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From to Common Sense Normal Modal K r i p k e S e m a n t i c s Completeness Mehul Bhatt and Correspondence Theory Oliver Kutz Thomas Schneider Lecture 9 Department of & Research Center on Spatial Cognition (SFB/TR 8) University of Bremen

Examples of Modal Examples of Modal Logics

Classic Distinctions between Modalities Modern interpretations of modalities ‣ Mathematical Logic: ‣ Alethic modality: necessity, possibility, contingency, impossibility ‣ The logic of proofs GL: [] A means: In PA it is provable that ‘A’. ‣ distinguish further: logical - physical - metaphysical, etc. ‣ Computer Science: ‣ Temporal modality: always, some , never ‣ LTL: Formal Verification ‣ Deontic modality: obligatory, permissible ‣ X A : in the next moment ‘A’ ‣ Epistemic modality: it is known that ‣ A U B: A is true until B becomes true ‣ Doxastic modality: it is believed that ‣ G = ‘always’ , F = ‘eventually’, ‣ liveness properties state that something good keeps happening: Technically, all these modalities are treated ‣ G F A or also G (B -> F A) in the same way, by using unary modal operators ‣ Linguistics / KR / etc. : Some History A Hilbert system for Modal Logic K

‣ The following is the standard Hilbert system for the modal logic K.

‣ Modern modal logic typically begins with the systems devised by Axioms !1 → (!2 → !1) C. I. LEWIS, intended to model strict implication and avoid the (!1 → !2) → (!1 → (!2 → !3)) → (!1 → !3) paradoxes of material implication, such as the ‘ex falso quodlibet’. !1 → !1 ∨ !2 ‣ “ If it never rains in Copenhagen, then Elvis never died.” !2 → !1 ∨ !2 (!1 → !3) → (!2 → !3) → (!1 ∨ !2 → !3) ‣ (No variables are shared in example => relevant implication) (!1 → !2) → (!1 →¬!2) →¬!1 two classical tautologies ‣ For strict implication, we define A ~~> B by [] (A --> B) ¬¬!1 → !1 ! ∧ ! → ! instead of ! " p in INT ‣ These systems are however mutually incompatible, and no base 1 2 1 logic was given of which the other logics are extensions of. !1 ∧ !2 → !2 !1 → !2 → !1 ∧ !2 ‣ The modal logic K is such a base logic, named after , □(! → ") → (□! → □") new axiom of and which serves as a minimal logic for the class of all its (normal) Box Distribution extensions - defined next via a Hilbert system. ! ! → ! ! Rules 1 1 2 !2 □! new rule of Necessitation

Table 1. AFregesystemsforthemodallogic#

modal logic axioms #4 # + □! → □□! KB # + ! → □♢! GL # + □(□! → !) → □! $4 #4+ □! → ! Some More Modal frege systems $4GrzKripke$4+ Semantics□(□(! → □!) → !) → □!

