Modal Logic PHI 134
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Module 3 Basic Syntax and Semantics for Modal Sentential Logic G J Mattey June 11 2009 Contents 1 Syntax of Modal Sentential Logic 11 Expressions ofMSL 12 Rules of Formation for MSL 2 Informal Interpretations of MSL Operators 21 EI as a Necessity Operator 22 0 as a Possibility Operator 23 3 as a Strict Implication Operator 3 InterDe nition of Modal Operators 31 Reduction to One Primitive Modal Operator 32 Other Modal Operators 321 Intensional Disjunction 322 Impossibility 323 Consistency 324 Strict Equivalence 4 Basic Semantics for MSL 41 TruthValue Assignments 42 Formal Semantics for MSL 421 Generalized ValuationFunctions 422 The Accessibility Relation 423 Semantical Rules 43 MetaLogical Properties and Relations in MSL 431 Semantical Entailment in a Frame 432 Semantical Equivalence in a Frame 433 Validity in a Frame 5 Conclusion In this module we will begin our examination of Madal Sentential Lagic or MSL Modal Sentential Logic is an extensian of nonmodal Sentential Logic All the sentences of SL are sentences of MSL and some sentences of MSL are not sentences of MSL The rst section of the module will describe the syntax of MSL The set of expressions of SL will be expanded by the addition of modal operators and the set of sentences composed of those expressions will be expanded by the addition of sentences containing modal operators The brief second section shows how each of the modal operators can be de ned in terms of the others The third and nal section sets out the basic elements of semantics of MSL In later modules the basic semantics will be re ned in order to produce various semantical systems of modal logic 1 Syntax of Modal Sentential Logic Historically a number of different modal sentential languages have been constructed by taking different modal operators as primitive1 Most versions of modal logic syntax add to the expressions of SL two one place operators the box EI and the diamond ltgt 2 We will treat these operators as primitive and add to the set of primitive operators the twoplace strict implication operator 3 A sentence whose main logical operator is a modal operator will be called a madal sentence3 11 Expressions of MSL We extend the expressions of SL to obtain the vocabulary of MSL o An in nitely large set ofsentence letters A B C ZA1 B1 ZlA2 B2 o A sentential constant J o Two punctuation marks and o A set of ve tmth tnctianal apemtars A V D and E o A set of three madal apemtars EI ltgt 3 Every expression of SL is an expression of MSL but not every expression of MSL is an expression of SL 12 Rules of Formation for MSL Corresponding to the three new modal operators are three new rules of formation rules 8 through 10 H All sentence letters are sentences of MSL N J is a sentence ofMSL 9 If a is a sentence of MSL then a is a sentence of MSL 5 If a and are sentences of MSL then a A is a sentence of MSL 1An operator is primitive when it is speci ed in the original list of expressions of the vocabulary to which it belongs An operator is de ned when it does not appear in the original list but is instead speci ed through the use of a de nition Examples will be given in the next section beginning on page 3 2Hughes and Cresswell in A New Intraduetian t0 Madal Lagic use L and M respectively 3The main logical operator of a sentence is the operator featured in the last rule of formation that generates the sentence E If a and are sentences of MSL then a V is a sentence of MSL 0 If a and are sentences of MSL then a 3 is a sentence of MSL gt1 If a and are sentences of MSL then a E is a sentence of MSL 9quot If a is a sentence of MSL then Bar is a sentence ofMSL 0 If a is a sentence of MSL then Oar is a sentence ofMSL H 53 If a and are sentences of MSL then a 3 is a sentence of MSL 11 Nothing else is a sentence of MSL We will stipulate that as an abbreviation outermost parentheses may be omitted from any sentence after the application of the formation rules has been completed Since the formation rules for MSL include all those for SL every sentence of SL is a sentence of MSL But the MSL rules of formation allow the generation of sentences of MSL that are not sentences of SL Thus MSL is an extension of SL 2 Informal Interpretations of MSL Operators In the period before contemporary formal semantics was developed modal operators were taken to repre sent modal terms in natural languages such as English4 This can be called their intended interpretation In themselves the modal operators and sentences formed from them have no meaning but since the devel opment of possibleworlds semantics they have been given precise meanings by formal semantical rules Before introducing these rules we will discuss the intended interpretations of the modal operators 21 EI as a Necessity Operator In the most common informal interpretation the box EI is supposed to represent necessary