General Notation and Terminology
2.1. NOTATION AND TERMINOLOGY FOR FORMULAE AND PROOFS
Definition 2.4 (Projection Functions). Let T1× T2be a type. We define the projection
functions π1: T1× T2→ T1and π2: T1× T2→ T2by
π1hx,yi =de f x (8)
π2hx,yi =de f y (9)
Definition 2.5 (First-Order Terms). Let Term0be a denumerable set of (atomic) first-
order terms (also called individual variables), and for each k ≥ 0 let Funck be a finite
set of function symbols of arity k. We define the set Term of terms inductively:
(1) x ∈ Term if x ∈ Term0;
(2) f (x1,..., xk) ∈ Term if f ∈ Funckand {x1,..., xk} ⊂ Term.
Definition 2.6 (Second-Order Logical Formulae). For each n ≥ 0 let Predn be a de-
numerable set of first-order predicate symbols of arity n. We define the set Form2 of
(second-order) logical formulae inductively:
(1) Px1,..., xk ∈ Form2 (called an atomic formula, or an atom, with arity k) if P ∈
Predk and {x1,..., xk} ⊂ Term for k ≥ 0;
(2) ⊥ ∈ Form2;
(3) A ∧ B ∈ Form2if A, B ∈ Form2;
(4) A ∨ B ∈ Form2if A, B ∈ Form2;
(5) A ⊃ B ∈ Form2if A, B ∈ Form2;
(6) ∀x.A ∈ Form2if x ∈ Term0and A ∈ Form2;
(7) ∃x.A ∈ Form2if x ∈ Term0and A ∈ Form2;
(8) ∀P.A ∈ Form2if P ∈ Predkand A ∈ Form2;
(9) ∃P.A ∈ Form2if P ∈ Predkand A ∈ Form2.
A formula is in the first-order fragment Form1 of Form2 iff there are no quantifiers on
predicates (cases 8 and 9). A formula is in the propositional fragment Prop of Form1iff
it contains no quantifiers (cases 6 through 9), and all predicate symbols have arity 0.
Notation 2.7. Second-order quantifiers may be written as ∀Pk.A or ∃Pk.A to make the arity of P explicit.
Remark 2.8. We opt to use the term “formula” rather than the common term “well- formed formula”, because all expressions matching the grammar defined above are, by
24 2. GENERAL NOTATION AND TERMINOLOGY
definition, formulae, and other expressions containing propositional variables and con- nectives that do not match the grammar are not formulae.
Notation 2.9 (Substitution). The expression [t/s]E denotes the result of substituting every occurrence of s with t in expression E. (It is assumed that both E ∈ T and [t/s]E ∈ T, for some type T.)
Notation 2.10 (Vector). We use ¯x to denote a vector of variables x1,..., xn. In a
(multi)set context, ¯x denotes a set of variables {x1,..., xn} in Term0.
In a slight abuse of notation, ¯x#X is used as shorthand for ¯x ∩ X = ∅ and ¯x ⊆ X for ¯x ∩ X= ¯x.
We use [¯y/ ¯x]A to denote the substitutions [y1/x1], . . . , [yn/xn]A, where ¯y and ¯x are
disjoint (and thus the order of substitution is irrelevant).
Notation 2.11. We combine like quantifier symbols and write ∀x1,..., xk.A as short-
hand for ∀x1,∀x2,...,∀xk (and similar for existential quantifiers). When the number of
variables is indeterminate, we write ∀ ¯x.A (or ∃ ¯x.A).
