LOGICS OF TIME AND COMPUTATION

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1 LOGICS OF TIME AND COMPUTATION

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3 CSLI Lecture Notes No. 7 LOGICS OF TIME AND COMPUTATION Second Edition Revised and Expanded Robert Goldblatt CSLI CENTER FOR THE STUDY OF LANGUAGE AND INFORMATION

4 Copyright 1992 Center for the Study of Language and Information Leland Stanford Junior University Printed in the United States CIP data and other information appear at the end of the book

5 To my daughter Hannah

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7 Preface to the First Edition These notes are based on lectures, given at Stanford in the Spring Quarter of 1986, on modal logic, emphasising temporal and dynamic logics. The main aim of the course was to study some systems that have been found relevant recently to theoretical computer science. Part One sets out the basic theory of normal modal and temporal prepositional logics, covering the canonical model construction used for completeness proofs, and the filtration method of constructing finite models and proving decidability results and completeness theorems. Part Two applies this theory to logics of discrete (integer), dense (rational), and continuous (real) time; to the temporal logic of henceforth, next, and until, as used in the study of concurrent programs; and to the prepositional dynamic logic of regular programs. Part Three is devoted to first-order dynamic logic, and focuses on the relationship between the computational process of assignment to a variable, and the syntactic process of substitution for a variable. A completeness theorem is obtained for a proof theory with an infinitary inference rule. There is more material here than was covered in the course, partly because I have taken the opportunity to gather together a number of observations, new proofs of old theorems etc., that have occurred to me from time to time. Those familiar with the subject will observe, for instance, that in Part Two proofs of completeness for various logics of discrete and continuous time, and for the temporal logic of concurrency, as well as the discussion of Bull's theorem on normal extensions of S4.3, all differ from those that appear in the literature. In order to make the notes effective for classroom use, I have deliberately presented much of the material in the form of exercises (especially in Part One). These exercises should therefore be treated as an integral part of the text. Acknowledgements. My visit to Stanford took place during a period of sabbatical leave from the Victoria University of Wellington which was supported by both universities, and the Fulbright programme. I would like to thank Solomon Feferman and Jon Barwise for the facilities that were made available to me at that time. The CSLI provided generous access to its excellent computer-typesetting system, and the Center's Editor, Dikran Karagueuzian, was particularly helpful with technical advice and assistance in the preparation of the manuscript. vu

8 Preface to the Second Edition The text for this edition has been increased by more than a third. Major additions are as follows. 7, originally concerned with incompleteness, now discusses a number of other metatheoretic topics, including first-order definability, (in)validity in canonical frames, failure of the finite model property, and the existence of undecidable logics with decidable axiomatisation. 9 now includes a study of the " branching time" system of Computational Tree Logic, due to Clarke and Emerson, which introduces connectives that formalise reasoning about behaviour along different branches of the tree of possible future states. Completeness and decidability are shown by the method of filtration in an adaptation of ideas due to Emerson and Halpern. In 10 dynamic logic is extended by the concurrency command a fl/3, interpreted as "a and (3 executed in parallel". This is modelled by the use of "reachability relations", in which the outcome of a single execution is a set of terminal states, rather than a single state. This leads to a semantics for [ a ] and < a > which makes them independent (i.e. not interdefinable via negation). The resulting logic is shown to be finitely axiomatisable and decidable, by a new theory of canonical models and filtrations for reachability relations. A significant conceptual change involves the definition of a "logic" (p. 16), which no longer includes the rule of Uniform Substitution. Logics satisfying this rule are called Uniform, and are discussed in detail on page 23. The change causes a number of minor adaptations throughout the text. A notable technical improvement concerns the completeness proof for S4.3Dum in 8 (pp ). The original Dwm-Lemma has been replaced by a direct proof that non-last clusters in the filtration are simple. This has resulted in some re-arrangement of the material concerning Bull's Theorem, and a simplification of the completeness theorem for the temporal logic of concurrency in 9 (pp ). Other small changes include additional material about the Diodorean modality of spacetime (p. 45), and a rewriting of the basic filtration construction for dynamic logic (p. 114) using a uniform method of proving the first filtration condition that obviates the need to establish any standardmodel conditions for the canonical model. Reformatting the text has provided the opportunity to make numerous changes in style and expression, as well as te, correct typos. I will be thankful for, if not pleased by, information about any further such errors. vuw. ac.nz Vlll

9 Contents Preface to the First Edition vii Preface to the Second Edition viii Part One: Prepositional Modal Logic 1 1. Syntax and Semantics 3 2. Proof Theory Canonical Models and Completeness Filtrations and Decidability Multimodal Languages Temporal Logic Some Topics in Metatheory 48 Part Two: Some Temporal and Computational Logics Logics with Linear Frames Temporal Logic of Concurrency Prepositional Dynamic Logic 109 Part Three: First-Order Dynamic Logic Assignments, Substitutions, and Quantifiers Syntax and Semantics Proof Theory Canonical Model and Completeness 162 Bibliography 169 Index 175

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11 Part One Prepositional Modal Logic

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13 1 Syntax and Semantics BNF The notation of Backus-Naur form (BNF) will be used to define the syntax of the languages we will study. This involves specifying certain syntactic categories, and then giving recursive equations to show how the members of those categories are generated. The method can be illustrated by the syntax of standard propositional logic, which has one main category, that of the formulae. These are generated from some set of atomic formulae (or propositional variables), together with a constant _L (the falsum), by the connective > (implication). In BNF, this is expressed in one line as < formula > ::= < atomic formula > ± < formula >->< formula > The symbol ::= can be read "comprises", or "consists of", or simply "is". The vertical bar is read "or". Thus the equation says that a formula is either an atomic formula, the falsum, or an implication between two formulae. The definition becomes even more concise when we use individual letters for members of syntactic categories, in the usual way. Let $ be a denumerable set of atomic formulae, with typical member denoted p. The set of all formulae generated from $ will be denoted Fma(<?), and its members denoted A, AI, A', B,... etc. The presentation of syntax then becomes Atomic formulae: p <2> Formulae: A Fma(4>) A::=p\L\A-+ A Technically, the recursive equation governs a non-deterministic rewriting procedure for generating formulae, in which any occurrence of the symbol to the left of the ::= sign can be replaced by any of the alternative expressions on the right side. Thus the two occurrences of A in the expression A» A may themselves be replaced by different expressions, and so stand for different formulae. In some BNF presentations, this is emphasised by

14 4 Prepositional Modal Logic using subscripts to distinguish different occurrences of a symbol. Then the above equation is given as Modal Formulae The language of propositional modal logic requires one additional symbol, the "box" D. The BNF definition of the set of modal formulae generated by <? is Atomic formulae: p 6 $ Formulae: A Fma(<I>) A::=p\±\A 1 ->A 2 \DA Possible readings of It is necessarily true that A. It will always be true that A. It ought to be that A. It is known that A. It is believed that A. It is provable in Peano Arithmetic that A. After the program terminates, A. Other connectives These are introduced by the usual abbreviations. Negation: ->A is A» _L Verum: T is -i-l Disjunction: AI VA 2 is (->Ai) > A? Conjunction: Equivalence: A\ A AI AI <-» A-i is is ->(Ai > ->Az) (.Ai» A%) A (A 2 "Diamond": O is Notational Convention In the case that n = 0, the expression just denotes the formula B. Bo A... A B n -i -» B

15 Exercises Syntax and Semantics 5 (1) Decide what OA means under each of the above readings of D. (2) Which of the following should be regarded as true under the different readings of D? OA-+A HA -» cm OT DA-> OA HA V D--A D(A -» )-> (HA -» OB) <X4 A <XB -> O(A A B) Subformulae The finite set Sf(A) of all subformulae of A 6 Fma($) is defined inductively by Sf(p) = {p} Sf(±) = {-L} Sf(A 1 -> Ai) = {A l -* A 2 } U Sf(A 1 ) U Sf(A 2 ) Sf(OA) = {DA} U Sf(A) Schemata We will often have occasion to refer to a schema, meaning a collection of formulae all having a common syntactic form. Thus, for example, by the schema OA-+A we mean the collection of formulae {OB -+B:B(E Fma($)}. Uniform Substitution The notion of a schema can be made more precise by considering uniform substitutions, as follows. Let A and B be any formulae, and p an atomic formula. By the uniform substitution of B for p in A we mean the procedure of replacing each and every occurrence of p in A by B. A formula A' is called a substitution instance of A if it arises by simultaneous uniform substitution for some of of the atomic formulae of A, i.e. if there exist some finitely many atomic formulae pi,...,p n, and formulae Bi,...,B n, such that A' is the result of

16 6 Prepositional Modal Logic simultaneously uniformly substituting?i for pi in A, and BI for p 2 in A, and..., and B n for p n in A. Let.EU = {A' : A' is a substitution instance of A}. Then a schema may be denned as a set of formulae that is equal to EA for some formula A. For example, if A is the formula Dp > p, with p atomic, then SA is what was described above as "the schema DA >.A". Frames and Models A frame is a pair f (S,R), where S is a non-empty set, and R a binary relation on 5: in symbols, R C S x S. A $-model on a frame is a triple M (R, S, V), with V : $ > 2 s. Hence V is a function assigning to each atomic formula p 0 a subset V (p) of 5. Informally, V(p) is to be thought of as the set of points at which p is "true". Generally we drop the prefix $- in discussing models, provided the context is clear. The relation "A is true (holds) at point s in model M", denoted M\=,A, is denned inductively on the formation of A Fma(<!>) as follows. M K P iff s V(p) M s -L (i.e. not M \= s J-) M K (Ai -> A 2 ) iff Ai f= s A! implies M \= s A 2 M (= DA iff for all t 5, s#i implies M\= t A Exercises 1.2 (1) A4 =.-.4 iff A4, A Work out the corresponding truth conditions for A AS, AVB, A <-> B. (2) X (= s OA iff there exists t. S with srt and AI (=«A. Motivations 1. Necessity. Following the dictum of Leibnitz that a necessary truth is one that holds in all "possible worlds", S may be thought of as a set of such worlds, with srt when t is a conceivable alternative to s, i.e. a world in which all the necessary truths of s are realised. DA then means "A is necessarily true", while OA means "A is possible", i.e. true in some conceivable world.

17 1 Syntax and Semantics 7 2. Different notions of necessity can be entertained. Thus logical necessity may be contrasted with physical necessity, the latter taking OA to mean U A is a consequence of the laws of physics". Under this reading, srt holds when t is a scientific alternative to s, i.e. a world in which all scientific laws of s are fulfilled. Hence in our world, D(x < c) is true under the physical reading, where c is the velocity of light and x the velocity of a material body. On the other hand it is logically possible that (x < c) is false. 3. In deontic logic, D means "A ought to be true". srt then means that t is a morally ideal alternative to s, a world in which all moral laws of s are obeyed. If s is the actual world, few would maintain that srs under this interpretation. On the other hand, any world is a logical, and scientific, alternative to itself. 4. Temporal Logic. Here the members of 5 are taken to be moments of time. If srt means "t is after (later than) s", then DA means "henceforth A", i.e. "at all future times A", while OA means "eventually (at some future time) A". Dually, if srt means that t is before s, then D means "hitherto", and so on. Natural time frames (S, R) for temporal logic are given by taking S as one of the number sets w (natural numbers), Z (integers), Q (rationals), or R (reals), and R as one of the relations <,<,>,>. Another interesting possibility is to consider various orderings on the points of four-dimensional Minkowskian spacetime (cf. page 45, and Goldblatt [1980]), or even more general non-linear "branchings" in time (Rescher and Urquhart [1971]). 5. Program states. Reading D as "after the program terminates", S is to be regarded as the set of possible states of a computation process, with srt meaning that there is an execution of the program that starts in state s and terminates in state t. A non-deterministic program may admit more than one possible "outcome" t when started in s. Then OA means "every terminating execution of the program brings about j4", while OA means that the program enables A, i.e. "there is some execution that terminates with A true". At the level of prepositional logic, the notion of state is formally taken to be primitive, as in the theory of automata, Turing machines, etc. A natural concrete interpretation of the notion is possible in quantificational logic, as will be seen in Part Three. Valuations and Tautologies Given a #-model M, and a fixed s S, define true false if s otherwise

18 8 Prepositional Modal Logic Then the function V s : $» {true, false} is a valuation of the atomic formulae, a notion familiar from prepositional logic. Using the standard truth-tables for prepositional connectives, V g is extended to assign a truthvalue to any formula not containing the symbol D. Thus a model on a frame gives rise to a collection {V s : s S} of valuations of <?, while, conversely, such a collection defines the model in which V(p) = {s : V 8 (p) = true}. A formula A is quasi-atomic if either it is atomic (.4 #), or else it begins with a D, i.e. A = OB for some B. If $' is the set of all quasiatomic formulae, then any formula A is constructible from members of $ q U {-L} using the connective >. Hence by using the truth-table for», any valuation V : $ q -* {true, false} of the quasi-atomic formulae extends uniquely to a valuation V : Fma($) > {true, false} of all formulae. A formula A is a tautology if V(A) = true for every valuation V of its quasi-atomic subformulae. Exercise 1.3 Any tautology is a substitution instance of a tautology of prepositional logic (i.e. a D-free tautology). Truth and Validity Formula A is true in model M, denoted M (= A, if it is true at all points in M, i.e. if M\= S A for all s S. A is valid in frame F = (S, R), denoted F (= A, if M (= A for all models M = (S, R, V) based on F. If C is a class of models (respectively, frames), then A is true (respectively, valid) in C, C \= A, if A is true (respectively, valid) in all members of C. A schema will be said to be true in a model (respectively, valid in a frame) if all instances of the schema have that property. More generally, we will use the notations M f= F and F = F, where F C Fma, to mean that all members of F are true in M, or valid in F.

19 1 Syntax and Semantics Exercises 1.4 (1) The following are true in all models, hence valid in all frames. DT D(A -> B) -> (HA -> DB) 0(A -+ )-> (DA -» 05) > OB) O(AVB)<^(OAV OB) (2) Show that the following do not have the property of being valid in all frames. D4-* A DA -+ DOA D(A -> )-> (DA -» 05) OT D(Q4 -» B) V D(DB -» A) D(4 V 5) -> DA V D5 D(DA ^ A)->DA (N.B. some instances of these schemata may be valid, e.g. when A is a tautology. What is required is to find a counterexample to validity of at least one instance of each schema.) (3) Show that OT and the schema DA» OA have exactly the same models. (4) Exhibit a frame in which DJ. is valid. (5) In any model M, (i) if A is a tautology then M \= A; (ii) if M \= A and M (=.4 -> B, then At = B; (iii) if M h ^ then M \= DA. (6) Items (i)-(iii) of the previous exercise hold if M is replaced by any frame f. Ancestral (Reflexive Transitive Closure) Let T = (S, R) be a frame. Define on S the relations R n C S x S, for n > 0, and R*, as follows. sr t iff s = t sr n+1 t iff 3u(sR n u & urt)

20 10 Prepositional Modal Logic Exercises 1.5 (1) R 1 = R. (2) sr*t is Bn > 0 3s 0,, 3s n e 5 with so = s, s n =, and for all i < n, SiRs i+1. (3) R* is reflexive and transitive. (4) If T is any reflexive and transitive relation on S with R C T, then.r* C T. That is,.r* is the smallest reflexive and transitive relation on S that contains R. (5) If SCI, and.r = {(s, s + 1) : s 5}, what is fl*?.r* is often known as the ancestral of.r (from the case that R is the "parent of relation). In view of exercise (4), it is also known as the reflexive transitive closure of R. The notion will play an important role in the logic of programs in Parts Two and Three. Generated Submodels If M = (S, R, V) and t S, then the submodel of M. generated by t is where M t = (S t,r*,v*), 5* - {u e S : tr*u} The structure J* = (S*, R*) is the subframe off = (S, R) generated by t. Exercises 1.6 (1) If R is transitive, then 5* = {t} U {u : tru}. (2) 5* is the smallest subset X of S that contains t and is closed under R, in the sense that u X and urv implies v e X. To evaluate the truth of formula A at point t may require investigating the truth of certain subformulae B of A at various.r-alternatives v of t. But then to determine the truth- value of B at v may require looking at alternatives of v. And so on. 5* comprises all points generated by this process. It is evident that evaluating truth at t will only involve points that are each obtainable from t by finitely many "^-alternations". This is embodied in the

21 1 Syntax and Semantics 11 Submodel Lemma 1.7. If A 6 Fma(<l>), then for any u 5*, M* \= U A iff M K A. Proof. By induction on the formation of A. The case A = p e # follows from the definition of V*, and the case A = _L is immediate. The inductive cases A = (B» D) and yl = D.B are given as exercises. Corollary 1.8. (1) M \= A implies M* \= A. (2) M \= A iff A is true in all generated submodels of M. (3) f (= yl iff A is valid in all generated subframes of T. p-morphisms Let Mi = (Si,Ri,Vi) and M 2 = (S 2,R 2,V 2 ) be models, and / : 5i -» S 2 a function satisfying srit implies f(s)r 2 f(t); f(s)r 2 u implies 3t(sR\t & f(t) u); s Vi(p) iff /(a) e Vb(p). Then / is called a p-morphism from.mi to MI- A function satisfying the first two conditions is a p-morphism from frame (Si,Ri) to frame (S2,R 2 ). p-morphism Lemma 1.9. If A Fma(<I>), then for any s e Si, Proof. Exercise. Mi\=.A iff M,\= f(.)a. If there is a p-morphism / : T\ > F 2 that is surjective (onto), then frame FI is called a p-morphic image of J 7!. p-morphism Lemma If J- 2 is a p-morphic image of F\, then for any formula A, Fi f= A implies F 2 (= A. Proof. Suppose A is false at some point t in some model M 2 (F 2, V 2 ) based on f 2. Take a surjective p-morphism / : Si > S 2 and define a model Mi = (J 7!, Vi) by declaring «Vi(p) iff /(*) Va(p). Then / is a p-morphism from Mi to ^2- Choosing any s with /(s) = i, the first p-morphism Lemma 1.9 gives A false at s in the model Mi based

22 12 Prepositional Modal Logic Exercise 1.11 Let T\ = ({0,1},R) and F-z = ({0},E), where in each case R is the universal relation 5x5. Show that FI\= A implies J^ = -A> (o>, <) =>1 implies F± (= 4. The curious appellation "p-morphism" derives from an early use of the name "pseudo-epimorphism" in this context, and seems to have become entrenched in the literature. Conditions on R The following is a list of properties of a binary relation R that are denned by first-order sentences. 1. Reflexive: Vs(sRs) 2. Symmetric: VsVt(sRt -> trs) 3. Serial: Vs3t(sRt) 4. Transitive: VsVWu(sRt A tru -» sru) 5. Euclidean: VsVtVu(sRt A sru -»tru) 6. Partially functional: VsVtVu(sRt A sru -> t = u) 1. Functional: Vs3\t(sRt) 8. Weakly dense: VsVt(sRt -> 3u(s.Ru A uflt)) 9. Weakly connected: VsVtVu(sRt A sfiw -> tru Vt = uv urt) 10. Weakly directed: VsVtVu(sRt A sflu -* 3v(tRv A w/zw)) Corresponding to this list is a list of schemata: 1. CU-^ 2. A-+OOA 3. D.4 -» OA 4. D OA <-> DA n(^ad^-^ 10. OD^-*DO^ Theorem Let T = (5, fi) be a frame. Then for each of the properties 1-10, ifr satisfies the property, then the corresponding schema is valid int.

23 1 Syntax and Semantics 13 Proof. We illustrate with the case of transitivity. Suppose that R is transitive. Let M be any model on f. To show that M \= HA -» DDA, take any s in M with M f= g DA We have to prove M K HOA, which means or, in other words, srt implies M \=t OA, srt implies (tru implies M (= A). So, suppose srt. Then if tru, we have sru by transitivity, so M \= u A, since M (= s DA by hypothesis. The other cases are left as exercises. Theorem If a frame F (S, R) validates any one of the schemata 1-10, then R satisfies the corresponding property. Proof. Take the case of schema 10. To show R is weakly directed, suppose srt and sru. Let M be any model on f in which V(p) {v : urv}. Then by definition, urv implies M [= p, so M $= u Op, and hence, as sru, M (= s OOp. But then as schema 10 is valid in.f, M. h«^ ^Pi so ^ s^» -M N< ^P- This implies that there exists a v with trv and M (= p, i.e. w V(p), so uflz; as desired. Next, the case of schema 8. Suppose srt. Let M be a model on f with V(p) = {v : t / v}. Then M ^t P, so A'l ^a Dp- Hence by validity of schema 8, M \ s OOp, so there exists a u with sru and Ai fc u Dp. Then for some v, urv and Ai ^t, p, i.e. t; = i, so that u.ri, as needed to show that R is weakly dense. Exercises 1.14 (1) Complete the proofs of Theorems 1.12 and (2) Give a property of R that is necessary and sufficient for F to validate the schema A > DA Do the same for D-L.

24 14 Prepositional Modal Logic First-Order Definability Theorems 1.12 and 1.13 go a long way toward explaining the great success that the relational semantics enjoyed upon its introduction by Kripke [1963]. Frames are much easier to deal with than the modelling structures (Boolean algebras with a unary operator) that had been available hitherto, and many modal schemata were shown to have their frames characterised by simple first-order properties of R. For a time it seemed that prepositional modal logic corresponded in strength to first-order logic, but that proved not to be so. Here are a couple of illustrations. (1) The schema W : D(DA -» A) -> HA is valid in frame (5, R) iff (i) R is transitive, and (ii) there are no sequences SQ,...,s n,... in S with s n Rs n +i for all ra>0. (for a proof cf. Boolos [1979], p.82). Now it can be shown by the Compactness Theorem of first-order logic that there exists a frame satisfying (i) and (ii) that is elementarily equivalent to (i.e. satisfies the same first-order sentences as) a frame in which (ii) fails. Hence there can be no set of first-order sentences that defines the class of frames of this schema. (2) The class of frames of the so-called McKinsey schema M: noa-*ena is not defined by any set of first-order sentences (Goldblatt [1975], van Benthem [1975]). (Both of the above schemata will figure in the discussion of incompleteness in 7, where there is also a further consideration of the question of firstorder definability.) Subsequent investigations demonstrated that prepositional modal logic corresponds to a fragment of second-order logic (Thomason [1975]). Undefinable conditions There are some naturally occurring properties of a binary relation R that do not correspond to the validity of any modal schema. One such is irreqexivity, i.e. Vs~*(sRs). To see this, observe that the class of all frames validating a given schema is closed under p-morphic images (1.10), but the class of irreflexive frames is not so closed. For instance, it contains (u>, <), but not its p-morphic image ({0},{(0,0)}) (cf. Exercise 1.11).

25 Exercise Syntax and Semantics 15 Show that neither of the following conditions correspond to any modal schema. Antisymmetry: VsVt(sRt A trs > s = t), Asymmetry: VsVt(sRt > -<trs). Historical Note The concepts of necessity and possibility have been studied by philosophers throughout history, notably by Aristotle, and in the middle ages. The contemporary symbolic analysis of modality is generally considered to have originated in the work of C. I. Lewis early this century (cf. Lewis and Langford [1932]). Lewis was concerned with a notion of strict implication. He defined "A strictly implies B" as 1(^4 A -i-b), where I is a primitive impossibility operator (later he expressed this as -<O(A A ~<B), where O expresses possibility). He defined a series of systems, which he called SI to 55, based directly on axioms for strict implication. The standard procedure nowadays is to adjoin axioms and rules for D, or O, to the usual presentation of prepositional logic. This approach to modal logic was first used in a paper by Godel [1933]. The model theory described in this section is due to Kripke [1959, 1963]. To learn about the history of modal logic, the reader should first consult the interesting Historical Introduction to Lemmon [1977], where further references may be found.

