Modal Logic PHI 134
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T and Equivalent Systems G J Mattey May 8 2005 Some kinds of modalities demand that what is necessary at a world be true at that same world This requirement is met by the T systems The axiomatic system is also known in the literature as M It was rst investigated by Kurt Godel and G H von Wright The T systems are extensions of the Dsystems and therefore are extensions of the K systems as well As with the other two families of systems we will discuss the semantical system rst then the derivational system then brie y the axiom system and nally applications 1 The Semantical System TI The semantical system T I is just like the system KI except for the requirement that R be re exive That is each world must be accessible to itself We may express the re exive character of a relation R as follows R is re exive if and only if HxRxx We can de ne a T I frame as a set W R such that Hww E W gt wa A T I interpretation meets the following condition HwBwI gt wa A consequence of the re exivity of the accessibility relation in TIframes will be called the characteristic consequence of the T systems Dar FT 1 Proof If V1EIa wT then for all worlds w accessible to w v1a wiT By the re exivity of R v1a wT The proof of the entailment can also be given using a metalogical derivation Sketch ofa semantical proof that Dar IT1 a l v1larw T Assumption 2 Hwwa Re exivity of R 3 wa 2 H E 4 Hwiwa gt v1aw T l SREIC 5 wa ev1aw T 4HE 6 v1arwT 35 gtE All T I frames are DIframes If a world w belongs to an interpretation 1 then w is accessible to itself If a world in an interpretation is accessible to itself then there is a world itself which is accessible to it More generally any relation that is re exive is also serial The converse does not hold since the accessible world required by seriality need not be the home world so not all DIframes are T I frames It is easy to see that all DIentailments are T I entailments 717 Wyn FBI 0 gt 7177Yn39T1 0 Since the class of T I frames is a subset of the class of DIframes any entailment that holds in all DIframes also holds in all T I frames So the semantical system D is contained in the semantical system TI Moreover T I is a stronger system than D in that some T I entailments are not DIentailments because some DIframes are not T I frames So T I is an eXtension of D Speci cally Dar K D a Proof Let W in a frame Fr contain two worlds w1 and wz such that Rwlwz and szwl R is therefore serial Now let v1a w1 F and v1a wz T Iffollows from SREI that V1EIa w1 T since a is true at all accessible worlds ie at wz But we have stipulated that v1a w1 F So there is an interpretation in a DIframe which blocks the entailment T has some consequences for iterated modalities We can assert the following metatheorem though the converses do not hold Elna T1 Ba and 001 T1 0001 Note that corresponding results hold for arbitrarily long strings of the same modal operator Proof of the rst entailment Suppose for an arbitrary TIframe world w in the frame and interpretation I based on that frame that v1IEIa w T By re exivity wa so SREIC V1EIa w T Proof of the second entailment Suppose for an arbitrary TIframe world w in the frame and interpretation I based on that frame that V1ltgta w T Since wa there is a world accessible to w at which Oar is true so by SRo V1ltgtltgta w T Exercise Prove that the converse entailments do not hold A result involving the 3 operator is that an a 3 T1 Proof Suppose that for an arbitrary T I frame world w in the frame and interpre tation based on the frame v1a w T and V1a 3 w T Then by SR 3 at any world w accessible to w if v1a w T then v1 wi T By re exivity w is one ofthe ws So if v1a w T then v1 w T Therefore v1 w T 2 The Derivational System TD The derivational system TD adds two rules to the derivational rules for KD with the aim of allowing the derivation of a from Ba As before the derivational rule will closely follow the semantical rule In the semantical system if a necessitysentence Bar is given the value T at world w not only is a true at any arbitrary accessible world but it is true at w as well The easiest way to represent this feature of T I is by allowing the elimination of the El operator within the current strict scope line or where Ela occurs outside all strict scope lines El Elimination la or IE We assert without proof that the derivational system TD resulting from adding this rule to the derivational rules of KD is complete with respect to the semantical system T I We also assert without proof that the derivational system is sound This claim can be motivated by the way in which a derivation mirrors the semantical reasoning used in the metalogical derivation above To prove Ela I TD a 1 Bar Assumption 2 a 1 El E An alternative rule for systems with 0 as primitive would be a rule of 0 introduction We call this a rule of strong 0 introduction because it does not require any result within a further strict scope line Strong 0 Introduction Oar SltgtI The rule is sound Ifa is true at a world so is Oar since the world is accessible to itself We can prove that El Elimination is a derived rule given that Strong 0 Introduc tion and Duality are rules of inference El Elimination as a derived rule 1 Bar Assumption 2 a Assumption 3 Dar 1 Reiteration 4 ltgtar 2 S 1 5 Ela 4 Duality 6 a 25 E Exercise Prove that the rule Strong 0 Elimination is derived from El Elimination given Duality The rules for TD allow are strong enough to make the characteristic rules of D derived rules We will show that Weak ltgt Elimination is derivable in T I Weak 0 Introduction