Definition 2.12 (Free Variables). Let A ∈ Form2. We use FV(A) to denote the set of
free (individual) variables in A. It is defined formally below:
(1) FV(Px1,..., xk)=de f Ski=1FV0(xi), where P ∈ Predkand
FV0(x)=de f { x } iff x ∈ Term0; Sk i=1FV0(xi0) if x= f (x 0 1,..., x 0 k) where f ∈ Funck. (2) FV(⊥)=de f ∅; (3) FV(A ∧ B)=de f FV(A) ∪ FV(B); (4) FV(A ∨ B)=de f FV(A) ∪ FV(B); (5) FV(A ⊃ B)=de f FV(A) ∪ FV(B);
(6) FV(∀x.A)=de f FV(A) \ {x}, where x ∈ Term0; (x is called bound in A);
(7) FV(∃x.A)=de f FV(A) \ {x}, where x ∈ Term0; (x is called bound in A);
(8) FV(∀P.A)=de f FV(A), where P ∈ Predk;
2.1. NOTATION AND TERMINOLOGY FOR FORMULAE AND PROOFS 25
Remark 2.13. The codomain of FV is Term0(the set of individual variables). A sim-
ilar function FV2could be defined with a codomain of Var (the set of propositional vari-
ables), but this is not needed here.
Remark 2.14. We will adopt the usual conventions about free and bound occurrences of variables in quantified formulae.
Definition 2.15 (Defined Connectives). Unless otherwise noted, the following con- nectives are defined in terms of other connectives:
¬A=de f A ⊃ ⊥
>=de f ⊥ ⊃ ⊥
A ≡ B=de f (A ⊃ B) ∧ (B ⊃ A)
where A ∈ Form2. Likewise, rules for these connectives in a calculus will be assumed
derivable in the usual way.
Remark 2.16. This work will examine translations of propositional calculi (or the propositional fragments of calculi) between formalisms. Some of the work will use first- or second-order logic to provide a semantic basis for the translation.
Definition 2.17 (Implicational Fragment). The implicational fragment of formulae in Form1(resp. Prop), denoted by Form1/ ⊃ (resp. Prop/ ⊃), is the set of all formulae in
Form1(resp. Prop) that do not have A ∧ B or A ∨ B as subformulae.
Definition 2.18 (Formula Size). Let A ∈ Form2. The size of A, written as |A|, is defined
inductively: (1) |Px1,..., xk|=de f 0 if P ∈ Predk; (2) |⊥|=de f 1; (3) |A ∧ B|=de f |A|+ |B| + 1; (4) |A ∨ B|=de f |A|+ |B| + 1; (5) |A ⊃ B|=de f |A|+ |B| + 1;
(6) |∀α.A=de f |A|+ 1, where α ∈ Term0+ Predk;
(7) |∃α.A=de f |A|+ 1 where α ∈ Term0+ Predk;
26 2. GENERAL NOTATION AND TERMINOLOGY
Definition 2.19 (Prime Formulae). A formula A ∈ Form2is prime iff A is atomic or
A= ⊥.
Remark 2.20. The distinction between a prime formula and an atomic formula is somewhat arbitrary here.
Definition 2.21 (Sequents). Let L denote a type of formulae, e.g. Prop or Form1.
The set SeqL of L-sequents of L-formulae is a pair in L∗× L∗, written (schematically)
asΓ⇒∆, where Γ,∆ ∈ L∗. “⇒” is called the sequent arrow. When L is obvious from or irrelevant to the context, we shall simply refer to L-sequents as “sequents”.
Notation 2.22. Empty antecedents or succedents in sequents are generally written as a blank space, rather than using the empty set symbol, e.g. “ ⇒∆” rather than “∅ ⇒ ∆”. (In some cases, the empty set symbol will be used for clarity.)
Remark 2.23 (Order of Representation). We note that while antecedents, succedents and hypersequents are multisets, their representations are in a particular order, as a list. For the purposes of the functions given in this thesis that have multisets in their domains or codomains, we will assume that there is a particular order of those multisets that is determined by their representations, and thus without loss of generality we will treat them as if they as if they are indexed multisets, where the index of an element corresponds to the order of its occurrence in a representation.
