Intermediate logics and negative variants

Recall that anintermediate logicis defined as a consistent set of formulae that containsIPLand is closed under the rules of modus ponens and uniform substi- tution, whereconsistent simply means ‘not containing⊥’.

Intermediate logics ordered by inclusion form a complete lattice whose meet operation amounts to intersection and whose join operation, also calledsum, is defined as follows: if Λi, i ∈ I is a family of weak logics, then Σi∈IΛi is the

logic axiomatized by S

i∈IΛi, that is, the closure of Si∈IΛi under the rules

of modus ponens and uniform substitution. For more details, see Chagrov and Zakharyaschev (1997). The sum oftwointermediate logic Λ and Λ0will also be denoted by Λ + Λ0_.

In our investigation, however, we will meet several logics that, just likeInqL, arenot closed under uniform substitution. We shall refer to such logics asweak intermediate logics.

Definition 3.2.1. We say that a set of formulas L is a weak logic in case

IPL⊆L and Lis closed under modus ponens. If, additionally L⊆CPL, then we callLa weakintermediate logic.

Both weak logics and weak intermediate logics ordered by inclusion form complete lattices as well, where again meet is intersection and the join (or sum) of a family is the weak logic axiomatized by the union, i.e. the closure of the union under modus ponens.

IfLis a weak logic, we writeϕ≡Lψ in caseϕ↔ψ∈L.

Definition 3.2.2. LetK be a class of Kripke models (resp. frames). If Θ is a set of formulae and ϕ is a formula, we write Θ|=Kϕ in case at any point in

any model inK (resp. any model based on a frame inK), the following holds: if all formulasθ∈Θ are forced, then so isϕ.

We denote by Log(K) the set of formulae that are valid on each model (frame) inK, that is: Log(K) ={ϕ| |=K ϕ}.

It is straightforward to check that for any classKof Kripke models,Log(K) is a weak logic, and that for any classKof Kripke frames,Log(K) is an intermediate logic.

Notation. For any formulaϕ, we denote byϕn _{the formula obtained fromϕ}

by replacing any occurrence of a propositional letter with its negation.

Definition 3.2.3 (Negative variant of a logic). If Λ is an intermediate logic, we define its negative variant Λn _as:

Λn ={ϕ|ϕn∈Λ}

In general, Λn will not be an intermediate logic: in fact, for any intermediate logic Λ we have¬¬p→p∈Λn _{for any propositional letterp; so, Λ}n _{will not be}

closed under uniform substitution unless Λn _{=CPL, whereCPLdenotes classical}

logic.

It is, however, the case that Λn_{is always a weak intermediate logic containing}

Λ; this is the content of the following remark.

Remark 3.2.4. For any intermediate logic Λ, its negative variant Λn _{is a weak}

intermediate logic including Λ.

Proof. Fix an intermediate logic Λ. Since Λ is closed under uniform substitution,

ϕ∈Λ impliesϕn_{∈Λ and soϕ∈Λ}n_{. This shows Λ⊆Λ}n_.

Moreover, if bothϕandϕ→ψbelong to Λn_{, then bothϕ}n _{and (ϕ→ψ)}n ₌ ϕn _→ψn_{are in Λ which is closed under modus ponens; therefore,ψ}n _{∈Λ, which}

means thatψ∈Λn_{. This shows that Λ}n _{is closed under modus ponens.}

Finally, ifϕ∈Λn _thenϕn _{∈Λ⊆CPL; but then, sinceCPL is substitution-}

closed,ϕnn _{∈CPLand therefore alsoϕ∈CPLbecause the double negation law}

holds inCPL. This shows that Λn_{⊆CPLand therefore that Λ}n _{is indeed a weak}

intermediate logic.

The following immediate observation will turn out useful later on.

Remark 3.2.5. If a logic Λ has the disjunction property, then so does Λn_.

For, ifϕ∨ψ∈Λn_{, thenϕ}n_∨ψn _{∈Λ; thus, by the disjunction property, at least}

one ofϕn _andψn _{must be in Λ, which means that at least one ofϕandψmust}

be in Λn_.

Definition 3.2.6 (Negative valuations). Let F be an intuitionistic frame. A valuationV is callednegativein case for any pointwinFand for any proposition letterp:

We will call a modelnegative in case its valuation is negative. Observe that if

M is a negative model, for any pointwand formulaϕwe haveM, wϕ ⇐⇒

M, wϕnn.

Definition 3.2.7(Negative variant of a model). IfM = (W, R, V) is a Kripke model, we define the negative variantMn _{ofM to be modelM}n _{= (M, R, V}n₎

where

Vn(p) :={w∈W|M, w¬p}

that is, Vn _{makes a propositional letter true precisely where its negation was}

true in the original model.