‣ Hilbert systems for other modal logics are obtained by adding axioms. ‣ A Kripke frameTable consists 2. Frege of systems a set W for, the important set of `possible modal logics worlds’, and a modal logic axioms R between worlds. A valuation β assigns propositional variables to worlds. A pointed model M is a frame, K4 K + ✷p ✷✷p x → together with a valuation and a distinguished world x. KB K + p ✷✸p As in classical logic, if we augment frames with assignments,wearriveatthe GL K + ✷(→✷p p) ✷p Mx = p q Mx = p and Mx = q → → notion of a model. | ∧ ⇐⇒ | | S4 K4+ ✷p p Mx = p q Mx = p or Mx = q → Definition 12. A model| ∨ for⇐⇒ the modal language| is| a pair (!, # ) where S4Grz S4+✷(✷(p ✷p) p) ✷p M = p q if M = p then M = q → → → x | → ⇐⇒ x | x | – ! =($, %) is a frame and ‣ More generally, in a fixed language, the class of all normal modal logics is defined Mx = p Mx = p – # : Var ($ ) |is a¬ mapping⇐⇒ assigning￿| to each propositional variable & a as any set of formulae that !→ # Mx = ✷p for all xRy : My = p set # (&) of worlds| ( ($ )⇐⇒denotes the power set of| $ ). ‣ (1) contains K (2) is closed under substitution and (3) Mx = #✸p exists xRy : My = p ‣ In particular, any normal extension of K contains the Axiom of Box-Distribution: With the notion of| models⇐⇒ we can now define the| semantics of modal for- ✷(p q) (✷p ✷q) mulas: → → → Definition 13. Let ', ( be modal formulas, let ) =($, %, # ) be a model and * $ be a world. Inductively we define the notion of a formula to be satisfied ∈ in ) at world *: – ),* = & if * # (&) where & Var , – ),* ∣= ' if not∈ ),* = ', ∈ – ),* ∣= ¬' ( if ),* =∣ ' and ),* = ( – ),* ∣= ' ∧ ( if ),* ∣= ' or ),* =∣ ( – ),* ∣= □'∨ if for all +∣ $ with (*,∣ +) % we have ),+ = '. ∣ ∈ ∈ ∣ Amodalformula' is satisfiable if there exists a model ) =($, %, # )and aworld* $ such that ),* = '.Dually,' is a modal if for every ∈ ∣ model ) =($, %, # )andevery* $ we have ),* = '. ∈ ∣ It can be shown that the Frege system from the previous sectionisindeeda proof system for the modal logic ,,i.e.,itissoundandcompleteforallmodal tautologies.

14 L10.2 Modal Tableaux

(P) all propositional tautologies

Modal Sat / Taut / Validity A Tableaux(K) (φ ψ) system( φ ψ) for Modal Logic K ￿ → → ￿ → ￿ φφ ψ L10.2 Modal Tableaux (MP) → ‣ Hilbert systemsψ are generally considered difficult to use in a ‣ Modal Sat: A modal formula is satisfiable if there practical way. φ exists a pointed model that satisfies it. (G) (P) all propositional tautologies ‣ There areφ many proof systems for Modal Logics. One of the most popular￿ ones are Semantic Tableaux: (K) (φ ψ) ( φ ψ) ‣ Modal Taut: A￿ formula→ is→ a modal￿ → tautology￿ if it is ‣ refutation based proof system satisfied in all pointed models. Figure 1: Modal logic K φφ ψ ‣ highly developed optimisation techniques (MP) → ψ T ‣ isallows system toK extractplus models (T) φdirectlyφ from proofs ‣ Modal Validity: A formula is valid in a class of ￿ → φ S4 ‣ ispopular system Tinplus particular (4) for￿φ Description￿￿φ Logic based formalisms frames (G)if it a modal tautology relative to that class of → frames. ￿φ ‣ often used for establishing upper bounds for the complexity of a SAT problem for a logic. Figure 2: Some other modal logics Check validity of Figure✷(p 1: Modalq) ( logic✷p K✷q) Box Distribution → → → is a finite sequence of natural numbers. In addition, every formula on the T is system K plus (T) φ φ tableau has a sign Z F, T that indicates the truth-value we currently ￿ → ∈ { } S4 is system T plus (4) ￿φ ￿￿φ expect for the formula in our reasoning. That is, a formula in the modal → tableaux is of the form σZA Figure 2: Some other modal logics where the prefix σ is a finite sequence of natural numbers, the sign Z is in F, T , and F is a formula of modal logic. At this point, we understand a { } isA a finiteTableaux sequence of natural system numbers. Infor addition, Modal every formula Logic on the K prefix σ as a symbolicA Tableaux name for a world insystem a Kripke structure. for Modal Logic K tableau has a sign Z F, T that indicates the truth-value we currently ∈ { } expect for the formula in our reasoning. That is, a formula in the modal Definition 1 (K prefix accessibility) For modal logic K, prefix σ￿ is accessible ‣ In prefixed tableaux, every formula starts with a prefix and a sign tableaux is of the form from prefix σ if σ‣￿ isA of basic the form semanticσn for tableaux some natural for K number is givenn. as follows: Z σZA ‣ We introduce prefixes (denotingModal Tableaux possible worlds) that keep track of accessibility. L10.3  ‣ For every formula of a class α with a top level operator and sign (T or ‣ Prefixes (denoting possible worlds) keep track of accessibility. ‣ Formulae in the tableaux are labelled with T or F. where the prefix σ is a finite sequence of natural numbers, the sign Z is in F for true and false) as indicated, we define two successor formulas α1 and ‣ A prefix is a finite sequence of natural numbers F, T , and F is a formula of modal logic. At this point, we understand a α2: ‣ We differentiate the following four kinds of formulas: { } prefix‣ Formulaeσ as a symbolic in a tableaux name forare alabelled world inwith a Kripke T or F. structure. α α1 α2 β β1 β2 ν ν0 π π0 TA B TA TB TA B TA TB T A TA T A TA ∧ ∨ ￿ ♦ Definition 1 (K prefix accessibility) For modal logic K, prefix σ is accessible FA B FA FB FA B FA FB ￿ ∨ ∧ F ♦A FA F ￿A FA from prefix σ if σ is of the form σn for some natural number n. FA B TA FB TA B FA TB ￿ → → F A TA TA T AEveryFA combinationFA of top-level operator and sign occurs in one of the ¬ ¬ For every formula of a class α with a top level operator and sign (T or above cases. Tableau proof rules by those classes are shown in Figure 3.A For the followingConjunctive cases of formulas we defineDisjunctive one successor formulaUniversal Existential F for‣ trueExample and false) 1 4 7 as9 is indicated, accessible wefrom define 1 4 7 twowhich successor is accessble formulas from 1α 41 etc.and tableau is closed if every branch contains some pair of formulas of the form α2: ‣ These tables essentiallyσTAand encodeσFA.A theproof semanticsfor modal of the logic. logic formula consists of a closed tableau LECTURE NOTES FEBRURARY 18, 2010 α α1 α2 β β1 β2 starting with the root 1FA. TA B TA TB TA B TA TB ∧ ∨ FA B FA FB FA B FA FB σα σβ σν 1 σπ 2 ∨ ∧ (α) (β) (ν∗) (π) FA B TA FB TA B FA TB σα σβ σβ σ ν σ π → → 1 1 2 ￿ 0 ￿ 0 F A TA TA T A FA FA σα ¬ ¬ 2 1 For the following cases of formulas we define one successor formula σ￿ accessible from σ and σ￿ occurs on the branch already 2 σ￿ is a simple unrestricted extension of σ, i.e., σ￿ is accessible from σ and no other prefix on the branch starts with σ￿ LECTURE NOTES FEBRURARY 18, 2010 Figure 3: Tableau proof rules for QML

The tableau rules can also be used to analyze F A A as follows: ￿ → ♦ 1 F A A (1) ￿ → ♦ 1 T ￿A (2) from 1 1 F ♦A (3) from 1 stop No more proof rules can be used because the modal formulas are ν rules, which are only applicable for accessible prefixes that already occur on the branch. If we drop this restriction, we can continue to prove and close the tableau: 1 F A A (1) ￿ → ♦ 1 T ￿A (2) from 1 1 F ♦A (3) from 1 1.1 TA (4) from 2 1.1 FA (5) from 3 ∗ But this is bad news, because the formula A A that we set out to ￿ → ♦ prove in the first place is not even valid in K. Consequently, the side condi- tion on the ν rule is necessary for soundness! As an example proof in K-tableaux we prove A B) (A B): ￿ ∧ ￿ → ￿ ∧

LECTURE NOTES FEBRURARY 18, 2010 Modal Tableaux L10.3 L10.4 Modal Tableaux

ν ν0 π π0 1 F ( A B) (A B) (1) ￿ ∧ ￿T ￿A→TA￿ ∧ T ♦A TA A Tableau1 T A Bsystem for(2) Modal from 1 Logic K A Tableau system for Modal Logic K ￿ ∧ ￿F ♦A FA F ￿A FA L10.4 Modal Tableaux 1 F (A B) (3) from 1 Every combination￿ ∧ of top-level operator and sign occurs in one of the ‣ We give an example derivation of a valid formula: ‣ A tableau Tis nowA expanded according to the following rules. above cases.1 Tableau￿ proof rules by those classes(4) from are 2shown in Figure 3.A ‣ A proof1 startsT ￿ Bwith assuming the falsity of a(5) formula, from 2and succeeds if every 1 F (￿A ￿B) ￿(A B) (1) tableaubranch is closedof the iftableau every closes, branch i.e. contains contains some a direct pair contradiction. of formulas of the form ∧ → ∧ 1 1.1 FA B (6) from 3 1 T A B (2) from 1 σTAand σFA.Aproof for modal logic formula consists of a closed tableau ￿ ∧ ￿ ∧ 1 F (A B) (3) from 1 Conjunctivestarting with the rootDisjunctive1FA. Universal Existential ￿ ∧ 1.1 FA (7) from 6 1.1 FB (8) from 6 1 T ￿A (4) from 2 1 T B (5) from 2 1.1 σα1.1 TA (9) from 4σβ 1.1 TB (10)σν from1 5 σπ 2 ￿ (α) (β) (ν∗) (π) 1.1 FA B (6) from 3 σα1 7 and 9σβ1 σβ2 10σ and￿ν0 8 σ￿π0 ∗ ∗ ∧ σα2 1.1 FA (7) from 6 1.1 FB (8) from 6 Let us prove1 the converse ￿(A B) (￿A ￿B) in K-tableaux: σ￿ accessible from σ and σ￿ occurs∧ on the→ branch∧ already 1.1 TA (9) from 4 1.1 TB (10) from 5 2 σ￿ is a simple unrestricted extension of σ, i.e., σ￿ is accessible from σ and no other prefix 1 F ￿(A B) (￿A ￿B) (1) 7 and 9 10 and 8 on the branch starts with∧ σ￿ → ∧ ∗ ∗ 1 T ￿(A B) (2) from 1 ∧ Let us prove‣ This the converseshows K-validity(A B of:) (✷AA ✷BB) in K✷-tableaux:(A B) 1 F A FigureB 3: Tableau proof rules(3) from for QML 1 ￿ ∧ → ￿ ∧ ￿ ￿ ∧ ￿ ∧ → ∧ 1 F (A B) ( A B) (1) The tableau rules can also be used to analyze F A A as follows: ￿ ∧ → ￿ ∧ ￿ 1 F ￿A (4) from 3 1 F ￿B (5)￿ from→ ♦ 3 1 T (A B) (2) from 1 ￿ ∧ 1.1 FA (6) from1 4F A 1.1AFB(1) (10) from 5 1 F ￿A ￿B (3) from 1 ￿ → ♦ ∧ 1.1 TA B (7) from1 2T A 1.1 TA(2) fromB 1(11) from 2 ∧ ￿ ∧ 1.1 TA (8) from1 7F ♦A 1.1 TA(3) from 1(12) from 11 1 F ￿A (4) from 3 1 F ￿B (5) from 3 1.1 TB (9) from 7stop 1.1 TB (13) from 11 1.1 FA (6) from 4 1.1 FB (10) from 5 1.1 TA B (7) from 2 1.1 TA B (11) from 2 A Tableau6 and 8 system for Modal10 and 13 Logic K ∧ ∧ (Again) No more∗ proof rules can be used because∗ the modal formulas are ν rules, 1.1 TA (8) from 7 1.1 TA (12) from 11 which are only applicable for accessible prefixes that already occur on the 1.1 TB (9) from 7 1.1 TB (13) from 11 Let us‣ tryWe to give prove a refutation￿(A Bof )a satisfiable,￿A ￿ butB: non-valid formula: ‣ A Kripke frame consists of a set W, the set of `possible worlds’, and a branch. If we drop this∨ restriction,→ ∨ we can continue to prove and close the 6 and 8 10 and 13 tableau: ∗ binary relation R between worlds.∗ A valuation β assigns 1 F ￿(A B) ￿A ￿B (1) 1 propositional variables to worlds. A pointed model Mx is a frame, ∨ 1 →F A ∨ A (1) Let us try to provetogether￿(A withB) a valuation￿A ￿ andB: a distinguished world x. 1 T ￿(A B) ￿ → ♦ (2) from 1 ∨ → ∨ ∨ 1 T ￿A (2) from 1 1 F ￿A ￿B (3) from 1 1 F ￿(A B) ￿A ￿B (1) ∨ 1 F ♦A (3) from 1 M∨ x =→p q ∨ Mx = p and Mx = q 1 F ￿A (4) from 3 1 T ￿(A B)| ∧ ⇐⇒ (2) from| 1 | 1.1 TA (4) from 2 B A M∨ = p q M = p or M = q 1 F ￿B (5) from 3 1 F A xB (3) fromx 1 x 1.1 FA (5) from 3 1.1 1.2 ￿ ∨ ￿ | ∨ ⇐⇒ | | 1.1 FA (6) from 4 1 F AMx = p q (4)if fromMx = 3 p then Mx = q ￿ | → ⇐⇒ | | 1.2 FB ∗ (7) from 5 1 F ￿B Mx = p (5)M fromx = p 3 But this is1.1 badTA news,B because the formula(8)￿ fromA ♦ 2A that we set out to 1.1 FA | ¬ ⇐⇒ (6) from￿| 4 ∨ → Mx = ✷p for all xRy : My = p prove in the1.2 firstTA placeB is not even valid in K(9). Consequently, from 2 the side condi- 1.2 FB | ⇐⇒ (7) from 5 | tion on the ν rule is∨ necessary for soundness! 1.1 TA B Mx = ✸p (8)exists fromxRy 2 : My = p ∨ | ⇐⇒ | ‣ AsThis an shows example K-satisfiability proof in K of:-tableaux ✷(A we proveB) ￿✸A A￿B) ✸ ￿B(A B): 1.2 TA B (9) from 2 1.1 TA(10) from 8 1.1 TB (11) from 8 1.2 ∨TA(12)∧ from¬∧ 9 ∧ 1.2→¬TB (13)∧ from 9 ∨ 10 and 6 open open 13 and 7 ∗ LECTURE NOTES FEBRURARY∗ 18, 2010 1.1 TA(10) from 8 1.1 TB (11) from 8 1.2 TA(12) from 9 1.2 TB (13) from 9 10 and 6 open open 13 and 7 This tableau does not close but remains open, which is good news because ∗ ∗ the formula we set out to prove is not valid in K. This tableau does not close but remains open, which is good news because the formula we set out to prove is not valid in K.