truth5 For ex ample we might wish to represent the necessary truth of the truthfunctionally valid SL sentence A A A by a ixing a box to it IA A A 22 0 as a Possibility Operator The diamond generally is taken to represent possibility A simple example is a case where it is a ixed to the left of a sentence letter such a A In the semantical system SI all sentence letters are true on at least one interpretation so we may wish to say that A is at least possibly true which can be expressed as follows 0A 4Or they were interpreted using matrices as described in Module 1 5Analogous modal operators obeying exactly the same semantical rules have been used to represent knowledge belief obli gation future times and other notions These informal interpretations will be discussed extensively in Module 6 and following modules On a row of a truthtable on which A is false its negation A is true So we might say that the negation is possibly true in which case A is possibly false We can state that A is possibly true and possibly false in this way 0A A ltgtA In such a case the sentence is interpreted in English as representing a state of affairs that is cantingent 23 as a Strict Implication Operator The twoplace shhook operator is intended to represent strict implication This is a relation between two sentences that holds when it is impossible for the rst to be true and the second false When the truthvalues are assigned as in the semantics for SL this amounts to the same thing as saying that the rst semantically entails the second In SL the sentence A A B semantically entails A So we may wish to write AAB 3A In the next section these informal notions will be given a rigorous treatment But it can be seen already that the modal operators can be used to express in MSL some of the most important semantical properties and relations of SL sentences6 This was Lewis s original goal in developing modal logic7 3 InterDe nition of Modal Operators The modal language MSL presented here contains a modest number of modal operators in its vocabulary Each of the operators in the vocabulary of MSL is treated as primitive or unde ned The vocabulary could just as well have been stated using just one primitive modal operator The other operators could then be de ned in terms of that one primitive operator Despite the fact that EI ltgt and 3 are primitive in MSL we will take a look at how they could be de ned in terms of one another Then we will examine some other modal operators that can be de ned 31 Reduction to One Primitive Modal Operator The de nitions are motivated by the intended interpretations of the operators Thus if we think of the El as expressing necessity the 0 as expressing possibility and the as expressing negation we might say that EIA means that A is necessary Given that what is necessary is not possibly not the case it seems that EIA should be taken as expressing the same thing as ltgtA It turns out that the two sentences are equivalent given the semantics to be developed later On the other hand with the same intended interpretations we might wish to say that ltgtA expresses that same thing as IA What is possible is not necessarily not the case Again these two sentences will proved to be equivalent in the semantical system we will develop The symbol for strict implication is intended to indicate the impossibility that the antecedent is true and the consequent false Thus A 3 B could be de ned as ltgtA A B or as IA D B All three of these will prove to be semantically equivalent More generally with 0 as primitive Dar df ltgtar 6As will be seen they can also express semantical properties and relations of MSL sentences 7For example The strict implication p 3 q means It is impossible that p be true and q false 3 A Survey afSymbalic Lagic pp 332333 01 3 df ltgta With EI as primitive Oar df Elar a 3 df Elar D We could also take the shhook as a primitive operator de ning the box and the diamond in terms of it Exercise De ne the two primitive oneplace operators in terms of the shhook challenging 32 Other Modal Operators A number of other modal operators may be de ned in MSL or taken as primitive 321 IntensionalDisjunction In his earliest papers on modal logic written in 1912 Lewis worked with an operator he called intensional or dilemmatic disjunction8 An intensional disjunction a X is true just in case the falsehood of one of the disjuncts logically or strictly implies the truth of the other9 Another way Lewis put it is that it is impossible for both disjuncts to be false10 Lewis s stock example of an intensional disjunction was Either Matilda does not love me or I am beloved We may symbolize this sentence in nonmodal Predicate Logic using a oneplace predicate Lx for x loves me and the letter m for Matilda Then we have Lm V 3xLx The falsehood of Lm implies the truth of Lm which in turn implies the truth of 3xLx The falsehood of 3xLx implies the truth of Lm This