A straightforward inductive proof yields the following result. Proposition 3.2.8. For any model M, any pointwand formulaϕ:

M, wϕn ⇐⇒ Mn, wϕ

Remark 3.2.9. For any modelM, its negative variantMn is a negative model.

Proof. Take any pointwofM and formulaϕ. According to the previous propo- sition and recalling that in intuitionistic logic triple negation is equivalent to single negation, we haveMn_{, w}

p ⇐⇒ M, w¬p ⇐⇒ M, w¬¬¬p ⇐⇒

Mn_{, w}

¬¬p.

Definition 3.2.10. Let K be a class of intuitionistic Kripke frames. Then we denote bynK the class of negative K-models, i.e., negative Kripke models whose frame is inK.

Proposition 3.2.11. For any classK of Kripke frames,Log(nK) =Log(K)n_. Proof. If ϕ6∈Log(K)n_{, i.e. ifϕ}n _{6∈Log(K), then there must be a K-model M}

(i.e., a model based on aK-frame) and a pointw such that M, w6ϕn_{. But}

then, by proposition 3.2.8 we haveMn, w6ϕ, and thusϕ6∈Log(nK) sinceMn

is a negativeK-model.

Conversely, if ϕ6∈ Log(nK), let M be a negative K-model and w a point in M with M, w 6ϕ. Then since M is negative, M, w6 ϕnn. Therefore, by proposition 3.2.8,Mn, w6ϕn. ButMn shares the same frame ofM, which is aK-frame: soϕn_{6∈Log(K), that is,ϕ6∈Log(K)}n_.

The following result states that for any intermediate logic Λ, its negative variant Λn _{is axiomatized by a system having Λ and all the atomic double negation}

formulas¬¬p→pas axioms, and modus ponens as unique inference rule. Proposition 3.2.12. If Λ is an intermediate logic, Λn _{is the smallest weak}

logic containing Λ and the atomic double negation axiom ¬¬p → p for each propositional letterp.

Proof. We have already observed (see remark 3.2.1) that Λn is a weak logic containing Λ; moreover, for any letter p we have¬¬¬p → ¬p∈ IPL ⊆ Λ, so each atomic double negation formula is in Λn.

To see that Λnisthe smallestsuch logic, let Λ0be another weak logic contain- ing Λ and the atomic double negation axioms. Considerϕ ∈ Λn: this means that ϕn _{∈ Λ. But clearly, ϕ is derivable by modus ponens from ϕ}n _{and the}

atomic double negation axioms for letters inϕ: hence, as Λ0 contains Λ and the atomic double negation formulas and it is closed under modus ponens,ϕ∈Λ0. Thus, Λn _⊆Λ0_.

With just a slight abuse of notation, we can thus identify Λn _{with a derivation}

system as follows.

Definition 3.2.13. We denote by Λn _{the following derivation system.}

Axioms:

• all formulas in Λ

• ¬¬p→pfor all lettersp∈ P

Rules:

• Modus ponens

If Θ is a set of formulae andϕis a formula, we will write Θ|=Λn ϕin caseϕis derivable from the set of assumptions Θ in the system Λn_.

In the following, we will think of Λn _{either as set of formulas or as a deriva-}

tion system depending on which approach is more convenient.

As we saw, negative variants are not in general closed under uniform substi- tution; there is, however, a special kind of substitutions under which theyare

closed. This is a general version of the aforementioned (and not yet proven) fact thatInqLis closed under substitution of the atoms by assertions.

Definition 3.2.14(Negative substitutions). We say that a substitution ( )∗ is

negative in a weak logicLin casep∗≡L¬¬(p∗) for all lettersp.

For instance, a map that substitutes each atom by a negation is always negative. Now, the point is that in a negative variant Λn_{, atoms are essentially negations.}

Thus, if we want to preserve validity, it is crucial that we substitute them by formulas that enjoy the same property, namely, formulas that are equivalent to negations in Λn_.

Proposition 3.2.15 (Closure of negative variants under negative substitu- tions). Let Λ be an intermediate logic. If ϕ ∈ Λn _{and ( )}∗ _{is a substitution}

Proof. For the sake of simplicity, let us omit the brackets when dealing with multiple substitutions and write, for instanceϕnn∗ for ((ϕn)n)∗.

First observe that since ( )∗ is negative in Λn,ϕ∗≡Λn ϕnn∗: for, the latter formula is obtained from the former by putting a double negation in front of each substitutep∗ of a letterp.

Now, since ϕ ∈ Λn _{we have ϕ}n _{∈ Λ; then since ϕ}nn∗n _{is a substitution}

instance of ϕn _{we have ϕ}nn∗n _{∈ Λ, whence ϕ}nn∗ _{∈ Λ}n_{. But we saw that} ϕnn∗_≡

Λn ϕ∗, so ϕ∗∈Λn and we are done.