Definition 2.24 (Polarity). A formula has a polarity that is either positive or nega- tive, depending on where it occurs in the structure of a compound formula or a sequent:
(1) if A ∧ B has a positive (resp. negative) polarity, then both A and B have a positive (resp. negative) polarity;
(2) if A ∨ B has a positive (resp. negative) polarity, then both A and B have a positive (resp. negative) polarity;
(3) if ∀ ¯x.A has a positive (resp. negative) polarity, then A has a positive (resp. nega- tive) polarity;
(4) if ∃ ¯x.A has a positive (resp. negative) polarity, then A has a positive (resp. nega- tive) polarity;
(5) if A ⊃ B has a positive (resp. negative) polarity, then A has a negative (resp. positive) polarity and B has a negative (resp. positive) polarity.
2.1. NOTATION AND TERMINOLOGY FOR FORMULAE AND PROOFS 27
(6) if A occurs in the antecedent (resp. succedent) of a sequent, then A has a negative (resp. positive) polarity.
Remark 2.25. The location of a component in a hypersequent has no bearing on its polarity.
Definition 2.26 ( ∧∧ and ∨∨ operators). The symbols ∧∧ and ∨∨ are normally used as alternative symbols to denote the iterated conjunction and disjunction of formulae in a multiset, i.e. ∧ ∧ {A1,..., An}=de f n ^ i=1 Ai ∨ {A∨ 1,..., An}=de f n _ i=1 Ai
(Unless otherwise noted, ∧∧ ∅=de f > and ∨∨ ∅=de f ⊥.) Here they will be used to denote functionsfrom multisets of formulae into formulae of specific forms. This allows the use of ∧∧−1and ∨∨−1as notation to denote their respective inverse functions from formulae of those forms into multisets of formulae. That is,
∧
∧ {A1,..., An}= A1∧... ∧ An ⇐⇒ ∧∧−1 A1∧... ∧ An= {A1,..., An}
and similarly for ∨∨ and ∨∨−1.
Remark 2.27. We note that the order of formulae for the ∧∧ and ∨∨ operators is ir- relevant with respect to the semantics of the logics we will be examining in this thesis (discussed in Chapter 3). However, conjunction and disjunction are commutative and associative in these logics, so we therefore allow ourselves to choose an arbitrary repre- sentative of their iterated operations. It is routine to check that all formulae obtained this way are equivalent.
Notation 2.28 (Precedence). For clarity, we may omit brackets around subformulae. The precedence of logical connectives not delineated by brackets should be read as fol- lows:
(1) ⊃ is right associative, so A ⊃ B ⊃ C is read as A ⊃ (B ⊃ C); (2) ∧ and ∨ are left associative, so A ∧ B ∧ C is read is (A ∧ B) ∧ C;
(3) ¬ applies to the formula immediately to the right of it, so that ¬A ⊃ B is read as (¬A) ⊃ B;
28 2. GENERAL NOTATION AND TERMINOLOGY
(4) ∧ and ∨ have equal precedence over ⊃, so that A ∧ B ⊃ C ∨ D is read as (A ∧ B) ⊃ (C ∨ D).
Notation 2.29. When hypersequents or labelled sequents are translated into formulae in SPSF, numeric subscripts will sometimes be added to logical symbols ∧ and ∨ (and their iterated forms ∧∧ and ∨∨ ) to differentiate explicit connectives from implicit connec- tives, e.g. to distinguish the translations of formulae A ∨ (B ∨ C) from A, B ∨ C in the succedent of a sequent, we might use A ∨0(B ∨0C) and A ∨1(B ∨0C). The subscripts are
useful for pattern matching the arguments of the inverse translation functions. Otherwise the subscripts have no meaning.
Terminology 2.30 (Formalism). We will use the term formalism to denote the gen- eral type of data structure, e.g. hypersequent or simply labelled sequent. (In the context of calculi or proofs, formalism will denote the calculi of that data structure, or proofs in such calculi.)
Terminology 2.31 (Framework). We will use the term framework to denote a set of calculi of a particular formalism, where the calculi share a set of base rules.
Definition 2.32 (Metaformula). A metaformula is a either a metavariable, or a for- mula with metavariables as subformulae.
Notation 2.33 (Rules). The names of rules for calculi will be denoted using a sans- serif font, e.g. L∧.
