Non-Classical Logics

Deductions in Modal Logic

Introduction
	Before Kripke came up with a decent semantics for modal logics we were forced to deal with them only using the tools of proof-theory. We'll be looking at how this is done in the next two lectures. We'll begin by looking at proofs using natural deduction.
Natural Deduction
	Standard Rules We'll assume that we have most of the rules that were introduced in the introductory logic course. Replacement rules: ~~p : : p (DN) Double Negation p v q : : q v p (Com) v Commutation p & q : : q & p (Com) & Commutation p v (q v r) : : (p v q) v r (Assoc) v Association p & (q & r) : : (p & q) & r (Assoc) & Association p v (q & r) : : (p v q) & (p v r) (Dist) v Distribution p & (q v r) : : (p & q) v (p & r) (Dist) & Distribution ~(p & q) : : ~p v ~q (DeM) DeMorgans Law ~(p v q) : : ~p & ~q (DeM) DeMorgans Law p v p : : p (Idem) v Idempotence p & p : : p (Idem) & Idempotence p É q : : ~p v q (IMP) Material Implication p É q : : ~q É ~p (Cont) Contraposition (p & q) É r : : p É (q É r) (Exp) Exportation p É (q É r) : : q É (p É r) (Perm) Permutation p º q : : (p É q) & (q É p) (Equiv) Equivalence p º q : : (p & q) v (~p & ~q) (Equiv) Equivalence p : : p & (q v ~q) (TConj) Tautologous Conjunct p : : p v (q & ~q) (TConj) Contradictory Disjunct Inference rules: p & q (Simp) Simplification p p (Conj) Conjunction ... q p & q p É q (MP) Modus Ponens, or p (AA) Affirming the Antecedent q p É q (MT) Modus Tollens, or ~q (DC) Denying the Consequent ~p p É q (HS) Hypothetical Syllogism, or q É r (ChArg) Chain Argument p É r p v q (DS) Disjunctive Syllogism, or ~p (DC) Denying a Disjunt q p (Add) Addition, or p v q (DA) Disjunctive Addition p v r (CD) Constructive Dilemma p É q r É s q v s p (CP) Conditional Proof ç ... ç q ç p É q ~p (RAA) Reductio ad Absurdum ç ... ç q & ~q ç p Modal Rules Replacement rules: ~<>~p : : []p ~[]~p : : <>p ~[]p : : <>~p ~<>p : : []~p Inference rules: []p (NE) Necessity Elimination p p (PI) Possibility Introduction <>p []p (MRT) Modal Reiteration T p []p (MRT) Modal Reiteration S4 []p <>p (MRT) Modal Reiteration S5 <>p _______ (NI) Necessity Introduction ç ... ç p ç []p The NI rule is an application of a *null assumption* deduction, which is like a CP but with no initial assumption to be discharged at the end. Since there are no assumptions, and the result is proved, the idea is that the consequence must be a necessary truth - it doesn't depend on any contingencies. Here is an example of the use of these inference rules. We wish to prove that ([]p & []q) É [](p & q), a theorem in T: 1. []p & []q Assumption \| 2. []p 1, Simp \| 3. []q 1, Simp \| ___4. Null Ass \| \| 5. p 2, MRT \| \| 6. q 3, MRT \| \| 7. p & q 5, 6, Conj \| 8. [](p & q) 4-7, NI \| 9. ([]p & []q) É [](p & q) 1-8, NI Prove the S4 theorem []p É [][]p: 1. []p Assumption \| ___2. Null Ass \| \| 3. []p 1, MRS4 \| 4. [][]p 2-3, NI 5. []p É [][]p 1-4, CP Prove the S5 theorem <>p É []<>p: 1. <>p Assumption \| ___2. Null Ass \| \| 3. <>p 1, MRS5 \| 4. []<>p 2-3, NI 5. []p É []<>p 1-4, CP Now that we know what is meant by a deduction the formal definition can be introduced. A deduction is a finite sequence of formulae each of which (a) is a premise, or (b) is an assumption which is discharged before the end of the sequence, or (c) not preceded by an undischarged null assumption and follows by replacement or inference rules from previous flae that do not occur in the scope of a previously discharged assumption, or (d) is preceded by an undischrged null assumption and either (i) follows by Replacement or Inference Rules from previous formulae that occur after the nearest un discharged null assumption, or (ii) follows by a modal reiteration rule from a previous formula that does not occur in the scope of a previously discharged assumption, or (e) follows from an immediately preceding discharge of an assumption by the CP Rule, the RAA rule, or the NI rule If a sequence of formulae A₁, A₂, ..., A_n, is a deduction and contains the premisses A_i, ..., A_j, then the deduction is a Proof of the Validity of the argument: A_i ... A_j A_n
Axiomatizations
	Axioms for PL We'll use axiom schemas (or schemata, if you wish to be pedantic) rather than axioms. A schema uses variable for propositions and we get axioms by making substitutions into the axiom schemas. Each axiom schema gives an infinite number of axioms. We'll start with the schemata that describe the standard propositional logic, called PL. The system of schemas is called PS. A1: A É (B É A) A2: (A É (B É C)) É ((A É B) É (A É C)) A3: (~A É ~B) É (B É A) Give examples of axioms derived from these. We need some way to get from one formula to another. For this we use rules of inference. In this case we need just one rule: the familiar rule of Modus Ponens (MP). We say that B follows from A, A É B by MP. We'll build up a concept of proof as we go. Let it be noted here that we're not going to be interested in proofs that take us from premisses to conclusions; we'll only care about proofs that give us theorems in the various systems (i.e. that don't rely upon other formulae being assumed.) So, define a proof as: A finite sequence of formulae such that each formula is either an axiom or follows from previous formulas by MP Here's an example: 1. p É ((p É p) É p) A1 2. (p É ((p É p) É p)) É ((p É (p É p)) É (p É p)) A2 3. ((p É (p É p)) É (p É p)) 1, 2, MP 4. p É (p É p) A1 5. p É p 3, 4, MP More definitions: If <A₁, ..., A_n> is a proof then it is a proof of* A_n* If there is a proof of A then A is a theorem* of PL.* From the proof above we can say that p É p is a theorem of PS, which we can write as \|- _PS (p É p) We can leave the system name off if it's understood. Note that every wff in a proof is a theorem. Write the Modus Ponens rule as: R1: \|- A, \|- (AÉ B) Þ \|- B Note that the only logical signs that appear so far are ~ and É. We can introduce the others by definitions: Def v: (A v B) =: (~A É B) Def &: (A & B) =: ~(A É ~B) Def º: (A º B) =: (A É B) & (B É A) All theorems of PS are PL tautologies by the truth table semantics- so PS is sound wrt that semantics All truth table tautologies of PL are theorems of PS, so PS is complete wrt the semantics. Proof Schemas for PL A proof schema* is a finite sequence of formula schemas such that each is either an axiom schema or follows from previous formula schemas by MP* We can write the previous proof as a proof schema just by switching from propositional letters to propositional variable letters; i.e. replace p by A. At the end we find that we have a theorem schema - A É A We prefer to talk about proof schemas and theorem schemas because they are more general. Keep a list of Theorem Schemas TS1: A É A Since we know that we can create a proof for a theorem schema, we can use them in proofs without qualms. Thus we need a new definition: A proof schema* is a finite sequence of formula schemas such that each is either an axiom schema or a theorem schema or follows from previous formula schemas by MP* Try some proofs: In this way we find a number of theorem schemas which are very useful in proofs TS1: A É A Self-implication TS2: A v ~A Law of excluded middle (LEM) TS3: ~A É (A É B) Duns Scotus law TS4: (B É C) É ((A É B) É (A É C)) Implicative Syllogism (Imp Syl) TS5: ~~A É A Double negation (DN) TS6: A É ~~A Double negation (DN) TS7: ~~(A É ~~A) É A or ~(A & ~A) Law of non-contradiction (LNC) TS8: (A & B) É A Simplification (Simp) TS9: (A & B) É B Simplification (Simp) TS10: A É (A v B) Addition (Add) TS11: B É (A v B) Addition (Add) TS12: ~(A & B) É (~A v ~B) De Morgan's Law (DeM) TS13: ~(A v B) É (~A & ~B) De Morgan's Law (DeM) TS14: (~A v ~B) É ~(A & B) De Morgan's Law (DeM) TS15: (~A & ~B) É ~(A v B) De Morgan's Law (DeM) TS16: A É ((A É B) É B) Modus Ponens (MP) TS17: (A É B) É (~B É ~A) Contraposition (Contra) TS18: A É (B É (A & B)) TS19: (A É B) É ((A É C) É (A É (B & C))) TS20: (A É C) É ((B É C) É ((A v B) É C)) The following rule of inference, called the Adjunction rule (Adj), can also be shown to apply in PS: R2: \|- A, \|- B Þ \|- (A & B) Given that we accept a new rule of inference we have to change the definition of a proof: A proof schema is a finite sequence of formula schemas such that each is either an axiom schema or a theorem schema or follows from previous formula schemas by MP or adjunction For example 1. ~~A É A TS5 2. A É ~~A TS6 3. (~~A É A) & (A É ~~A) 1, 2, Adj 4. ~~A º A 3, Def º TS21: ~~A º A And from TS17 and A3 we get TS22: (A É B) º (~B É ~A) These sorts of equivalences can be used in proofs to replace parts of formula schemas with their equivalents. We make this another rule of inference, the substitutivity of material equivalents: R3: \|- (C º D), \|- A Þ \|- A(D/C) where A(D/C) means that all, some, or no instances of the formula C occurring as a subformula of A are replaced by D In the same way we can come up with many other useful rules of inference. For example: R4: \|- (A É B), \|- ~B Þ \|- ~A Modus Tollens R5: \|- A Þ \|- (A v B) disjunctive addition R6: \|- (A v B), \|- ~A Þ \|- B disjunctive syllogism R7: \|- (A & B) Þ \|- A simplification R8: \|- (A É B), \|- (B É C) Þ \|- (A É C) chain argument R9: \|- ~(A & B), \|- A Þ \|- ~B affirm the negadjunct R10: \|- ~~A Þ \|- A double negation R11: \|- ((A & B) É C) Þ \|- (A É (B É C)) exportation R12: \|- (A É (B É C)) Þ \|- ((A & B) É C) importation R13: \|- (A & (B v C)) Þ \|- ((A & B) v (A & C)) distribution &v R14: \|- (A v (B & C)) Þ \|- ((A v B) & (A v C)) distribution v& R15: \|- (A v B), \|- (A É C), \|- (B É D), Þ \|- (C v D) complex dilemma Axioms for MPL We're only going to be concerned with the new axioms and rule of inference required for the different normal modal logics that we've been looking at. It's convenient to treat the axiom systems last. First we'll introduce another definition - just for the sake of convenience: Def <>: <>A =: ~[]~A The rule of inference we introduce is one that will introduce a modality where it doesn't already appear. Write the Nexessitation rule as: MR1: \|- A Þ \|- []A Now to introduce the axiom schemata for the normal modal logics that we've been looking at. I'll use the Girle numbering for the schemata and also mention their usual names (which are suggestive, but would be ambiguous between axiom and system if we used them.) It's convenient to list them here. A4: ([]A É A) (T or reflexivity) A5: [](A É B) É ([]A É []B) (K or distribution of [] over É) A7: ([]A É [][]A) (S4 or KK-Thesis or positive introspection) A8: (A É []<>A) (Platonic Thesis) A9: (<>A É []<>A) (S5 or negative introspection) A10: ([]A É ~[]~A) (D or consistency) To start with we note that all the logics include the axioms A1, A2, A3, so they are supersets of PS. Given the rules of inference Modus Ponens and Necessitation, we then have K = PS + A5 T = K + A4 S4 = T + A7 Br = T + A8 S5 = T + A9 K4 = K + A7 KBr = K + A8 K5 = K + A9 DT = K + A10 D4 = DT + A7 DBr = DT + A8 D5 = DT + A9 The following are all provable in T. Try proving them. MTS1: []( Aº B) É ([] A º []B) MTS2: []( A& B) º ([] A & []B) MTS3: []A º ~<>~A MTS4: []~ A º ~<> A MTS5: ~<>(A v B) º (~<> A & ~<>B) MTS6: <>( A v B) º (<> A & <>B) MTS7: []( A É B) É (<> A É <>B) MTS8: ([] Av []B) É []( Av B) MTS9: <>(A& B) É (<>A& <>B)