Intermediate Logics 3.1 Overview
3.5. PARTIALLY SHIELDED FORMULAE
Int∗/Geointo corresponding predicate formulae for intermediate Kripke models. We also introduce a calculus for G3c/PSF(Figure 3.2 on page 57) for manipulating sequents of formulae inPSF, and extend it with rules that preserve the admissibility of structural rules and cut but allow it to be used for deriving the translations of formulae in intermediate logics.
A discussion of embedding modal formulae into classical predicate logic, such that the resulting formulae express the Kripke semantics of the original modal logics, can be found in [BdY01]. A general discussion of embedding non-classical formulae into first- order classical logic can be found in [ONdG01].
Notation 3.32. The expression A{x¯} denotes a formula A such that FV(A)⊆ x¯. The expressionΓ{x¯}denotes a multiset of formulae such that for allA∈Γ,FV(A)⊆x¯.
The expression A{x¯y¯}denotes a formulaAsuch thatFV(A)⊆x¯∪y¯, where ¯xand ¯yare not necessarily disjoint, since the variables are named. We make no assumption about the order of variables within the vector.
For simplicity, Ax¯ will be used when it is clear from the context that it denotes a formulaAsuch thatFV(A)⊆x¯rather than a formula of the formAx¯such thatFV(Ax¯)= x¯. (For example, in the presentation of the calculusG3c/PSFin Figure 3.2 on page 57.)
Definition3.33 (Strict Partially Shielded Formula). We define the setSPSFof strict
partially shielded formulaeinductively: (1) Px∈SPSFifP∈Pred1;
(2) ⊥ ∈SPSF;
(3) A∧B∈SPSFifA,B∈SPSF; (4) A∨B∈SPSFifA,B∈SPSF;
(5) ∀y.(Rxy∧A{y})⊃B{y} ∈SPSFifA,B∈SPSF.
Note that R is a fixed binary predicate symbol in Pred2, withRxy∈RF (also known as
R-formulae) called theshieldof the formula in case 5. AlthoughRxyis a subformula in that case,Rxy<SPSF.
Lemma3.34. Let A∈SPSF. Then|FV(A)| ≤1.
54 3. INTERMEDIATE LOGICS
Formulae in SPSFcan be used to adequately express the translation of intuitionistic formulae into their corresponding first-order formulae for intuitionistic Kripke models, for example, A⊃(B⊃ A) corresponds to∀y1.((Rxy1∧Ayˆ 1)⊃(∀y2.(Ry1y2∧Byˆ 2)⊃Ayˆ 2)), using the translation function from formulae toSPSFis given later in Chapter 6. However, the frame axioms for this class of Kripke models—reflexivity and transitivity—and for classes that correspond to stronger logics in Int∗/Geo, cannot be expressed inSPSF. So we extend the definition:
Definition 3.35 (Partially Shielded Formulae). We define the set PSF of partially
shielded formulaeinductively: (1) A∈PSF, ifA∈SPSF; (2) Rxy∈PSF, whereRxy∈RF; (3) > ∈PSF(see the remark below); (4) (A∧B)∈PSFifA,B∈PSF; (5) (A∨B)∈PSFifA,B∈PSF; (6) ∀y.¯(P{x¯y¯} ∧A1{x¯y¯} ∧...∧An{x¯y¯})⊃B{x¯y¯} ∈PSFwheren≥0, ifP,A1,...,An,B∈ PSF,Pis atomic and ¯y∩FV(P),∅; (7) ∀y.A¯ ∈PSFifA∈PSF; (8) ∃y.A¯ ∈PSFifA∈PSF;
In case 6, P is called the shield of the formula. Note that case 5 of the definition for SPSFis also a special case of case 6.
Remark3.36. >is explicitly included inPSFas a primitive symbol because⊥ ⊃ ⊥(as
>is normally defined) is not inPSF.
Definitions 3.33 and 3.35 are syntactic: although A∨B is classically equivalent to
¬A⊃B, the latter is not inSPSForPSF. Proposition3.37. SPSF⊂PSF⊂Form1.
Proof. From Definitions 3.33 and 3.35.
Terminology3.38. Amultiset is inSPSF, written asSPSF∗, iffall of the formulae in
3.5. PARTIALLY SHIELDED FORMULAE 55
Notation3.39. A sequent (Γ⇒∆)∈SPSF∗, iff Γ∈(SPSF+RF)∗and∆∈SPSF∗. This
corresponds to a basic relational sequent, as will be shown later in Chapter 8. (Note that this is a slice abuse of notation, asRformulae are not inSPSF.)
