Intermediate Logics
3.2. INTERMEDIATE LOGICS
3.2.2. Intermediate Logics. Intermediate logics (also called superintuitionistic log- ics) are logics between Int and Cl, inclusive. We use Int∗to denote the class of all inter- mediate logics.
There are uncountably many logics in Int∗ [Göd33a, Jan68b]. The set of logics in Int∗ form a distributive lattice where join corresponds to set inclusion [CZ97].
Definition 3.2 (Disjunction Property). A logic S has the disjunction property when S ` A ∨ B iff S ` A or S ` B. An intermediate logic is called constructive if it has the disjunction property.
Not all intermediate logics are constructive. (Clearly Cl is an example.)
Following are some of the well-known intermediate propositional logics which will be discussed later in this thesis. Intermediate predicate logics will not be discussed. The applications for these logics will not be discussed explicitly, though we cite sources where the applications are discussed.
Classical Logic: has the following alternative axiomatisations:
Cl=de f Int+ A ∨ (A⊃ B) (10)
=de f Int+ ¬¬A⊃ A (11)
Cl/⊃ =de f Int+ ((A⊃ B)⊃ A)⊃ A (12)
where Cl/ ⊃ is the implicational fragment of Cl. (Clearly, LEM, A ∨ ¬A is in- cluded in A ∨ (A ⊃ B).)
Jankov-De Morgan Logic: was introduced by Jankov [Jan68a]. It has the follow- ing alternative axiomatisations:
Jan=de f Int+ ¬¬A ∨ ¬A (13)
=de f Int+ ¬(A ∧ B)⊃ (¬A ∨ ¬B) (14)
Jan is also known as the logic of “weak excluded middle” (after the corre- sponding formula in the first axiomatisation) or as LQ e.g. [Avr91a] or KC e.g. [CZ97].
40 3. INTERMEDIATE LOGICS
Gödel-Dummett Logic: Dummett [Dum59] generalised the logics Gk (below) to
infinite-valued logics. It has the following alternative axiomatisations:
GD=de f Int+ (A⊃ B) ∨ (B⊃ A) (15)
=de f Int+ (A⊃ (B∨C))⊃ ((A⊃ B) ∨ (A⊃ C)) (16)
GD/⊃ =de f Int/⊃ + ((A⊃ B⊃ C)⊃ (((B⊃ A)⊃ C)⊃ C) (17)
where GD/ ⊃ is the implicational fragment [Bac68, Bul62, Dum59]. GD has also been known in the literature as G e.g. [BCF03a] Gω e.g. [Got01], or LC (for the logic of “linear chains”, referring to the Kripke semantics) e.g. [Avr91a].
Gödel Logics: Gödel [Göd33a] introduced a class of (finite) many-valued logics Gk. The class is defined inductively [BCF03a]:
G1=de f GD (18)
Gk+1=de f GD+ A1∨ (A1⊃ A2) ∨ . . . ∨ ((A1∧... ∧ Ak) ⊃ Ak+1) (19)
where k ≥ 1. For example,
G2= GD + A ∨ (A⊃ B) (20)
G3= GD + A ∨ (A⊃ B) ∨ ((A ∧ B)⊃ C) (21)
Clearly G2= Cl, and G3= Sm (given below, with an alternative axiomatisation).
A survey of applications of these logics can be found in [BCF03a].
Logics of Bounded Depth: is a class of logics with Kripke frames of bounded depth. Each logic BDk =de f Int+ BDk, where the the characteristic axiom BDk
is defined inductively [CZ97]:
BD1= A1∨ ¬A1 (22)
3.2. INTERMEDIATE LOGICS 41
For example,
BD1= Int + A ∨ ¬A (24)
BD2= Int + A ∨ (A⊃ (B∨ ¬B)) (25)
Clearly BD1= Cl. An alternative axiomatisation uses generalisations of Peirce’s
Law of the characteristic axiom [van02]:
BD01= ((A1⊃ A0) ⊃ A1) ⊃ A1 (26)
BD0k+1= ((Ak+1⊃ BD0k) ⊃ Ak+1) ⊃ Ak+1 (27) which can be used to define the implicational fragments BDk/⊃. BDk has also
been known in the literature as BHk (for “bounded height”) and LPk (in refer-
ence to the generalisation of Peirce’s Law, e.g. [van02]).
Logics of Bounded Width: is a class of logics with Kripke frames of bounded width. The axiomatisation of BWk [CZ97] is by:
BWk=de f Int+ k _ i=0 (Ai⊃ _ j,i Aj) (28) For example, BW1= Int + (A⊃ B) ∨ (B⊃ A) (29) BW2= Int + (A⊃ B∨C) ∨ (B⊃ A ∨C) ∨ (C ⊃ A ∨ B) (30)
Clearly BW1= GD. BWk is also known in the literate as BAk (for “bounded
anti-chains”) [van02].
Logics of Bounded Top Width: is a class of logics with Kripke frames of bounded width at the top of their trees. The axiomatisation of BTWk is [CZ97, van02]:
BTWk=de f Int+ ^ 0≤i< j≤k ¬(¬Ai∧ ¬Aj) ⊃ k _ i=0 ¬Ai⊃ _ j,i ¬Aj (31)
In [FM93], the axiomatisation is given as
BTWk= Int + ¬A1∨ k−1 _ i=2 i ^ j=1
¬Aj⊃ ¬Ai+1
∨ k−1 ^ i=1 ¬Ai⊃ ¬¬Ak (32)
42 3. INTERMEDIATE LOGICS
From the semantics given below, we will see that BTW1= Jan.