account of intensional disjunction suggests the following de nition in terms of the strict implication operator which Lewis in fact adopted a X df ar 3 We can also de ne a X in terms of the other two primitive operators of the language MSL a X df ltgta a X df Elar V The reader might notice that the rst de nition parallels precisely the de nition in nonmodal Sentential Logic of V in terms of and D The de nition of 3 in terms of and V is paralleled by the following de nition ar 3 df aX 322 Impossibility By 1914 Lewis employed a oneplace primitive operator which was intended to express impossibility11 From this starting point we can give the following de nitions 8 Implication and the Algebra of Logic and A New Algebra of Implications 9The symbol X is peculiar to this text There is no standard symbol for intensional disjunction 10 The Calculus of Strict Implication 1914 11 The Matrix Algebra for Implications as well as in A Survey afSymbalic Lagic 1918 Note that the symbol in the present typeface is slightly thicker than the symbol for negation ltgta df a Dar df ar a 3 df a 01 X df 01 323 Consistency At the same time Lewis introduced the impossibility operator he introduced a de ned operator for consis tency o Two sentences 1 and are said to be consistent just in case it is possible that they both be true Consistency can be de ned in terms of other operators as follows a o df 01 A 01 0 df E01 A 01 0 df 01 3 01 0 df 01 X 01 0 df 01 A 324 Strict Equivalence The nal modal operator introduced by Lewis was strict equivalence symbolized here by 12 We shall here only give the de nition of strict equivalence in terms of the three primitive modal operators of this text 01 df ltgta A ltgt A a arS 3 df ar3 D 3ar ar dfar 3 A 3ar Strict equivalence is the only one of the operators primitive or de ned that cannot be made primitive without some loss of expressive power This is analogous to a result for SL according to which the material biconditional 5 even with is not su icient to de ne the other nonmodal operators13 Exercise Given the informal interpretations of the operators in question show how to de ne the shhook in terms of the circle 4 Basic Semantics for MSL The semantics for Modal Sentential Logic is a generalization of the semantics for nonmodal Sentential Logic The semantical rules of SI assign a truthvalue to a nonatomic sentence based on the truthvalues of its components From the truthvalue assignment TVA made to the sentence letters by the interpretation the truthvalue of the sentence results immediately from the application of the semantical rules Thus if an interpretation assigns to the sentenceletter A the value T then the value of A is F etc In the last 12Lewis s symbol was He used that symbol for both strict equivalence and de nitional identity which we would now say is mixing the objectlanguage with the mewlanguage 13See Geolfrey Hunter Metalagic 1996 p 69 module it was proved that the operators and their semantical rules of SI are truthfunctional We will now say that in such a case the operators and the semantical rule governing them are directly truthfunctional For modal sentences sentences whose main operator is modal the situation is dilTerent the standard semantical rules for modal operators are nut directly truthfunctional We cannot obtain the truthvalue of EIA on an interpretation simply from a single value for A resulting from the TVA made by the interpreta tion The semantical rules for sentences whose main operator is a oneplace modal operator are not directly truthfunctional in the way the oneplace negation operator is Nonetheless the determination of the value of the modal sentence EIA will turn out to be a function of possibly more than one truthvalue assigned by the given interpretation to A and so semantics for modal sentential logic may be said to be indirectly truthfunctional in a way to be described fully below 41 TruthValue Assignments We may depict in the standard truthtable format a partial TVA for an SL sentence So for example if an interpretation I assigns T to A and F to B we have a table that looks like this A B T F And by SRA we get A B A A B T F F where the value of the conjunction is F because the assignment to one of its conjuncts is F Note that this determination is entirely selfcontained No reference is made to any other TVA but the one which assigns T to A and F to B To make another assignment would be to give a diITerent interpretation since there is no distinction in SI between an interpretation and the truthvalue assignments it makes The truth and falsehood of a sentence is relative to a single given truthvalue assignment and hence to a single given interpretation A diITerent interpretation with a diITerent TVA gives a diITerent result for the sentence A A B ABAAB TTT Once again no reference is made to any other truthvalue assignment It is as if each TVA represents a world of its own In terms of the