LECTURE NOTES FEBRURARY 18, 2010 LECTURE NOTES FEBRURARY 18, 2010 Modal Sat / Taut / Validity Completeness (Sketch)

‣ Modal Sat: A modal formula is satisfiable if there ‣ Soundness: Every K-provable formula is valid in all exists a pointed model that satisfies it. frames.

‣ Modal Taut: A formula is a modal tautology if it is ‣ Completeness: Every K-valid formula is K-provable. satisfied in all pointed models. ‣ Lindenbaum Lemma: Every consistent set of formulae can be extended to a maximally one. ‣ Modal Validity: A formula is valid in a class of ‣ Canonical Models: Construct worlds, valuations, frames if it a modal tautology relative to that class of and accessibility from the MCSs frames. ‣ Truth Lemma: Every consistent set is satisfied in the canonical model.

Canonical Models & Truth Lemma Canonical Models & Truth Lemma

‣ Worlds are maximally consistent sets MCSs ‣ Worlds are maximally consistent sets MCSs ‣ Valuations are defined via membership in the MCSs ‣ Valuations are defined via membership in the MCSs ‣ Accessibility is defined as follows ‣ Accessibility is defined as follows

X R Y iff for every formula A we have X R Y iff for every formula A we have [] A ! X implies A ! Y <> A ! Y implies A ! X

‣ or equivalently ‣ Existence Lemma: For any MCS w, if <> ! w then there is an accessible state v such that ! v. X R Y iff for every formula A we have Note: this is the main difference to the <> A ! Y implies A ! X classical completeness proof. Canonical Models & Truth Lemma Characterising Modal Logics

‣ Worlds are maximally consistent sets MCSs ‣ Most standard modal logics can be characterised via frame validity in ‣ Valuations are defined via membership in the MCSs certain classes of frames. ‣ Accessibility is defined as follows ‣ A logic L is characterised by a class F of frames if L is valid in F, and any non-theorem φ " L can be refuted in a model based on a frame in F. X R Y iff for every formula A we have <> A ! Y implies A ! X modal logic characterising class of frames K all frames ‣ Truth Lemma: In the canonical model M we have K4 all transitive frames M, w ⊧ iff ! w. KB all symmetric frames 1 Proof is almost immediate GL R transitive, R− well-founded S4 all reflexive and transitive frames from Existence Lemma and 1 S4Grz R reflexive and transitive, R− Id well-founded the Definition of R −