Logics of Bounded Cardinality: is a class of logics with Kripke frames of bound- ed size. The axiomatisation of a logic BCk is defined as BCk =de f Int+ BCk,
where the characteristic axiom BCk is defined inductively [CZ97]:
BC0=de f A0 (33)
BCk+1=de f BCk∨ (A0∧... ∧ Ak) ⊃ Ak+1 (34)
for k ≥ 1. Alternatively it can be defined is [van02]:
BCk=de f _
0≤i< j≤k
Ai≡ Aj (35)
For example,
BC1= Int + A ∨ (A⊃ B) (36)
BC2= Int + A ∨ (A⊃ B) ∨ ((A ∧ B)⊃ C) (37)
Clearly the logics BC1= Cl, and Gk = GD + BCk.
Greatest Semiconstructive Logic: A semiconstructive logic is a logic where a proof of a disjunction A ∨ B requires a proof of A or a proof of B, but the proof of Aor B is not necessarily constructive. The greatest (or maximal) semiconstruc- tive logic GSc [FM93] is axiomatised by:
GSc=de f Int+ BD2+ ¬A ∨ (¬A⊃ ¬B) ∨ (¬A⊃ ¬¬B) (38)
GSc is also known as GS or BD2F2 in [AFM99b]. Axiomatisations for several
families of semiconstructive logics are given in [FM93]. For brevity, we will not discuss them here.
Smetanich Logic: Sm is the logic obtained by combining GD and BD2, or equiv-
alently [DN10]:
Sm=de f Int+ (¬B⊃ A)⊃ (((A⊃ B)⊃ A⊃ A) (39)
It is equivalent to G3, and is also known in the literature as LC2(cf. [DN10]) or
3.2. INTERMEDIATE LOGICS 43
Kreisel-Putnam Logic: was introduced by Kreisel and Putnam [KP57] as a coun- terexample to a conjecture by Łukasiewicz that Int was the maximally consistent logic with the disjunction property [Łuk52]. It is axiomatised by
KP=de f Int+ (¬A⊃ B∨C)⊃ (¬A⊃ B) ∨ (¬A⊃ C) (40)
Exactly seven intermediate logics are interpolable [Mak79], all of them are noted above. They form the lattice in Figure 3.1, based on set inclusion [DN10].
BD2 //GSc ""E E E E E E E E Int <<y y y y y y y y ""E E E E E E E E Sm //Cl Jan //GD <<y y y y y y y y
Figure 3.1. The lattice of the seven interpolable intermediate logics.
Semantics for these logics will be given in Section 3.3 below. There are several other well-known intermediate propositional logics that we will not cover here, largely because the do not have Kripke models which are axiomatised by first-order formula, and so are not relevant to the methods discussed in this thesis.
A more detailed introduction to the philosophical motivations and history of intuition- istic and some intermediate logics can be found in [van02]. A detailed exposition of the semantics of intermediate logics can be found in [CZ97].
3.2.3. Modal Interpretation. An interpretation of Int in terms of the modal logic S4 was given by Gödel [Göd33b]. We note the translation here, as it is used later as a basis for adapting calculi for model logics into calculi for Int.
44 3. INTERMEDIATE LOGICS
Definition 3.3 (Gödel Translation). The Gödel translation of formulae [Göd33b] of formulae in Int to formulae in S4 is the function given below:
P= P where P is atomic (A ∧ B)= A∧ B
(A ∨ B)= A∨ B (A ⊃ B)= (A⊃ B) Lemma 3.4. Int ` A iff S4 ` A.
Proof. Cf. [MT48, Art01]
This can be extended to various intermediate logics, so that Jan corresponds to S4.2, GD corresponds to S4.3 and Cl corresponds to S5 [DN10].
This interpretation will be used later to adapt some calculi for S4 into calculi for Int.
3.3. Kripke Semantics
3.3.1. Preliminaries. Kripke semantics are a kind of relational semantics that were introduced for modal logics in [Kri59a, Kri59b], and for Int were in [Kri65]. We give the notation and terminology of Kripke semantics that is relevant to this thesis below.
Definition 3.5 (Kripke Frame and Model). A Kripke frame M is a structure hW, Ri, where W is a non-empty set of points, sometimes called “nodes”, “individuals”, “worlds” or “states”, R ⊆ W2 is a binary relation between points. We may abbreviate (x, y) ∈ R as Rxy. A Kripke model M= hW,R,vi is a model with a Kripke frame hW,Ri and an interpretation function v. of type W × Var → Bool, where Var is the set of propositional variables, and Bool is the set of Boolean values {0, 1}.
A Kripke model belonging to a particular class K (e.g. intuitionistic Kripke models), will also have an extended interpretation function v0 of type W × Prop → Bool that extends v to cover all propositional formulae. (Because the definition of v0 will depend on the class of Kripke models that M belongs to, and not the specific model M, so it is omitted from the signature.) Using v0, we define the forcing relation: