2 One could interpret this parameter as determining the Kaplanian character of the connectives [27].Alternatively, one could interpret it as the content of the connectives determined by their character given a particular conversational context.Officially, hyperlogic is neutral on what determines the interpretation of the connectives on a particular occasion of use.In particular, it is compatible with contextualist, relativist, expressivist, and even objectivist views about the connectives.What hyperlogic requires is simply the ability of hyperintensional operators to shift the hyperconvention parameter.To keep things simple, we will set aside issues around context-sensitivity so that we don't have to add the context parameter to the index.
At first, one might suspect the logic of hyperlogic is entirely uninteresting.How much could be valid in a framework with the expressive resources to talk about other logics?As it turns out, however, this initial impression is mistaken.To show this, I present a sound and complete proof system for hyperlogic.It involves two separate axiomatic systems that are recursively defined in terms of one another, each representing different kinds of consequences: one represents ordinary classical consequence (truth preservation relative to a classical interpretation of the connectives) while the other represents "universal" consequence (truth preservation relative to any interpretation of the connectives).This dual proof system contains rules for moving back and forth between these axiomatic systems.The result is an elegant, tractable, and nontrivial logic for hyperlogic.
There are several reasons independent of the semantics of metalogical claims to be interested in the logic of hyperlogic.For one, the main semantic innovation of hyperlogic, viz., to add a shiftable convention parameter for interpreting the logical connectives, is behind several "conventionalist" approaches to hyperintensionality in the literature, which model hyperintensional environments as convention-shifting operators (cf.[32, 36, 39, 40, 59]). 3This contrasts with approaches that introduce incomplete and/or inconsistent states (impossible worlds, truthmakers, situations, etc.) into standard possible worlds frameworks. 4Even if one thinks these conventionalist approaches are ultimately mistaken, one might still wonder how many hyperintensional phenomena can be explained in terms of it.Hyperlogic presents an encouraging answer for conventionalists about hyperintensionality.
In addition, hyperlogic provides a simple logic for "according to".For example, the following sounds fine to say: (10) Pluto is not a planet, but according to the folk definition of "planet," Pluto is a planet.
The phrase "according to the folk definition of 'planet"' seems to, in some sense, shift the interpretation of "planet" mid-sentence so that the second "Pluto is a planet" is interpreted via the folk definition of "planet" [33, p. 8].If so, it's natural to ask how this operator works and what logical principles govern it.As we'll see, hyperlogic offers a simple yet attractive answer to these questions.Finally, hyperlogic may provide insight into the problem of logical omniscience.Stalnaker [53, 54, 56] famously analyzed the content of an agent's mental state as a set of possible worlds, viz., those at which what the agent (actually) believes is true.While this view has its merits, it infamously predicts that agents' beliefs are closed under logical entailment.There is a vast disagreement in the literature over how to address this problem. 5Hyperlogic potentially provides a novel and attractive solution by analyzing mental content in terms of sets of world-hyperconvention pairs instead of sets of worlds.This new picture can validate certain modest closure principles without requiring beliefs be closed under classical consequence.It can thus avoid at least certain forms of logical omniscience while preserving the main features that initially motivated the Stalnakerian picture.
This paper is the first in a two-part series on the logic of hyperlogic.Part A focuses on a very general system for hyperlogic, which places no restrictions on the class of models.The logic of this system is fairly weak, and therefore constitutes a kind of minimal hyperlogic upon which stronger hyperlogics can be based.Part B explores stronger logics of this sort, as well as the logic of hyperlogic enriched further with hyperintensional operators.
Here is an outline of what is to come in this part.In Section A2, I give a brief overview of the syntax and semantics of hyperlogic.In Section A3, I present, and prove the completeness of, a proof system for the fragment of hyperlogic without propositional quantifiers.In Section A4, I extend these results to the language of hyperlogic with propositional quantifiers.I conclude in Section A5. §A2.Hyperlogic: syntax and semantics.We start by reviewing the syntax, semantics, and consequence relation(s) of hyperlogic as presented in [30].In Section A2.1, we introduce the language of hyperlogic.In Section A2.2, we clarify the notion of a hyperconvention and use it to state a semantics for hyperlogic.In Section A2.3, we identify two notions of consequences in hyperlogic and explain their relation.
The full language of hyperlogic extends L 0 in three ways.I will introduce each extension separately so that fragments of the full language can be studied independently.
First, hyperlogic adds an "entailment" operator ▷, where (φ 1 , ... , φ n ▷ ) represents the claim that φ 1 , ... , φ n (in that order) entail .This operator is left-multigrade, meaning it can take any finite number (possibly zero) of arguments on the left.We could make ▷ right-multigrade as well (e.g., to represent multiple-conclusion logics) without substantively affecting the results presented in what follows.But for notational ease, we assume a fixed arity of 1 on the right.
Second, hyperlogic adds propositional quantifiers ∀p and ∃p that bind into sentence position [18].When combined with the entailment operator, we can regiment laws of logic as universal entailment claims.For instance, we can regiment the law of double negation elimination as ∀p(¬¬p ▷ p).
Finally, hyperlogic adds operators similar to those found in hybrid logic.Hybrid logic extends propositional modal logic with (i) state terms 1 , 2 , 3 , ... (including state variables and state "nominals", i.e., constants), which double as terms denoting worlds and as atomic formulas that hold at their denotation; (ii) for each state term , an "according to" operator @ , which resets the world of evaluation to the world denoted by ; and (iii) for each state variable s, a binding operator ↓ s, which reassigns the denotation of s to the current world of evaluation [1, 11].Informally, we can read s as "s is actual," @ s as "according to s,...," and ↓ s as "where s is the current world,...." Instead of hybrid operators for worlds, hyperlogic introduces hybrid operators for interpretations of the base language, including the logical connectives.Thus, it introduces an infinite stock of interpretation variables IVar = {i 1 , i 2 , i 3 , ...} and interpretation nominals INom = {l 1 , l 2 , l 3 , ...}.We single out a designated nominal cl to stand for a classical (S5) interpretation of the connectives.An interpretation term is a member of ITerm := IVar ∪ INom.We use , κ, , etc. as metavariables over interpretation terms.Definition A2.4 (Hybrid language L H ).
Informally, we can read as " is correct," @ as "according to ,...," and ↓ i as "where i is the current interpretation,...." The binder ↓ allows us to define the following as abbreviations for the connectives under their classical interpretation.Definition A2.5 (Rigidly classical connectives).Where i is not in φ or , These "connectives" are interpreted classically even at nonclassical interpretations and even within the scope of "according to" operators.We will make extensive use of these rigidly classical connectives throughout, as it is in large part thanks to them that hyperlogic has a nontrivial logic.These three extensions can be freely combined: L QE is the quantified entailment language, L QH is the quantified hybrid language, and L HE is the hybrid entailment language.For convenience, we define the full language of hyperlogic as H := L QHE .Definition A2.6 (Substitution).We adopt the usual notions of "free" and "bound" variables (where i is bound by ↓ i and p is bound by ∀p and ∃p).We say 2 is free for 1 in φ if no free occurrence of 1 in φ is in the scope of ↓ 2 .In that case, we write φ[ 2 / 1 ] for the result of replacing every free occurrence of 1 in φ with 2 .Similarly, is free for p in φ if no free occurrence of p in φ is in the scope of ∀q or ∃q where q occurs free in , or a binder ↓ i where i occurs free in .If is free for p in φ, we write φ[ /p] for the result of replacing every free occurrence of p in φ with .Simultaneous substitution φ[ 1 /p 1 , ... , n /p n ] is defined likewise.