Correspondence Theory: Example Gödel–Tarski–McKinsey translation

‣ We sketch as an example the correspondence between the modal logic ‣ The Gödel–Tarski–McKinsey translation T, or simply Gödel translation, axiom that defines the logic K4 and the first-order axiom that characterises is an embedding of IPC into S4, or Grz. the class of transitive frames: T(p)=✷p Let W, R be a frame. R is transitive if x, y, z W.xRyand yRz imply xRz ￿ ￿ ∀ ∈ T( )= Theorem. ✷p ✷✷p is valid in a frame W, R iff R is transitive ⊥ ⊥ → ￿ ￿ T(ϕ ψ)=T(ϕ) T(ψ) ‣ Proof. ∧ ∧ ‣ (1) It is easy to see that the 4-axiom is valid in transitive frames. T(ϕ ψ)=T(ϕ) T(ψ) ∨ ∨ ‣ (2) Conversely, assume the 4-axiom is refuted in a model Mx = T(ϕ ψ)=✷(T(ϕ) T(ψ)) → → ✷p ✸✸ p p ∧ ¬ ¬ ‣ Here, the Box Operator can be read as `it is provable’ or `it is constructable’. x y z ‣ The frame can clearly not be transitive. p ✸ p ∧ ¬ Gödel–Tarski–McKinsey translation Rules: Admissible vs. Derivable

‣ Theorem. The Gödel translation is an embedding of IPC into S4 and Grz. ‣ The distinction between admissible and derivable rules was introduced by PAUL LORENZEN in his 1955 book “Einführung in die operative Logik I.e. for every formula ϕ IPC T(ϕ) S4 T(ϕ) Grz ∈ ⇐⇒ ∈ ⇐⇒ ∈ und Mathematik”. ‣ Informally, a rule of inference A/B is derivable in a logic L if there is an ‣ Applications: L -proof of B from A. ‣ modal companions of superintuitionistic logics ‣ If there is an L -proof of B from A, by the rule of substitution there also L NExt(S4):ρ(L)= A L T(A) is an L -proof of σ(B) from σ(A), for any substitution σ. For admissible ∈ { | ￿ } rules this has to be made explicit. ‣ A rule A/B is admissible in L if the set of theorems is closed under the rule, i.e. if for every substitution σ: L ⊢ σ(A) implies L ⊢ σ(B) . For this we usually write as: A B |∼

Rules: Admissible vs. Derivable CPC is Post complete ‣ A logic L is said to be Post complete if it has no proper consistent extension. ‣ Therefore the addition of admissible rules leaves the set of theorems of a logic intact. Whilst they are therefore`redundant’ in a sense, they can ‣ Theorem. Classical PC is Post complete significantly shorten proofs, which is our main concern here. ‣ Proof. (From CHAGROV & ZAKHARYASCHEV 1997) ‣ Example: Congruence rules. ‣ Suppose L is a logic such that CPC # L and pick some formula φ ! L - ‣ The general form of a rule is the following: CPC. φ1,...,φn ‣ Let M be a model refuting φ. Define a substitution σ by setting: φ if M = pi ‣ σ(pi):= ￿ | If our logic L has a ‘well-behaved conjunction’ (as in CPC, IPC, and most ￿ otherwise modal logics), we can always rewrite this rule by taking a conjunction ⊥ and assume w.l.o.g. the following simpler form: ψ ‣ Then σ(φ) does not depend on M, and is thus false in every model. φ ‣ We therefore obtain σ(φ) ! ⟘ ! CPC. ‣ We are next going to show that in CPC (unlike many non-classical logics) ‣ But since σ(φ) ! L, we obtain ⟘ ! L by MP, hence L is inconsistent. QED the notions of admissible and derivable rule do indeed coincide! CPC is 0-reducible CPC is Structurally Complete ‣ A logic L is said to be structurally complete if the sets of admissible and derivable rules coincide. ‣ Theorem. Classical PC is structurally complete. ‣ ‣ A logic L is 0-reducible if, for every formula φ " L, there is a Proof. variable free substitution instance σ(φ) " L. ‣ It is clear that every derivable rule is admissible. ‣ Theorem. Classical PC is 0-reducible. ‣ Conversely, suppose the rule: φ 1 ,..., φ n is admissible in CPC, but not derivable. φ ‣ Proof. ‣ This means that, by the Deduction Theorem φ1 ... φn φ CPC ‣ Follows directly from the previous proof. QED ∧ ∧ → ￿∈ ‣ Since CPC is 0-reducible, there is a variable free substitution instance which is ‣ Note: K is Post-incomplete and not 0-reducible. false in every model, i.e. we have σ(φ1) ... σ(φn) σ(φ) CPC ∧ ∧ → ￿∈ ‣ This means that the formulae σ ( φ i ) are all valid, while σ ( φ ) is not. ‣ Therefore, we obtain: σ(φ1) ... σ(φn) CPC ∧ ∧ ∈ ‣ But σ ( φ ) CPC , which is a contradiction to admissibility. QED ￿∈