formal semantics we might say that there are at least two interpreta tions 1 and I such that vlAT and v1rBF while v1AT and vlrAT Then v1rA A BF and VI A A BT Although the determination of the truthvalues of sentences of SL depends entirely on an interpretation and its TVA other semantic properties of the sentence can be determined only by looking at more than one interpretation of it Whether a sentence is truthfunctionally valid for example requires that we look at the values for the sentence under diITerent TVAs Thus the sentence A A B is truthfunctionally indeterminate as we have already shown by producing a TVA under which it is true and a diITerent one in which it is false 14 14A sentence is tmth metianally indeterminate just in case there is an interpretation that assigns it the value T and an interpre tation that assigns it the value F All sentence letters are truthfunctionally indeterminate Now let us consider the disjunction A V A which is truthfunctionally valid We can say this because there are exactly two possible partial TVAs making an assignment to A producing the following two tables15 A AVA T T A AVA F T Granting that the assignments made to any other sentence letters are irrelevant to the truth or falsehood of A V A we can say that it is true on all interpretations which just are truthvalue assignments and so it is truthfunctionally valid Truthfunctional validity and the related notions are concepts which are not expressible in the syntax of SL itself but only in the metalanguage we use to talk about SL The formal metalinguistic notation for truthfunctional truth F51 A V NA uses a symbol that is not included in the syntax of SL But we can express something analogous in MSL IA V A One way to understand the modal operators is as simulating in the modal objectlanguage MSL metalogical properties and relations This is done by extending the objectlanguage SL to include modal sentences whase truthvalues an an interpretatian can depend an multiple truthvalue assignments Such an extension allows the representation in the object language MSL of semantical properties and relations that are more general than simple truth and falsehood Though there are three modal operators de ned in the syntax of MSL we shall limit our initial discussion to the oneplace modal operators EI and ltgt From a given sentence a we can form the necessitysentence Bar in MSL We will express in the formal semantics the informal meaning of the necessity sentence accord ing to which Bar is true just in case a is necessarily true A possibilitysentence Oar will be true in the formal semantics just in case it is possible that a is true Again it must be stressed that these modal operators can be given other readings From a formal standpoint to interpret necessity and possibilitysentences we need a way of making reference to multiple truthvalue assignments within a single interpretatian The semantics for Modal Sen tential Logic does not identify interpretations with TVAs in the manner of the semantics for nonmodal Sentential Logic In the semantics for MSL a single interpretation must be able to allow more than one distinct truthvalue assignment to the sentence letters This means that the application of an SL valuation function to a sentence a V1a is not adequate for the semantics for MSL To accommodate multiple truthvalue assignments we require what are most commonly known as pas sible warlds 16 We may think of possible worlds as locations with respect to which truthvalue assign ments are made Arbitrary worlds will be indicated by the metavariable W with or without primes or a lowercase italic alphabetic subscript Speci c worlds will be indicated by w with or without primes or 15Of course this result is usually depicted in a single table but two are used here to indicate the fact that it is not a row on a truthtable by itself but the matching of a truthvalue to a sentence which gives the desired result 15They have also been called assignments cases situations states of alfairs indices points etc Whatever their name they are nothing more than reference points for assignments of truthvalues to sentences positive integer subscripts We shall have recourse to alphabetic subscripts such as with Wi when we wish to talk about arbitrary worlds as will be done below Thus to build on our previous example we could say that a speci c interpretation I of A contains two partial truthvalue assignments We can call the rst partial TVA which assigned A the value T an assignment at w1 and the second partial TVA which assigned A the value F an assignment at wz W1 A AVA T T W2 A AVA F T 42 Formal Semantics for MSL As with Sentential Logic we will give a formal speci cation of the basic semantics for modal logic In later modules we shall make good use of the formal semantics given here 421 Generalized ValuationFunctions Now we can extend the notion of a valuationfunction to make it adequate for the formal semantics of MSL What is required is a twoplace function v1 which maps a pair consisting of a sentence and a warld onto truthvalues ie the set TF In the example just given we can say with respect to our interpretation I and its valuationfunction v V1A7 W1T7 V1A W2F It is easy to see by SR and SRV that for any world W v1A V A WT It might be helpful to recognize that we could have used twoplace valuation functions in the semantics for sentential logic That is instead of subscripting the v with a reference to the interpretation under which it makes its assignments we could have made that reference the second argument of the function V1ar Var I But this notation would not have allowed the generalization to the semantics for modal logic since it pre supposesa n t 1 between i and truthvalue assignments r 1 422 The Accessibility Relation At this point we are close to being able to state the speci c semantical rules for the determination of the truthvalues of necessity and possibilitysentences at a world But we require one more piece of machinery We must think of the possible worlds in an interpretation as ordered under a relation of accessibility The inclusion of this relation in the semantics for modal logic is the hallmark of modern possible worlds semantics We will say for example that on an interpretation I a speci c world wz is accessible to a speci c world w1 This can be depicted graphically with an arrow from w1 to wz w1 gtw2 The metavariable R will indicate an arbitrary twoplace relation of accessibility In keeping with our general practice we shall use R for a speci c accessibility relation Thus we can state what was just de picted graphically in symbolic terms as Rwlwz Every modal logic interpretation requires both a nonempty set of worlds and a relation of accessibility among the members of the set of worlds Unless restrictions are placed on the accessibility relation any world in a given interpretation may or may not be accessible to any other Together the set of worlds and the accessibility relation constitute a frame If we use W to indicate an arbitrary set of worlds a frame Fr can be represented as an ordered pair so that FrWR18 We add a valuationfunction V to a frame to get an interpretation IWRV The interpretation formed in this way is said to be based an the frame to which the valuation function is added We will say that a world is in an interpretation I when it is a member of the set W which itself is a member of I We can express a twoplace relation as a set of ordered pairs the rst of the two bearing the relation to the second In the case of accessibility the rst member of a pair will be a world and the second member a world accessible to the rst world An example of a frame is W17W27W3ls ltW1W2gt7ltW2W3gt7ltW3W1gtgt This may be represented graphically as follows W1W2W3 Insofar as we think of a possible world as representing a row of a truthtable we can think of each world in a frame as a row of a truthtable which does not have its values lled in Suppose we have a frame with two worlds w1 and wz such that Rwlwz A partial representation of the frame covering only the sentence letters A and B might look like the following w1 gt wz A B A B 7 7 7 7 An interpretation would give truthvalues to A and B One such interpretation which assigns T to A and F to B at w1 and T to bothA and B at wz can be represented as follows W1 W2 A B A B T F T T 17 Hughes and Cresswell use the metaphor of being able to see wz from W1 18Ordered ntuples are expressed with angled brackets and n elements separated by commas The accessibility relation is the basis for determining the truthvalue of modal sentences Speci cally a sentence of the form Oar is true at a world if and only if it is true at some accessible world and a sentence of the form Bar is true at a world if and only if it is true at all accessible worlds In the example just given the following values would be generated w1 gt wz A B A B T F T T EIA ltgtB T F In the case of Dar since there is only one world accessible to w1 and a is true there Bar is true at w1 There is no world accessible to wz at which B is true and so ltgtB is false at wz In some cases accessibility plays no role in the evaluation of modal sentences If as with a V ar there is no way to give the sentence the value F at any world it will have the value T at all worlds that could be accessible to a given world So Ela V ar will have the value T at all worlds The examples given here have been simple illustrations of the basic semantics We will now turn to a formal speci cation of the semantics 423 Semantical Rules Because valuationfunctions have been generalized to include a second argument referring to possible worlds we will have to restate the semantical rules for the truthfunctional operators All the rules for those operators operate locally so to speak in that they apply only at a speci c possible