A2.2. Semantics.
The main semantic innovation behind hyperlogic is to relativize truth to a "hyperconvention," i.e., a maximally specific interpretation of the base language.More precisely, a hyperconvention specifies a space of (coarse-grained) possible worlds propositions, assigns each propositional variable to a possible worlds proposition in the space, and assigns each connective in L 0 (or L E ) to an operation on propositions, i.e., a function from some proposition(s) to a proposition.Definition A2.7 (Hyperconvention).Let W = ∅ and ⊆ ℘ W .A -hyperconvention for L 0 (over W) is a function c with domain {¬, ◻, , ∧, ∨, →} ∪ Prop such that: i. c(p) ∈ for all p ∈ Prop.
We call the proposition space for c.We write c for the that c is defined over6 and △ c (with infix notation) for c(△).A hyperconvention for L (over W) is a -hyperconvention for L over W for some ⊆ ℘ W .We let H (L) W be the set of all hyperconventions for L over W. Throughout, I use "hyperconvention" to mean "hyperconvention for L E " if the language under discussion contains ▷, and "hyperconvention for L 0 " otherwise.
At the outset, we place no constraints on which operations can be assigned to a connective by a hyperconvention.The task of exploring how things change when we impose such constraints is taken up in Part B.
Just as a proposition is typically modeled as a set of worlds, a "convention" is modeled as a set of hyperconventions.