Admissiblity in CPC is decidable Admissible Rules in IPC and Modal K ‣ Corollary. Admissibility in CPC is decidable. ‣ ‣ Proof. Pick a rule A/B. This rule is admissible if and only if it as well as modal logics behave quite differently with is derivable if and only if A ! B is a tautology. respect to admissible vs. derivable rules (as well as many other meta-logical properties) ‣ Some Examples: Congruence Rules: ‣ E.g., intuitionistic logic is not Post complete. Indeed there is a continuum p q p q p q of consistent extension of IPC, namely the class of superintuitionistic ↔ ↔ ↔ p r q r p r q r p r q r logics; the smallest Post-complete extension of IPC is CPC. ∧ ↔ ∧ ∨ ↔ ∧ → ↔ → ‣ Unlike in CPC, the existence of admissible but not derivable rules is quite p q p q p q common in many well known non-classical logics, but there exist also ↔ ↔ ↔ r p r q r p r q r p r q examples of structurally complete modal logics, e.g. the Gödel-Dummett ∧ ↔ ∧ ∨ ↔ ∧ → ↔ → logic LC. ‣ We next give some examples for IPC and modal K. ‣ if these are admissible in a logic L (they are derivable in CPC, IPC, K), the principle of equivalent replacement holds i.e.: ‣ Finally, we will discuss how the sets of admissible rules can be presented in a finitary way, using the idea of a base for admissible rules. ψ χ L implies φ(ψ) φ(χ) L ↔ ∈ ↔ ∈ Admissible Rules in Modal Logic Admissible Rules in Modal Logic

‣ The following rule is admissible, e.g., in the modal logics K, D, K4, S4, GL. ‣ The following rules is admissible, e.g., in the modal logics K, D, K4, S4, GL. ‣ It is derivable in S4, but it is not derivable in K, D, K4, or GL. ‣ It is derivable in S4, but it is not derivable in K, D, K4, or GL.

✷p (✷) ✷p p (✷) p ‣ Proof. (Derivability in S4 and K): ‣ Proof. (Admissibility in K): ‣ It is derivable in S4 because ✷ p p is an axiom: ✷p p → → Assume (F, R), β,x = σ(p) for some frame (F, R). ‣ Assume a proof for p and apply MP once. Pick some￿ y F ,set￿￿G| = F y , F S = R y,￿∈ x , and γ(p)=∪β{(p)} for all p.Then: ‣ It is not derivable in K: The formula n p ! p ∪ {￿ ￿} x σ(p) G is refuted in the one point irreflexive frame. ¬p p ¬ ‣ (G, S), γ,y = ✷σ(p)whilstwestillhave y ✷σ(p) Note that the classical Deduction Theorem ￿ ￿ | ¬ does not hold in modal logic! (G, S), γ,x = σ(p) ¬ ￿ ￿ | ¬