world To the generalization of the truthfunctional rules we add the semantical rules or truthde nitions for sentences governed by the modal operators SRTVA If a is a sentenceletter then either V1ar WT or V1ar WF but not both SRJ For all I and all W in I v1J WF and v1J W T SR v1ar WT if and only if v1ar WF v1ar WF if and only if v1ar WT SRA V1a AB WT if and only if V1ar WT and V1B WT V1a AB WF if and only if V1ar WF or v1 WT SRV v1a VB WT if and only if v1a WT or v1 WT v1a V B WF if and only if v1a WF and v1 WF SRD V1a D B WT if and only if either V1ar WF or V1B WT V1a D B WF if and only if V1ar WT and v1 WF SRE V1a E B WT if and only if either V1ar WT and V1B WT or V1ar WF and V1B WF v1a E WF if and only if either V1a WT and v1 WF or V1a WF and v1 WT SRltgt V1ltgtar WT if and only if V1ar WiT at some world Wi in I such that RWWi V1ltgtar WF if and only if V1ar WiF at all worlds W in I such that RWWi SREI V1EIa WT if and only if v1a WiT at all worlds Wi in I such that RWWi V1EIa WF if and only if v1a WiF at some world Wi in I such that RWWi SR 3 V1a 3 WT if and only if either v1a WiF or V1B WiT at all worlds W in I such that RWWi V1a 3 WF if and only if both v1a WiT and V1B WiF at some world Wi in I such that wai These semantical rules and de nitions of frame and interpretation when added to the semantical rules for Sentential Logic yield a semantical system for modal logic that we will for the time being call the basic semantical system We will illustrate the use of the semantical system KI with an example Consider an interpretation I where W w1 wz Rw1w1 Rwlwz and v1A w1 T v1A wz T 0 W1 W2 A A T T On such an interpretation it follows that v1IA w1 T The reason is that worlds w1 and wz are all the worlds accessible to w1 and A is true at both of them The value of EIA at wz is not as straightforwardly determined because there are no worlds accessible to wz It is standard practice to say that in this case it is vacuausly the case that EIA is true at wz That is SREI is understood as saying that if there are any accessible worlds then a is true at such worlds Since the antecedent of the conditional is false in the present instance the conditional itself is taken as true The metalogical conditional implicit in SREI is treated as a material conditional Therefore v1IA wz T n w1 gt wz A A T T EIA EIA T T Further V1ltgtA w1 T since there is an accessible world w1 as well as wz at which A is true However V1ltgtA wz F There is no world accessible to wz at which A is true 0 W1 gt wz A A T T CA GA T F This shows the weakness of the basic semantics we have developed There are interpretations on which a sentence is necessarily true but not possibly true This is because the basic semantics allows what Hughes and Cresswell call deadend worlds worlds to which no world is accessible 12 43 MetaLogical Properties and Relations in MSL Given the truthde nitions for sentences of MSL we can de ne semantical properties and relations of MSL sentences analogous to those of SL sentences Because different restrictions on the accessibility relation in frames will generate different systems of modal logic we shall here give formal de nitions which are fully general and can apply to any semantical system which will be considered in what follows For this reason we will de ne the notions of semantical entailment and validity relative to a frame We can then later de ne them for more speci c modal systems We shall postpone the proof of modal forms Bivalence and TruthFunctionality as well as discussion of a modal form of Semantical Consistency until Module 5 when we consider the semantical system KI 431 Semantical Entailment in 3 Frame Modal semantics requires that we expand the de nition of semantical entailment to accomodate the inclusion in the semantics of possible worlds the accessibility relation and the twoplace valuationfunction In the modal semantics a sentence only has a truthvalue at a world Thus we will say that relative to a given frame the relation of semantic entailment holds between a set 71312 yn and a sentence a when all the interpretations based on the frame which make all the sentences 7 of the set true at a world also make the sentence a true at that world The strict de nition of semantical entailment in a frame Fr is as follows19 Semantical Entailment in 3 Frame Fr 71312 yn IFr ajust in case for any I based on Fr and any W in W in Fr if V1011 WT V1yz WT and and V101 WT then v1a WT For example consider the frame used earlier Fr w1wzw1w1 ltW1W2gt Suppose we want to determine whether EIA lFr 0A That is we want to determine whether there are any interpretations based on Fr such that at some world in W EIA is true while OA is false In fact there is such an interpretation the one we considered above It contains a world wz such that vIA wzT and vltgtA wzF w1 gt wz A B A B T F T T EIA 0A T F So the semantic entailment fails Now we will consider a semantical