Notation 2.34 (Derivations). Trees constructed according to the rules of a calculus are called derivation fragments. A derivation or proof has axioms as leaves. (Where the difference is obvious or irrelevant, we will use “derivation” to refer to both derivation fragments and derivations.)
In a derivation, rule names given in parenthesis, e.g. Γ⇒∆
Γ,Γ0⇒∆ (LW)
denote an application of a non-primitive (admissible or derived) rule. An exponent (a number, asterisk or plus) on a rule name in a derivation signifies multiple instances of a
2.1. NOTATION AND TERMINOLOGY FOR FORMULAE AND PROOFS 29
rule in much the same way an exponent does for multiple occurrences of a formula or multiset. An exponent of −1 indicates the inverted form of a rule (see below).
For rules with multiple premisses that follow a pattern, we will use angle brackets. For example, if a rule has n premisses of the formΓi⇒ Ai, then it may be shown schematically
(or even in a derivation) as
hΓi⇒ Aiini=1
Γ0⇒∆0
An end sequent (end hypersequent) is the root node of a derivation tree. An initial sequent (initial hypersequent) is a leaf node of a derivation tree, i.e., an axiom.
We use the notation GS `Γ⇒∆ to indicate that sequent Γ⇒∆ is derivable in calculus GS for logic S. Where the calculus is obvious from the context, we say simply `Γ⇒∆. This is extended to hypersequents and label sequents.
In derivation fragments, the notation .. .. δ
Γ⇒∆ ρ
denotes a derivation ofΓ⇒∆ from δ (often represented as an equation number or lemma). Notation 2.35 (Rule Instantiation). Let ρ be a schematic rule in a calculus. A (partial) instantiation σ is a substitution of metavariables such that σρ is a form of ρ with the substitution applied to it’s metavariables. σρ may be another schematic rule (for example, substituting an active formula with one of a specific form), or it may be an inference (an instance of ρ with no metavariables).
Terminology 2.36 (nth Premiss). For rules with more than two premisses, the nth premiss will refer to the nth premiss from the left.
Definition 2.37 (Invertibility). A rule in a calculus GS Γ0 1⇒∆ 0 1 ... Γ 0 n⇒∆0n Γ⇒∆ ρ
is invertible iff the rules
Γ⇒∆ Γ0 i⇒∆ 0 i ρ−1 i
30 2. GENERAL NOTATION AND TERMINOLOGY
Remark 2.38. This definition is applicable irrespective of the formalism of GS, e.g. sequent, hypersequent, labelled sequent, etc.
Notation 2.39. Inverted rules are denoted with the exponent −1, and a subscript for the ith premiss (when there are more than one premiss), e.g. ρ−i1
Definition 2.40 (Derivation depth). The expression h(δ) denotes the depth (or height) of the derivation, which is defined as following:
(1) Axioms (initial sequents or hypersequents, etc.) have a derivation depth of 0. (2) The derivation depth of a rule instance is equal to 1 plus the maximum derivation
depths of the premisses.
Remark 2.41. The term “depth” is used instead of “height”, since rules of lower depth are closer to the top of the derivation.
Definition 2.42 (Depth-Preserving Admissible). A rule is depth-preserving admis- sible Γ0 1⇒∆ 0 1 ... Γ 0 n⇒∆0n Γ⇒∆ ρ
in a calculus GS iff for all instantiations σ of ρ, whenever the maximum derivation depth of the premisses of σρ is h, then the conclusion of σρ can be derived at a derivation depth no greater than h.
Remark 2.43. This definition is applicable irrespective of the formalism of GS, e.g. sequent, hypersequent, labelled sequent, etc.
Definition 2.44 (Depth-Preserving Invertible). A rule ρ is depth-preserving invert- ible in a calculus GS iff ρ is invertible, and the inverted rule(s) are depth-preserving admissible in GS.
Notation 2.45 (Freshness). t#E indicates that expression t is not a subexpression of E, read as t is fresh for E.
The expression t ∈ E may be used to indicate that t is a subexpression of E, when it is clear from the context type t belongs to relative to E.