5.4 Natural deduction

that satisfies all formulas of Γ and every world in it. Fortunately, we have a much more usable approach, which is an extension, respectively adaptation, of the systems of natural deduction met in Chapters 1 and 2. Recall that we presented natural deduction proofs as linear representations of proof trees which may involve proof boxes which control the scope of assumptions, or quantifiers. The proof boxes have formulas and/or other boxes inside them. There are rules which dictate how to construct proofs. Boxes open with an assumption; when a box is closed – in accordance with a rule – we say that its assumption is discharged. Formulas may be repeated and brought into boxes, but may not be brought out of boxes. Every formula must have some justification to its right: a justification can be the name of a rule, or the word ‘assumption,’ or an instance of the proof rule copy; see e.g. page 13.

Natural deduction works in a very similar way for modal logic. The main di erence is that we introduce a new kind of proof box, to be drawn with dashed lines. This is required for the rules for the connective . The dashed proof box has a completely di erent role from the solid one. As we saw in Chapter 1, going into a solid proof box means making an assumption. Going into a dashed box means reasoning in an arbitrary accessible world. If at any point in a proof we have φ, we could open a dashed box and put φ in it. Then, we could work on this φ, to obtain, for example, ψ. Now we could come out of the dashed box and, since we have shown ψ in an arbitrary accessible world, we may deduce ψ in the world outside the dashed box.

Thus, the rules for bringing formulas into dashed boxes and taking formulas out of them are the following:

Wherever φ occurs in a proof, φ may be put into a subsequent dashed box.

Wherever ψ occurs at the end of a dashed box, ψ may be put after that dashed box.

We have thus added two rules, introduction and elimination:

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In modal logic, natural deduction proofs contain both solid and dashed boxes, nested in any way. Note that there are no explicit rules for , which must be written ¬ ¬ in proofs.

The extra rules for KT45 The rules i and e are su cient for capturing semantic entailment of the modal logic K. Stronger modal logics, e.g. KT45, require extra rules if one wants to capture their semantic entailment via proofs. In the case of KT45, this extra strength is expressed by rule schemes for the axioms T, 4 and 5:

An equivalent alternative to the rules 4 and 5 would be to stipulate relaxations of the rules about moving formulas in and out of dashed boxes. Since rule 4 allows us to double-up boxes, we could instead think of it as allowing us to move formulas beginning with into dashed boxes. Similarly, axiom 5 has the e ect of allowing us to move formulas beginning with ¬ into dashed boxes. Since 5 is a scheme and since φ and ¬¬φ are equivalent in basic modal logic, we could write ¬φ instead of φ throughout without changing the expressive power and meaning of that axiom.

Definition 5.20 Let L be a set of formula schemes. We say that Γ L ψ is valid if ψ has a proof in the natural deduction system for basic modal logic extended with the axioms from L and premises from Γ.

Examples 5.21 We show that the following sequents are valid:

1. |−K p q → (p q).