2.2 Modal Logic
2.2.2 Sequents
A sequent calculus is a set of rules and axioms by which deductions may be generated. A deduction δ of a sequent Γ ñ Σ relative to a calculus C is a tree whose leaves are axioms of C, is such that each node in the tree is generated from its predecessors by means of a rule of C, and whose root is Γ ñ Σ. If δ is a deduction of Γ ñ Σ relative to calculus C this is indicated by δ $c Γ ñ Σ. More generally $c Γ ñ Σ indicates that there is a deduction of Γ ñ Σ relative to C.
The sequent calculus for Classical Propositional Logic, cpl, is given in fig. 2.1. This calculus serves as the basis for all the modal sequent calculi considered in this chapter.
The modal logics described in section2.2.1 are captured by adding various com- binations of the rules from fig. 2.2 to cpl .3
Definition 3 (Adequacy). A class of frames S is adequate for a sequent calculus C iff for any sequent Γ ñ Σ, |ùS Γ ñ Σ iff $C Γ ñ Σ.
3There are many different ways of generating sequent calculi for the modal logics described in
section2.2.1. Many of these sequent calculi can be found in Wansing [73] and Poggiolesi [39].
Figure 2.1: Classical Propositional logic Structural Rules Id ϕ ñ ϕ Γ ñ Σ TL Γ, ϕ ñ Σ Γ ñ ϕ, Σ Γ, ϕ ñ Σ Cut Γ ñ Σ Γ ñ Σ TR Γ ñ ϕ, Σ Logical Rules Γ ñ ϕ, Σ L Γ, ϕ ñ Σ Γ ñ ψ, Σ Γ, ϕ ñ Σ LÑ Γ, ϕ Ñ ψ ñ Σ Γ, ϕ ñ Σ R Γ ñ ϕ, Σ Γ, ϕ ñ ψ, Σ RÑ Γ ñ ϕ Ñ ψ, Σ
Each of class of frames discussed in section 2.2.1 above is adequate with respect to a calculus definable from figs. 2.1 and 2.2. Each calculus adopts all of the rules of fig.2.1 and some subset of the rules of fig.2.2. The calculi that are adequate for each class of frames are given in the table below. The table also contains citations where the proofs of adequacy can be found.
Figure 2.2: Modal Sequent Rules Γ ñ ϕ k ˝Γ ñ ˝ϕ Γ, ϕ ñ Σ t Γ, ˝ϕ ñ Σ ˝Γ ñ ϕ 4 ˝Γ ñ ˝ϕ Γ ñ d ˝Γ ñ Γ ñ ϕ, ˝Σ b ˝Γ ñ ˝ϕ, Σ ˝Γ ñ ϕ, ˝Σ 5 ˝Γ ñ ˝ϕ, ˝Σ ˝∆ :“ t˝δ : δ P ∆u
Class of Frames Set of Rules Deducibibility Relation Reference
K k $k Sambin and Valentini [61]
D k ` d $d Valentini [71]
T k ` t $t Ohnishi and Matsumoto [35]
TR k ` t ` 4 $s4 Ohnishi and Matsumoto [35]
B k ` b $b Takano [70]
L k ` t ` 4 ` 5 $s5 Ohnishi and Matsumoto [35]
One of the philosophical motivations for exploring modal logics is to see to what extent the meaning of modal expressions can be explained by the inferences in which they feature. It has been proposed, by e.g. Belnap [4], Dummett [13], or Poggiolesi [39], that in order for a set of rules to determine the meaning of an expression the set must meet several constraints. Some of the features that philosophers have discussed are the following
1. Explicit: A rule R is explicit when only one occurrence of the expression in question occurs essentially in the conclusion. A set of rules that has this feature is not in danger of defining more than one status of a sentence at once. For instance, if the left of a sequent corresponds to truth and the right to falsity, the rule k offers an explanation of the truth of ˝ϕ inextricably from an explanation of its falsity.
2. Separated: A rule R is separated when the only expression that features essentially in that rule is the expression whose meaning is being explained. If a rule features two expressions essentially then it runs the risk of offering an explanation of those two expressions at once as opposed offering an explanation of just one of them.
3. Symmetrical: A set of rules S is symmetrical when for each rule R P S R either introduces a connective on the left or on the right and there is at least one rule for each. If the left and right side of a sequent correspond to some status a sentence may have, then symmetrical rules guarantee that for any status there is a way of determining whether a particular sentence has that status. Thus the status of a sentence would not, in principle, be underdetermined by the rules of S.
4. Non-Circularity: A rule R is non-circular if there is no essential occurrence of the connective being defined in the premises of R. A circular rule leaves open the possibility that there is no way to understand how to use a particular expression without already having some antecedent grasp of that expression.
As is noted by Dummett [13] ([13, pg.257]), in some cases this may not be problematic. But it is a good feature of a rule if it avoids the possibility of circularity altogether.
5. Uniqueness: Let S be a set of rules, and Srε1
{εs be the result of replacing ε1 for ε everywhere in S. S is said to uniquely chracterize an expression ε it introduces iff both
(a) $ Γ, ϕ ñ Σ iff $ Γ, ϕrε1
{εs ñ Σ (b) $ Γ ñ ϕ, Σ iff $ Γ ñ ϕrε1
{εs, Σ.
Proponents of proof-theoretic semantics generally hold some version of the claim that meaning is use. This could be stated in a weak form as the claim that the rules governing the behavior of an expression determine the meaning of the expression. In the case above, the purported rules that govern the meaning of ε are S. Suppose that uniqueness fails but that S is said to determine the meaning of ε. If uniqueness fails then there is a sequent, e.g. Γ, ϕ ñ Σ, that is provable though Γ, ϕrε1{εs ñ Σ is not. The only difference between Γ, ϕ ñ Σ and Γ, ϕrε1{εs ñ Σ is that ε1 replaces ε in the second. ε and ε1 cannot therefore have the same meaning. Since ε and ε1 have the same use (S and Srε1{εs respectively) S does not determine the meaning of ε.
6. Cut Admissibility: Let C be a calculus and Ccf be the calculus that is exactly like C except that it lacks the Cut rule. A logic is cut admissible when $C Γ ñ Σ iff $Ccf Γ ñ Σ. Cut admissibility is a desirable property for a calculus. In
general, it shows that & ñ , and so serves as proof of consistency. In most 45
settings it also establishes that only rules governing the expressions occurring in a sequent are used to determine whether or not there is a deduction of that sequent.4
7. Sub-formula Property: A calculus has the sub-formula property iff whenever $ Γ ñ Σ there is a deduction δ of Γ ñ Σ such that if ∆ ñ Λ occurs in δ then any sentence ϕ P ∆ Y Λ is a sub-formula of a sentence in Γ Y Σ. In most settings a calculus that is cut admissible has the sub-formula property.
None of the logics characterized above have explicit rules governing ˝. Every logic described above requires the k rule to govern the behavior of ˝, but this rule is not explicit. The rules 4, b, and 5 are also circular. Let C be a modal calculus characterized above with rules R governing ˝. If d is introduced to the calculus using the rules Rrd{˝s the sequent ˝p ñ dp is not deducible even though the sequent ˝p ñ ˝p is. Thus, all of these logics fail to uniquely characterize the modal expression ˝. It was shown by Ohnishi and Matsumoto [35] that the calculus for S5 is not cut admissible.