entailment which holds for all frames EIAAB entails EIA Suppose for an arbitrary frame Fr an arbitrary world W in Fr and an arbitrary interpretation I based on Fr that v1IA A B W T Then by SREI at all accessible worlds Wi v1A A B Wi T By SRA v1A Wi T Then again by SREI v1IA W T We may illustrate this reasoning by way of truthtables To do so we need a further notational device When we place an asterisk over an accessibility arrow it indicates that we are reasoning about all accessible worlds When an asterisk is placed below the arrow it indicates that we are reasoning about at least one accessible world The rst step is to assume that EIA A B is true at an arbitrary world W and infer thatA A B is true at all accessible worlds Wi 1ng keeping with our standard practice a speci c frame will be indicated by the nonbold Fr 13 W nAB AAB T T Next we infer that the truthvalue of A A B at Wi is T W Wi IA A B A A B T T A T Finally we infer that EIA is true at W w gt w IA A B A A B T T A T EIA T 432 Semantical Equivalence in a Frame We can use the de nition of entailment in a frame to form a de nition of equivalence in a frame Two sentences of MSL a and B are equivalent in a frame Fr just in case the set consisting of the one semantically entails in the frame the other and Viceversa Semantical Equivalence in a Frame Fr 1 is semantically equivalent to B in a frame Fr if and only if for all interpretations I based on Fr and all worlds W in FR v1a W v1BW The following meta theorem can then be proved a is semantically equivalent to B in frame Fr just in case a IFr B and B IFr 1 Exercise Prove the metatheorem just stated We will here illustrate with truthtables the semantical equivalence of IA and ltgtA w gt w IA T wuwai ltgtA ltgtA The following table re ects the fact that if a possibilitysentence Oar is false at a world then a is false at all accessible worlds by SRo w gt w ltgtA T ltgtA F A F A T EIA T 433 Validity in a Frame The limiting case of entailment in a frame is validity in a frame We have seen in the case of Sentential Logic that validity can be treated as semantic entailment by the empty set of sentences and here it understood as relativized to a frame 0 IFr a A sentence a is valid in a frame if and only if it is true at all worlds on all interpretations based on that frame Validity in Frame Fr IFr a i r for all W in W in Fr all I based on Fr and all V1 in I V1a WT To return to a previous example IA V A is valid in any frame whatsoever No matter how A is interpreted at any given world as T or F at that world the sentence A V A is true at every world that could be in a frame Therefore A V A is true at any world accessible to a given world W in any frame in which case IA V A is true at W So on any interpretation and any world based on any frame IA V A is true which was to be proved We can generalize this reasoning to conclude that the necessitation of every SI valid sentence of SL is valid in every frame in the basic semantical system W Wi A V A T IA V A T When entailment equivalence or validity holds for all frames we shall drop the reference to the frame Later we will develop a number of semantical systems by placing restrictions on the accessibility conditions Then we will introduce the notion of a Class Of frames containing the restricted accessibility relation as a member Accordingly we will speak of entailment equivalence and validity in all frames of a certain class 5 Conclusion It may be useful to re ect on the nature of the accessibility relation One way to think of it is as repre senting those situations that matter with respect to the modality in question For example we might think of something as necessary when it is inevitable in the sense that there is no alternative that matters to its being the case The alternatives that matter might be thought of as being accessible worlds Truth of the necessitysentence Ela would require truth of the embedded sentence a in all those worlds that matter A possibilitysentence would indicate that its embedded sentence is true in at least one of the worlds that matters We shall in what follows have much more to say about how accessibility captures our intuitions about the various kinds of modalities We are now nally in a position to see why the semantics for modal sentential logic is indirectly truth functional Although the truth of a necessity sentence or any other modal sentence is not directly a function of the assignments made to its components it is a function of the assignments made to its components at the accessible possible worlds And those assignments are ultimately based on truthvalue assignments to the sentenceletters from which the sentence is composed In Module 5 we will give a de nition of modal TruthFunctionality in connection with speci c semantical system and prove inductively that the system conforms to that de nition
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