Definition A2.8 (Convention
).An convention is a nonempty set of hyperconventions.We let C (L) W be the set of conventions ( for L) over W. We can think of a logic as a special type of convention that only concerns the interpretation of the connectives (and ▷, if present).Here, we need not take a stand on what features of entailment are essential to logic: the hyperconvention semantics can accommodate a range of views on this matter.
To define our models, we need to introduce the notions of an index and an index proposition.In the hyperconvention semantics, truth is evaluated relative to an index, i.e., a world-hyperconvention pair.Definition A2.9 (Index).Given a set H of hyperconventions over W, an index over H is a pair w, c where w ∈ W and c ∈ H .We let I H = W × H be the set of indices over H.
As a formal semantics, hyperlogic is neutral on how to understand what an index represents.Kocurek [30, p. 13682] interprets indices as worlds "under descriptions."On this picture, logic is not a feature of the world but a feature of our representation of it (cf.[32]).We could, however, instead hold that logic is genuinely part of the world.In that case, an index w, c represents a (perhaps logically impossible) world, where the w component determines all the nonsemantic facts while c determines the semantic facts.
Given this notion of an index, there are now three relevant notions of "proposition" to consider.First, there's the standard, coarse-grained notion of a proposition as a set of worlds, which is what hyperconventions assign to propositional variables, and operations on which they assign to connectives.Call this the intension of a formula relative to a hyperconvention.Second, there's a fine-grained notion of a proposition as a set of indices, which is the compositional semantic value of a formula.(Thus, semantic values are more fine-grained than intensions.)Finally, there is an intermediate notion of a "visible" index proposition, i.e., a function from hyperconventions to intensions in their proposition space.More precisely:

Definition A2.10 (Index proposition). Given a set of hyperconventions H over W, an index proposition over H is a set of indices
We let P H be the set of visible index propositions over H.We use X, Y, Z, ... for worlds propositions and P, Q, R, ... for visible index propositions.
Since a propositional variable's intension relative to a hyperconvention is always a world proposition from that hyperconvention's proposition space (i.e., c(p) ∈ c ), the (fine-grained) semantic value of a propositional variable is always a visible index proposition.Propositional quantifiers, therefore, range over visible index propositions (see [30, p. 13677]).
We are now ready to define our models and semantics more explicitly.A model in this semantics specifies (i) a set of states (or "worlds"), (ii) a set of conventions for interpretation terms to denote, (iii) a set of (visible) propositions for quantifiers to range over, and (iv) a valuation function.

Definition A2.11 (Hyperframes and hypermodels).
A hyperframe is a triple of the form F = W, D C , D P , where: is a convention domain; we define D H := D C to be the hyperconvention domain (in other words, D H is the set of hyperconventions that appear somewhere in D C ); ii. for all c ∈ D H and all X ∈ c , there is a P ∈ D P such that P(c) = X .
A valuation for F is a mapping V such that: Following [30], we only impose two minimal constraints on proposition domains at the outset (Part B will explore others).These minimal constraints effectively rule out distinct yet indiscernible hyperconventions (i.e., they ensure the soundness of PII in https://doi.org/10.1017/S1755020322000193Published online by Cambridge University Press Table A6 in Section A4.1).It would be interesting to see how the logic of hyperlogic changes if we drop those constraints.But I have yet to find a completeness proof that does without them, so I leave that task aside. 7finition A2.12 (Semantics).Where x is a (propositional or interpretation) variable and v is a possible value for that variable, let V x v be the valuation like V except that formulas, we write M, w, c , Γ to mean that M, w, c , for all ∈ Γ.When M is clear from context, we drop mention of it.
Note, the right-hand side of the semantic clause for △ should be read as requiring Also, following [30], we interpret iterated @-operators as redundant.Thus, @ 1 @ 2 φ is equivalent to @ 2 φ.This is how such operators standardly work in hybrid logic and it simplifies the semantics and logic greatly.This equivalence could be questioned, though, and it would be worth investigating a more general semantics where it doesn't hold.Doing so is beyond the scope of this paper, however.
Which logics can be represented as a hyperconvention on this semantics?Kocurek [30] proves a result that provides an answer to this question. 8Say a logic L over L 0 is a set of pairs of the form φ 1 , ... , φ n , where φ 1 , ... , φ n , ∈ L 0 (we allow the first element to be the empty tuple ).Intuitively, if φ 1 , ... , φ n , ∈ L, then φ 1 , ... , φ n , in that order, entail in L. Say a logic L is representable by a hyperconvention c over W if for any hyperframe This means that so long as the state space of a hypermodel is sufficiently large, one can represent any finitary logic over that state space.This includes many of the familiar logics in the literature (intuitionistic logic, Kleene's logic, paraconsistent logics, quantum logic, etc.).9