Admissible Rules in Modal Logic Admissible Rules in Modal Logic

‣ The following rules is admissible, e.g., in the modal logics K, D, K4, S4, GL. ‣ The following rule is admissible in every normal modal logic. ‣ It is derivable in S4, but it is not derivable in K, D, K4, or GL. ‣ It is derivable in GL and S4.1, but it is not derivable in K, D, K4, S4, . ‣ It is not admissible in some extensions of K, e.g.: K⨁ $

✷p ✸p ✸ p (✷) (✸) ∧ ¬ p ⊥ ‣ Proof. (Non-admissibility in K⨁ $): ‣ Löb’s rule (LR) is admissible (but not derivable) in the basic modal logic K. ‣ K⨁ $ is consistent because it is satisfied in the K ✷ one point irreflexive frame to the right. ⊕ ⊥ ‣ It is derivable in GL. However, (LR) is not admissible in K4. ‣ It follows in particular that a rule admissible in a logic L need not be admissible in its extensions. ✷p p (LR) → p Admissible Rules in IPC (KPR) is Not Derivable: Proof

‣ The following rule is admissible in IPC, but not derivable: ‣ Harrop’s rule is derivable in IPC if the following is a tautology: ‣ Kreisel-Putnam rule (or Harrop’s rule (1960), or independence of premise rule). ( p q r) ( p q) ( p r) p q r ¬ → ∨ → ¬ → ∨ ¬ → (KPR) ¬ → ∨ ‣ The following Kripke model for IPC gives a counterexample: ( p q) ( p r) ¬ → ∨ ¬ → ‣ (KPR) is admissible in IPC (indeed in any superintuitionistic logic), q r r q r q ¬p ¬ p¬ p¬ but the formula: ¬ ¬ ( p q r) ( p q) ( p r) ¬ → ∨ → ¬ → ∨ ¬ → ‣ is not an intuitionistic tautology, therefore (KPR) is not derivable, and IPC is not structurally complete. ‣ Note: IPC has a standard Deduction Theorem (only intuitionistically p q r valid axioms are used in the classical proof) ¬ → ∨

Decidability of Admissibility Some Notes on Bases

‣ Is admissibility decidable for IPC? RYBAKOV gave a first postive answer ‣ Is admissibility decidable? I.e. is there an algorithm for recognizing in 1984. He also showed: admissibility of rules? (FRIEDMAN 1975) ‣ admissible rules do not have a finite basis; ‣ Yes, for many modal logics, as Rybakov 1997 and others showed. ‣ gave a semantic criterion for admissibility. ‣ It is typically coNExpTime-complete (JEŘÁBEK 2007). ‣ Admissibility in intuitionistic logic can also be reduced to admissibility ‣ Decidability of admissibility is a major open problem for modal logic K. in Grz using the Gödel-translation. ‣ Recent results by WOLTER and ZAKHARYASCHEV (2008) show e.g. the ‣ IEMHOFF 2001: there exists a recursively enumerable set of rules as a undecidability of admissibility for modal logic K extended with the basis. universal modality. ‣ Without proof, we mention that the rule below gives a singleton basis for the modal logic S5. ✸p ✸ p (✸) ∧ ¬ ⊥ Summary Literature

‣ ALEXANDER CHAGROV & MICHAEL ZAKHARYASCHEV, Modal Logic, ‣ We have introduced the modal logic K and the intuitionistic Oxford Logic Guides, Volume 35, 1997. calculus IPC. ‣ MELVIN FITTING, Proof Methods for Modal and Intuitionistic Logic, ‣ Have shown how they can be characterised by certain classes Reidel, 1983.  of Kripke frames. ‣ TILL MOSSAKOWSKI, ANDRZEJ TARLECKI, RAZVAN DIACONESCU, What ‣ Discussed several proof systems for these logics. is a logic translation?, Logica Universalis, 3(1), pp. 95–124, 2009. ‣ Introduced translations between logics and discussed how ‣ MARCUS KRACHT, Modal Consequence Relations, Handbook of Modal these can be used to transfer various properties of logics. Logic, Elsevier, 2006. ‣ Discussed the difference between admissible and derivable ‣ ROSALIE IEMHOFF, On the admissible rules of intuitionistic rules in modal, intuitionistic, and classical logic. propositional logic, Journal of Symbolic Logic, Vol. 66, 281–294, 2001.