Remark3.40. The setsGFandLGF of guarded formulaeandloosely guarded for-
mulae[AvN96], have similarities to the setPSF. Recall that formulae inLGF are of the form∀y.¯(P1{x¯y¯} ∧...∧Pk{x¯y¯})⊃A{x¯y¯}, wherePi{x¯y¯}is an atom, and that formulae inGF are special cases of LGFwherek=1. The formula to the left of the implication is called theguard.
BecausePSFincludes non-atomic formulae in the scope of what would be called the guard, the term “shielded” is used instead. UnlikeGFandLGF, the formula to the right of the implication must also be inPSF.
Note also that GFand LGFallow predicates in Pred3 or higher arity as guards. The motivation for defining the setPSFrather than usingGForLGF, was that all subformulae of a formula inPSFare inPSF, and that geometric implications are inPSF.
We note that implications in SPSF (case 5) are intuitionistically equivalent to the guarded formula∀y.Rxy⊃(A{y} ⊃B{y}).
3.5.1. A Kripke Semantics for Partially Shielded Formulae. In later chapters of this thesis, we use translations of hypersequents and labelled sequents into sequents in PSF∗ to show the relationships between these formalisms, or to justify the soundness of these rules. This requires a definition of the interpretation of formulae inPSFwith respect to Kripke models.
Definition 3.41 (Satisfaction of a Formulae in PSF by a Kripke Model). Let M =
hW,R,vibe a Kripke model, and let D( ˆx) be an surjective function fromTerm0toW. We define thesatisfiabilityof a formulaA∈PSFby a Kripke model inductively:
(1) MRxˆyˆiff(x,y)∈R, whereD( ˆx)= xandD(ˆy)=y; (2) MPˆxˆiffv(x,P)=1, whereD( ˆx)=x; (3) M2⊥; (4) M>; (5) MA∧BiffMAandMB; (6) MA∨BiffeitherMAorMB;
56 3. INTERMEDIATE LOGICS
(7) M∀y.¯(P{x¯y¯} ∧A1{x¯y¯} ∧...∧An{x¯y¯})⊃B{x¯y¯}iffMP{x¯y¯} ∧A1{x¯y¯} ∧...∧An{x¯y¯} impliesMB{x¯y¯};
(8) M∀x.Aˆ ifffor allw∈W,M[ ˆw/xˆ]A, whereD( ˆw)=w;
(9) M∃x.Aˆ iffthere existsw∈W such thatM[ ˆw/xˆ]A, whereD( ˆw)=w. wherePis the propositional variable that corresponds to the unary predicate ˆP.
We extend this notion naturally for sequents inSPSF∗2andPSF2so thatMΓ⇒∆iff
M∧∧Γ ⊃ ∨∨∆.
Definition3.42 (Validity of a Formulae inPSFin a Class of Kripke Models). A for-
mulaA∈PSF isvalidfor a class of Kripke modelsK ifffor every Kripke modelM∈ K
there exists a mappingDfromTerm0toWsuch thatMA.
3.5.2. The calculus G3c/PSF. We introduce the calculusG3c/PSFin Figure 3.2 on the facing page.
Terminology3.43. In theL∀ ⊃(viz. R∀ ⊃) rule ofG3c/PSF(Figure 3.2 on the next
page), the active and principal formula Rx¯z¯is called the shield, the variables ¯zofR are calledbindable, and the variables ¯xare calledunbindable.
Remark 3.44. We (again) note that theL∀and R∀rules of G3c/PSFcannot be ap-
plied to shielded implications (or specifically, to all bound variables in shielded implica- tions) because the formulae in the premisses would not be partially shielded.
Below we discuss the properties of that calculus, including an embedding inG3cand cut elimination.
Remark3.45. Ideas forG3c/PSF—in particular for theL∀ ⊃rule that uses the shield
formula as a “key” to unlock the shielded formula in G3c/PSF were influenced by a general calculus for guarded logic that was developed in [DS06].