A2.3. Consequence.
There are two notions of consequences we can define in the hyperconvention semantics.First, there is a classical notion of consequence, i.e., truthpreservation relative to a classical interpretation of the connectives.Second, there is a "universal" notion of consequence, i.e., consequence no matter how we interpret the connectives. 10o define these notions more precisely, we need to define the notion of a "classical" interpretation of the connectives. 11finition A2.14 (Classical hyperconvention).Given a hyperframe iii. for all X 1 , ... , X n , Y ∈ c : A convention is classical for F if all of its member are.A (hyper)convention is classical for M if it's classical for the hyperframe M is based on.Finally, Note that classical hyperconventions interpret ◻ and as universal S5 modals.I suspect that the proofs of completeness presented in Sections A3 and A4 can survive if we weaken this requirement so that ◻ and can be interpreted as obeying other normal modal logics, assuming we make corresponding adjustments to the axioms (see footnote 16 for one possible strategy).But I won't take up this question here, as the proofs are already complex enough even assuming ◻ and are universal modals.
It is straightforward to verify the following: Henceforth, I will only consider classical hypermodels: when I say "hypermodel," I always mean "classical hypermodel."Classical/universal validity, equivalence, etc. are defined likewise.
a. Assume l (distinct from cl ) does not occur anywhere in Γ or in φ.Then Γ Proposition A2.17 essentially gives us a method of moving back and forth between classical and universal consequences.This will be the key to developing our proof system of hyperlogic in the next section.§A3.Completeness for the quantifier-free fragments.We now present some soundness and completeness results for the quantifier-free languages L H and L HE .In Section A3.1, we present an axiomatization for consequence in L H .In Section A3.2, we prove this system is sound and complete.In Section A3.3, we extend the axiomatization to L HE .