26 2 I Proof Theory Logics Given a language based on a countable set $ of atomic formulae, a logic is denned to be any set A C Fma($) such that A includes all tautologies, and A is closed under the rule of Detachment, i.e., if A, A -> B e A then B e A. Examples of Logics (1) PL = {A e Fmct($) :Aisa tautology }. (2) For any class C of models, or of frames (including the cases C = {M} and C = {f}), Ac = {A : C h A} is a logic. (3) Fma($) itself is a logic. (4) If {Ai : i & 1} is a collection of logics, then their intersection is a logic. Thus for any F C Fma(<i>) there is a smallest logic containing F, namely the intersection of the collection {A : A is a logic and r C A}. Note that PL is the smallest logic, and Fma($) the largest, in the sense that for any logic A, PLCAC Fma($). Tautological Consequence A formula A is a tautological consequence of formulae Ai,...,An if A is assigned true by every valuation that assigns true to all of AI,..., A n. In particular, a tautological consequence of the empty set of formulae is the same thing as a tautology. 16

27 2 Proof Theory 17 Exercise 2.1 Show that any logic A is closed under tautological consequence, i.e. if AI,..., A n e A, then any tautological consequence of AI,..., A n belongs to A Instead of denning a logic A to include all tautologies, it would suffice to include a set of schemata from which all tautologies can be derived by Detachment, e.g. the schemata » A. Theorems The members of a logic are called its theorems. We write \~A A to mean that A is a ^1-theorem, i.e., Ki A iff A A. Soundness and Completeness Let C be a class of frames, or of models. Then logic A is sound with respect to C if for all formulae A, \~A A implies C (= A. A is complete with respect to C, if, for any A, C \= A implies \~A A. A is determined by C if it is both sound and complete with respect to C. Deducibility and Consistency If r U {4} C Fma($), then A is A-deducible from F, denoted F \- A A, if there exist BO,., 5 n -i F such that \- A Bo -> (Si -» ( (Bn-i -» A) )) (in the case n = 0, this means that \~A A). We write F \f A A when A is not,4-deducible from F. A set r C Fma(<I>) is vl-consj'stent if F \/A -L. A formula ^4 is ^1- consistent if the set {.A} is.

28 18 Prepositional Modal Logic Exercises 2.2 (1) \- A A iff 0 hi A (2) If \- A A then r h^ A. (3) If yl C yl', then F\- A A implies T h^..4. (4) If A T then F \- A A. (5) If F C A and T (-^ /4, then 2\ h A A. (6) If T \- A A and {yl} h^ B, then T h^ 5. (7) Detachment: If T h^ A and r h^ yl -+ B, then T h^ B. (8) Deduction Theorem: F\J {A} \- A B iff F h A A-> B. (9) r 1-^ yl iff there exists a finite sequence A 0,..., A m = A such that for all t < m, either Ai F U A, or else ylfc = (^» ylj) for some j, k < i (i.e. ^4j follows from Aj and ^4^ by Detachment). (10) {A : F \- A A} is the smallest logic containing F U A. (11) Soundness: If M \= s T U A and T \- A A, then M \= s A. (12) If F C yl, then F is yl-consistent if, and only if, yl 7^ Fma($). (13) r 1 is yl-consistent iff there exists a formula A with F \/ A A. (14) F is yl-consistent iff there is no formula A having both F \~ A A and r \- A ^A. (15) r h^ A iff r U {->A} is not ^-consistent. (16) r U {A} is yl-consistent iff F \/ A ->A. (17) If F is yl-consistent, then for any formula A, at least one of F U {A} and P U {~<A} is yl-consistent. Maximal Sets Let M. = (S, R, V) be a model of a logic yl, i.e. M \= A. Associate with each s e S the set r a = {A<= Fma($) :M\=,A}. Then F s is yl-consistent (why?), and moreover, for each formula A, one of A and -<A is in F s. In the next section we will be building models for certain logics. Since we have only a syntactic structure, namely yl, to begin with, we will have to use syntactic entities, such as formulae or sets of formulae, as the points of our models. It turns out that the way to proceed is to use sets of formulae that enjoy the properties possessed by those sets F s naturally associated with points of a given yl-model. A set F C Fma(<l>) is defined to be A-maximal if