We do not present complexity or decidability results aboutG3c/PSFhere, as that is beyond what is needed for our purposes in this work. It is an area for future investigation. Clearly some of the formulaeAi{x¯y¯}in the principal formulae ofL∀ ⊃and R∀ ⊃can be formulae in RF. Where corresponding Ai{x¯z¯} formulae occur as side formulae in the antecedent, the ith premiss is an axiom and can be ignored. For example, we suppose
3.5. PARTIALLY SHIELDED FORMULAE 57 Γ,P⇒P,∆ Ax Γ,⊥ ⇒∆ L⊥ Γ⇒ >,∆ R> Γ,A,B⇒∆ Γ,A∧B⇒∆ L∧ Γ⇒A,∆ Γ⇒B,∆ Γ⇒A∧B,∆ R∧ Γ,A⇒∆ Γ,B⇒∆ Γ,A∨B⇒∆ L∨ Γ⇒A,B,∆ Γ⇒A∨B,∆ R∨ hΓ,Rx¯z,¯ ∀y....¯ ⇒Aix¯z,¯ ∆ini=1 Γ,Rx¯z,¯ ∀y....,¯ Bx¯z¯⇒∆ Γ,Rx¯z,¯ ∀y.¯(Rx¯y¯∧A1x¯y¯∧...∧Anx¯y¯)⊃Bx¯y¯⇒∆ L∀ ⊃ Γ,Rx¯z,¯ A1x¯z,...,¯ Anx¯¯z⇒Bx¯z,¯ ∆ Γ⇒ ∀y.¯ (Rx¯y¯∧A1x¯y¯∧...∧Anx¯y¯)⊃Bx¯y,¯ ∆ R∀ ⊃ Γ,∀x.A,¯ [¯y/x¯]A⇒∆ Γ,∀x.A¯ ⇒∆ L∀ Γ⇒[¯z/x¯]A,∆ Γ⇒ ∀x.A,¯ ∆ R∀ Γ,[¯z/x¯]A⇒∆ Γ,∃x.A¯ ⇒∆ L∃ Γ⇒[¯y/x¯]A,∃x.A,¯ ∆ Γ⇒ ∃x.A,¯ ∆ R∃
We omit the curly brackets for brevity, e.g. usingAx¯instead ofA{x¯}. All formulae
A,A1,...An,B,P∈PSF, withRbeing atomic, and ¯zisfreshfor the conclusion of theR∀,
L∃andR∀ ⊃rules, and∀y¯...is an abbreviation for∀y.¯(Rx¯y¯∧A1x¯y¯∧...∧Anx¯y¯)⊃Bx¯y¯. Figure3.2. The calculusG3c/PSFfor sequents of partially shielded formulae.
formulaeA1x¯y,...,¯ Amx¯y¯(form≤n) to be formulae inRF:
hΓ,Rx¯z,¯ A1x¯y,...,¯ Amx¯y,¯ ∀y....¯ ⇒Aix¯¯z,∆ini=1 Γ,Rx¯z,¯ ∀y....,¯ Bx¯¯z⇒∆ Γ,Rx¯z,¯ A1x¯y,...,¯ Amx¯y,¯ ∀y.¯(Rx¯y¯∧A1x¯y¯∧...∧Anx¯y¯)⊃Bx¯y¯⇒∆
L∀ ⊃
This allows us to give a simpler form of theL∀ ⊃rule below: Proposition3.46. TheL∀ ⊃0rule
Γ,Rx¯z,¯ A1x¯z,...,¯ Anx¯z,¯ ∀y¯...,Bx¯z¯⇒∆ Γ,Rx¯z,¯ A1x¯¯z,...,Anx¯z,¯ ∀y.¯(Rx¯y¯∧ ∧∧Aix¯y¯)⊃Bx¯y¯⇒∆
(L∀ ⊃0)
is derivable inG3c/PSF.
58 3. INTERMEDIATE LOGICS
Remark 3.47. TheL∀ ⊃0 rule is useful for cases where some of the Aix¯y¯ that are in negative positions of the principal formula are relational formulae. In these cases, the pre- misses that contain a relational formula in the succedent are axioms, and can be ignored. This is important for translations between partially shielded sequents and languages for relational sequents (such as the one used in this thesis) that do not allow relational formu- lae in the succedent.
We note that proofs inG3c/PSFcan be embedded in the systemG3c: Lemma3.48 (Embedding). IfG3c/PSF`Γ⇒∆, thenG3c`Γ⇒∆.
Proof. Straightforward. (The proof is written out in Lemma B.2 on page 237.) Corollary3.49 (Soundness). G3c/PSFis sound.
Proof. From Lemma 3.48 and the soundness ofG3c[TS00]. Definition 3.50. A first-order term t is free for x in A iff t does not contain a free
variable ythat would become bound by replacing xwithtinA. (In the case of formulae inPSF, which does not allow functions in terms, this means thattis not equal to a bound variable y in A such that replacing x with t would change a free variable into a bound variable.)
This notion is extended to multisets and sequents naturally. Lemma3.51 (Substitution). Variable substitution
Γ⇒∆
[t/x]Γ⇒[t/x]∆ [t/x]
where t is free for x inΓ,∆, is depth-preserving admissible inG3c/PSF.