A3.1. Proof systems for classical and universal reasoning.
The axiomatizations for classical and universal consequences in L H involve a kind of double recursion: they are not defined as separate systems with their own axioms and rules but rather interdefined with rules for moving between the two (cf.[14, 23, 24]; [22, sec.7.5]).
When reasoning within hyperlogic, it is often useful to switch back and forth between classical reasoning and universal reasoning, as these notions of consequences obey different rules.Consider the rule of necessitation: This rule is classically sound since classical hyperconventions interpret ◻ as a normal modal operator.But the rule is not universally sound: while @ cl (p ∨ ¬p) is universally valid, ◻ @ cl (p ∨ ¬p) is not since a hyperconvention could interpret ◻ abnormally.By contrast, consider the corresponding rule for @: If φ is provable, then @ φ is provable.This rule is not classically sound: while p ∨ ¬p is classically valid, @ (p ∨ ¬p) is not since may denote a nonclassical convention.Yet the rule is universally sound: if φ holds on any hyperconvention, then φ holds at every hyperconvention in the convention denoted by , i.e., @ φ holds.
For this reason, we will introduce two interdefined proof systems: ⊢ (for classical provability) and , (for universal provability).We call the collection of these two proof systems H, the minimal hyperlogic in L H . 12 Before giving the axioms and rules (Tables A1 and A2), let me explain some of the notation used to state them.
First, because the deduction theorem is classically sound (φ 1 , ... , → since nothing of that form is universally valid.So the "axioms" for , have formulas on the left. 13econd, we introduce the following abbreviations (where i isn't ): 12 Technically, we should subscript ⊢ and , to the proof system H to distinguish it from later proof systems.But we drop this subscript throughout for readability. 13A related deduction theorem is universally sound: φ 1 , ... , φn For technical reasons, however, it is easier to state the axioms without appeal to this "universal deduction theorem."This universal deduction theorem is derivable in H, so nothing is lost in this choice.
https://doi.org/10.1017/S1755020322000193Published online by Cambridge University Press Table A3.Some useful theorems and derivable rules for ⊢ in H.
H ⊢ (theorems and derivable rules) Some further notational conventions: We write ⊣⊢ and -, for co-provability.Where φ 1 , ... , φ n are formulas, we write φ for ( We use as a metavariable over unary connectives ({¬, ◻, }), ◯ over binary connectives ({∧, ∨, →}), and △ over connectives of any arity.(This will generally be clear from context.)The rigidly classical counterparts of , ◯, and △ are designated as ☀, •, and ▲ respectively. (For example, if = ¬, then ☀ = ∼; if ◯ = ∧, then • = &; etc.) Tables A1 and A2 contain the basic axioms and rules for each proof system.A proof is just a list of statements of the form φ 1 , ... , φ n ⊢ or φ 1 , ... , φ n , , each line of which is either an axiom or follows from previous lines via a rule.By induction on the length of proofs, both proof systems satisfy the following substitution principles.Tables A3 and A4 contain some useful theorems and derivable rules.Their derivations are left as exercises for the reader. 15Throughout, I suppress mention of axioms corresponding to classical propositional reasoning (Struct, MP, and Ded) and of RE, which is implicitly used frequently.I likewise suppress mention of S5 unless the application involves specifically modal reasoning. 16Also, by the U2C rule, all of the axioms for , can be imported into ⊢, so I use the same labels for both versions. 15Solutions can be found here: https://philpapers.org/archive/KOCSTQ.pdf. 16Many appeals to the S5 axiom apply to any normal modal logic.Other appeals to S5 could be dispensed with if we introduced a primitive = operator into the language, rather than defining it in terms of ∎ and ≡.Indeed, we could define ∎ in terms of = as follows: ∎ φ := (φ = (φ + ∼ φ)).This suggests that we could weaken Definition A2.

Theorems S5
, φ where φ is a substitution instance of an S5-theorem whose connectives are replaced with their rigidly classical counterparts where is free for i in φ VE ↓ ↓ i.φ-, ↓ j.φ[j/i] where j is free for i in φ and j is not free in where ′ is the result of replacing some occurrences of φ with φ ′ in A3.2.Soundness and completeness.We now set out to prove that H is sound and complete in L H -that is, ⊢ is sound and complete for classical consequence in L H and , is sound and complete for universal consequence in L H .The proof of soundness is straightforward, though it requires two lemmas (established by induction on formulas), which we'll appeal to later.

Lemma A3.20 (Unused variables).
For any F = W, D C , D P , any w ∈ W , any c ∈ D H , and any M and M ′ based on F, if V and V ′ agree on all free variables in φ (including propositional variables), then M, w, c , φ iff M ′ , w, c , φ.

Lemma A3.21 (Partial substitution). For any
, where φ ′ is the result of replacing some occurrences of 1 with 2 in φ.
We write Γ ⊢ φ to mean " 1 , ... , n ⊢ φ is a theorem of H for some 1 , ... , n ∈ Γ".Similarly for Γ , φ. conventions only need to interpret ◻ and as normal modalities if we extend the language with a primitive = and add the appropriate axioms governing = (e.g., to ensure ∎ validates the T axiom, we would need φ = , φ ≡ ).We would also need to revise some of the axioms, e.g., Bool.Verifying these changes would result in a sound and complete proof system is left for future research.

Theorem A3.22 (Soundness in L H
).For all Γ ⊆ L H and φ ∈ L H : We start by proving completeness for classical consequence via a canonical model construction.We then pair this completeness result with Proposition A2.17 to bootstrap our way to completeness for universal consequence.

Definition A3.23 (Lindenbaum set).
A set Γ L H+ is Lindenbaum if it is a maximal consistent set satisfying the following constraints: Enumerate the L H+ -formulas as φ 1 , φ 2 , φ 3 , ... .We define a sequence of sets Γ 1 , Γ 2 , Γ 3 , ... .First, Lastly, let's say l + or p + is "unused" if it is the first member of INom + or Prop + that has yet to appear in the construction.Then: The proof that Γ + is maximal is standard.The fact that Γ + satisfies (i)-(iii) follows from the construction of Γ + .We just need to show Γ + is consistent.It suffices to show that each Γ k is consistent.By "i is fresh," I mean it occurs nowhere in the relevant formulas.