Proof. By induction on the derivation depth. Notation3.52. For readability, we omit parentheses from instances of the substitution
rule.
Lemma3.53 (Weakening). The weakening rules
Γ⇒∆
Γ,A⇒∆ (LW) Γ ⇒∆
Γ⇒A,∆ (RW) are depth-preserving admissible inG3c/PSF.
3.5. PARTIALLY SHIELDED FORMULAE 59
Proof. By induction on the derivation depth.
Lemma 3.54 (Generalised axioms). Sequents of the form A,Γ⇒∆,A, where A is an
arbitrary formula inPSF, are derivable inG3c/PSF.
Proof. By induction on the size ofA. Lemma3.55 (Invertibility). The rules ofG3c/PSFare depth-preserving invertible.
Proof. Straightforward, ForL∀andL∀ ⊃,LWis used. For all other rules, by simultaneous
proof, using induction on the derivation depths.
Lemma3.56 (Nullary Connective Deletion). The following constant deletion rules
>,Γ⇒∆
Γ⇒∆ (L>)
Γ⇒∆,⊥
Γ⇒∆ (R⊥) are depth-preserving admissible inG3c/PSF.
Proof. By induction on the derivation depth. (See Lemma B.4 on page 238.) Lemma3.57 (Contraction). The contraction rules
Γ,A,A⇒∆
Γ,A⇒∆ (LC)
Γ⇒A,A,∆
Γ⇒A,∆ (RC) are depth-preserving admissible inG3c/PSF.
Proof. Straightforward simultaneous induction on the derivation depth. (The proof is
written out in Lemma B.5 on page 239.)
Theorem3.58 (Cut). The context-splitting cut rule
Γ⇒A,∆ Γ0,A⇒∆0
Γ,Γ0⇒∆,∆0 (cut)
is admissible inG3c/PSF.
Proof. By induction on the cut rank (a lexically-ordered pair consisting of the size of the cut formula and sum of the depths of the premisses). (The proof is given in Theorem B.6
60 3. INTERMEDIATE LOGICS
Corollary3.59 (Cut). The context-sharing cut rule
Γ⇒A,∆ Γ,A⇒∆
Γ⇒∆ (cut
0)
is admissible inG3c/PSF.
Proof. From Theorem 3.58, usingLCandRC.
3.5.3. Geometric Extension of G3c/PSF. In later chapters we will introduce trans- lations of hypersequents, simply labelled sequents and relational sequents intoPSF, such that a (hyper)sequent is valid in a class of Kripke models iff its translation into PSF is derivable in G3c/PSF. But in order to use G3c/PSF to validate sequents that are translations of formulae in an intermediate logic, the axioms corresponding to the prop- erties of the corresponding Kripke models must be included. For example,the sequent
Rxy,Ryz,Ax⇒Azis only derivable in G3c/PSFwhen some form of the transitivity and persistence axioms are included.
In [Min00], a translationφis given for a formulaAthat is true with respect to a set of axioms that isparametrised by the formulaeA. Adapted for the notation used here:
κA=de f ∀x.Rxx,∀xyz.Rxy∧ Ryz⊃ Rxz, n
[
i=1
{ ∀xy.Rxy⊃Pˆix⊃Pˆiy}
corresponds to the reflexivity, transitivity and persistence axioms of intuitionistic Kripke models, such that for all formulae A∈Prop,LJpm`κA⊃φAiffLJpm`A, where each
ˆ
Pi∈Pred1corresponds to a propositional variablePi∈Var1, for all atomic propositional variables that are subformulae ofA, andLJpmis a sound and complete sequent calculus forInt(see Figure A.4 on page 236).
This could be adapted to extend the proof theory of G3c/PSF, where of sequents of the form κX,Γ⇒∆, where κX,Γ,∆⊂PSF, and X⊂Var. The parametrisation could even be eliminated by extend the language ofPSF and the rules ofG3c/PSFfor second-order quantification and use a second-order definition of the persistence axiom:
3.5. PARTIALLY SHIELDED FORMULAE 61
where P∈Pred1. Work on a second-order extension of G3c/PSF is an area for future investigation. Another alternative is to define the theRformula as an abbreviation for
Rxy=de f ∀P1.(Rxy∧Px)⊃Py
It appears that reflexivity and transitivity follow from this defined relation, but it is unclear whether a calculus using this definition correspond withIntor a stronger logic. This too is an area for later research.
Because we are usingG3c/PSFto logics inInt∗/Geo, a simpler alternative is to ex- tend G3c/PSF withgeometric rules (Definition 3.60 below) that correspond to the ax- ioms in κ. Recall the work cited in Section 3.4 on page 51 that adding such rules to a calculus based on G3cdoes not affected the admissibility of cut, weakening or contrac- tion.
Definition3.60 (Geometric Rule). Ageometric rule[Neg03, Neg07, DN10] is a rule
of the form
[¯z/y¯] ¯A1,A¯0,Γ⇒∆ ... [¯z/y¯] ¯An,A¯0,Γ⇒∆ ¯
A0,Γ⇒∆ (56)
where the variables ¯z do not occur free in the conclusion, and each ¯Ai (in an abuse of notation) is a multisetPi1,...,Pik(i).
Here we show that geometric rules are derivable from geometric implications in the antecedent in G3c/PSF. This we can analyse the frame axioms ofInt∗/Geo and obtain geometric rules that can be added toG3c/PSFto obtain a calculus for the Kripke models that validate the corresponding logics.
Proposition3.61. If A is a geometric formulae, then A∈PSF.
Proof. By induction on the structure ofA. Lemma3.62. Let A0∈Geo. Then there is a geometric rule that corresponds to analys-
ing A0,Γ⇒∆inG3c/PSF.
Proof. Note that a formula in Geo is eitherGR or HR (Definition 3.28 on page 52). If
A0∈GR, then treat it as∀x.¯(> ⊃A0) in the procedure below.
Not all geometric implications are inPSF, e.g.∀x.¯((C∨D)⊃B). However, in [Pal02], it was shown that any set of geometric implications is intuitionistically equivalent to a
62 3. INTERMEDIATE LOGICS
set consisting of formulae of the form ∀x¯(A0⊃ ∃y.¯(A1∨...∨An)), where each Ai is a conjunction of atomic formulae, whichisinPSF.
We then transform that formula into a rule by analysing it: [¯z/y¯] ¯A1,∃y¯...,∀x¯...,A¯0,Γ⇒∆ [¯z/y¯]A1,∃y¯...,∀x¯...,A¯0,Γ⇒∆ L∧∗ ... [¯z/y¯] ¯An,∃y¯...,∀x¯...,A¯0,Γ⇒∆ [¯z/y¯]An,∃y¯...,∀x¯...,A¯0,Γ⇒∆ L∧∗ [¯z/y¯](A1∨...∨An),∃y¯...,∀x¯...,A¯0,Γ⇒∆ R∨∗ ∃y.¯(A1∨...∨An),∃y¯...,∀x¯...,A¯0,Γ⇒∆ L∃ ∃y.¯(A1∨...∨An),∀x¯...,A¯0,Γ⇒∆ (LC) ∀x.¯(A0⊃ ∃y.¯(A1∨...∨An)),A¯0,Γ⇒∆ (L∀ ⊃0)
This clearly matches the geometric rule schema.
Remark 3.63. An alternative to defining the language ofPSF is to combine the lan-
guage of SPSF (Definition 3.33 on page 53) with that of geometric formulae (Defini- tion 3.26). However, such a calculus would require two kinds ofL∀ ⊃andR∀ ⊃rules.
Definition 3.64. Let G3c/PSF∗ be the system obtained by adding to the rules of
G3c/PSFthe rules from Figures 3.3 and 3.5 on the facing page.
Rxx,Γ⇒∆
Γ⇒∆ refl
Rxz,Rxy,Ryz,Γ⇒∆ Rxy,Ryz,Γ⇒∆ trans Rxy,Px,Py,Γ⇒∆
Rxy,Px,Γ⇒∆ LF0
Rzx1,...,Rzxn,Γ⇒∆
Γ⇒∆ root
wherePxandPyare atomic inLF0 andzis not free inΓ,∆inroot. Figure3.3. Extension rules ofG3c/PSF∗ forInt.
Remark 3.65. The root rule (Figure 3.3) is considered an extension rule for Int by
Theorem 3.15 on page 46.
Therootrule can be shown eliminable forG3c/PSF∗ with the extension rules forInt
but not in the presence ofdirrule. Hence, the rule is given as primitive.
While theroot rule is not necessary for proving formulae in logics inInt∗/Geo, it is useful for proving properties about those logics. Hence the inclusion here.
We introduce rules corresponding to the semantics of the more general intermediate logics introduced earlier in Figure 3.4 on the next page, and give rules for some special cases of them in Figure 3.5 on the facing page.
3.5. PARTIALLY SHIELDED FORMULAE 63 hRxi+1xi,Rx1x2,...,Rxk−1xk,Γ⇒∆iki=−11 Rx1x2,...,Rxk−1xk,Γ⇒∆