Definition A3.30 (Canonical hyperconventions).
Where |κ| 1 ∈ Γ, the canonical κhyperconvention c κ over W Γ is defined as follows: iii.If @ κ cl ∈ Γ, then each c κ (△) is defined as in Definition A2.14.Otherwise: For any ∈ ITerm + , define the canonical -convention as Note that C is well-defined by the following lemma: Proof.The left-to-right direction follows since Γ differentiates terms.The right-toleft direction follows from SubId and Corollary A3.27.

A3.3. Axioms for ▷.
It is straightforward to extend H into L HE .The two additional axioms needed are stated in Table A5.The resulting system is called H ▷ .To prove the completeness of H ▷ , we simply amend Definition A3.30 so that c κ (▷)(X 1 , ... , X n , Y ) is defined as: The proof goes through as in Section A3.2, adding the relevant inductive steps for ▷.
Completeness with propositional quantifiers.We now extend these results to languages with propositional quantifiers.In Section A4.1, we state the additional axioms and rules governing quantifiers.In Section A4.2, we revise the proof of completeness from Section A3.2.In Section A4.3, we consider how these results are affected when ▷ is introduced.

A4.1. Axiomatizing quantifiers.
The new axioms and rules for the quantifiers are stated in Table A6.We call the resulting system QH.This axiomatization makes use of the following abbreviations (where p is not free in φ): If V (κ) = {c 1 } and V ( ) = {c 2 }, the truth conditions reduce to the following: Intuitively, Eφ says that φ denotes a world proposition that "exists" according to the current hyperconvention.This formula is not trivially satisfied: if, say, W / ∈ c , then M, w, c E(p ⊃ p) since for no Q ∈ D P does Q(c) = W .
Note that Elim ∀ does not allow substituting any for p (even if p is free for ), since need not denote an existent proposition according to a hyperconvention.For example, if ∅ / ∈ c , then M, w, c , ∀p ◆ p (since P(c) ∈ c for any P ∈ P D H ) even https://doi.org/10.1017/S1755020322000193Published online by Cambridge University Press Table A6.Axioms and rules for provability in L QH .

QH
All the axioms and rules in H, plus: ] where q is free for p in φ Vac ∀ φ , ∀p φ where p does not occur free in φ though M, w, c ◆ (q & ∼ q) (since (q & ∼ q) M,c = ∅ regardless of c).However, since V (p) is always a visible proposition (V (p)(c) ∈ c ), instantiation with other propositional variables is allowed.Note also the PII axiom, which is effectively the principle of the identity of indiscernibles for hyperconventions: if c 1 and c 2 have the same proposition space, and interpret the propositional variables and connectives the same way, then c 1 = c 2 .The soundness of PII is ensured by the two minimal constraints on proposition domains in Definition A2.11. 17he proofs of Lemmas A3.20 and A3.21 remain unchanged.In addition, we have the following (which is needed to prove the soundness of Elim ∀ ): Lemma A4.36 (Propositional relabeling).If q is free for p in φ, then φ[q/p] M = φ M p V (q) .Similarly for simultaneous substitution.
each finite Γ 0 ⊆ Γ.Still, as we'll see in Part B, there are natural restricted classes of hypermodels on which PII is sound as is. 19Over such classes, completeness can be restored.§A5.Conclusion.Hyperlogic is a hyperintensional system that is designed to regiment, and facilitate reasoning about, metalogical claims within the object language.This is achieved by introducing a multigrade entailment operator, propositional quantifiers, and modified hybrid operators into the language.To interpret these claims, we introduced hyperconventions, i.e., maximally specific interpretations, into points of evaluation.While one might suspect that the logic of hyperlogic is uninteresting, as we've seen, this suspicion is incorrect.We presented dual axiomatic systems for both classical and universal consequences in a number of fragments of hyperlogic and proved their soundness and completeness.
The minimal logic of hyperlogic explored in this paper is fairly weak and assumes next to nothing about the possible interpretations of the connectives.It also does not yet include hyperintensional operators like belief operators or counterfactuals.In Part B of this series, we begin to fill these gaps by exploring stronger logics that can be obtained by imposing various restrictions on the class of hypermodels and also by adding hyperintensional operators to the language.

Table A4 .
Some useful theorems and derivable rules for , in H.