as a derived rule D a Bar II or IE Oar 1 Exercise Show that EI Introduction is derivable in T I with Duality Because of the semantical result 1 a 3 IT1 which was proved earlier we can add a derived rule for system TD and stronger systems 3 Elimination S 3E Proof of 3 Elimination as a derived rule is based on the derivation of Ela 3 from a 3 and the elimination of the El by the TD rule of El Elimination 3 Elimination as a derived rule a a 3 D aD SR 3 EIaD EII aD EIE DE 3 The Axiom System T The axiom system T is obtained by adding to the axioms of K the further axiom I T Ba 3 a This axiom is clearly valid in T by the same reasoning that showed that Dar IT1 1 Assume that Bar is true at an arbitrary world w in an arbitrary interpretation I based on a T I frame Then a is true at all accessible worlds By re exivity a is true at w and so by SRD Dar 3 a is true at w which was to be proved 4 Applications of the T Systems 41 Alethic Modal Logic The T systems are more adequate than are the weaker Dsystems for representing consequence relations involving logical possibility and necessity As noted in a previous module we want to be able to say that what is logically necessary is true and that what is true is logically possible Both these desiderata are satis ed in the T systems With hypothetical necessity it also seems that the T systems are more desirable than are the weaker systems Given that Ela holds at a world it is natural to suppose that the condition governing the necessity holds at that world and hence that a is true there For example suppose that the El operator is meant to signify physical necessity where the condition is that a set of laws of nature hold If it is true in a given world that those laws hold it seems that they should apply to that world Now we might want to consider worlds which are subject to laws diITerent from the laws governing the world in question Then it seems right to say that when those laws are expressed by EI the sentence Ela should be true at those worlds but not at the world where the laws do not hold 42 Conditional Logic The T systems overcome a problem for the logic of strict implication It was noted in an earlier module that in the semantical rules for K there are frames with worlds where a and a 3 are true but is false D1 is no better o r in this respect These are precisely worlds which are not accessible to themselves Imposing the re exivity requirement on accessibility in T I now allows the entailment ma 3 T1 If the the shhook is understood as indicating a local relation of implication the reasoning just used in the case of hypothetical necessity applies Suppose we take a 3 to indicate that at all worlds where a condition holds if a is true then is true If that sentence holds at the home world then it is natural to take it that the condition holds there as well Then if a is true there should also be true there So the only time the T systems might be considered too strong for this application is when the shhook is intended to express a condition that does not hold in the home world It is hard to say what such an application would be 43 Deontic Logic The T systems are too strong for deontic logic given standard views about obli gation and permission With respect to morality and law what is obligatory is in many cases not what actually holds and what actually holds is often not permitted And even if in fact the real were to conform to the ideal we would not want to say this is so as a matter of the logic of obligation 44 Doxastic Logic The T systems are also too strong for doxastic logic Even if it is a matter of the logic that all valid sentences are believed it is not a matter of logic that every belief no matter what its content is true 45 Epistemic Logic It does seem to be part of the notion of knowledge that what is known is true so the characteristic consequence in T Kma IT a should hold Further there is a consequencerelation holding in T that is relevant to epistemic logic Kma tTIFxJKma If a subject knows that a then for all he knows he knows that a This is an immediate result of the use of the epistemic correlate of Strong 0 Introduction On the semantic side the consequence holds because the world at which the sub ject knows is an accessible world to itself and accessible worlds eXpress what is compatible with what a subject knows So the fact that the subject has knowl edge that a is compatible with what the subject knows An equivalent result is Kma IT1 KmKman It seems that my knowing that 1 implies that I do not know that I do not know that a 46 Temporal Logic A temporal logic of the past and future might not be appropriately based on T What is the case at all future times does not have to be the case now though it Informal Introduction to Modal Logic G J Mattey 1 Motivation for Modal Logic One of the rst things learned by beginning logic students is the de nition of a valid argument The standard de nition of a valid argument runs along these lines An argument from a set of premises to a conclusion is valid if and only if it is not possible for all the premises of the argument to be true and for the conclusion to be false Alternatively one might say that an argument is valid just in case it is that necessarily if the premises are true then the conclusion is true The concept of a valid argument lies at the heart of logic The de nition of the concept of validity in turn depends essentially on the modal concepts of possibility and necessity These concepts are called modal because they indi cate a way or mode in which the truthvalues of the premises are connected with the truthvalues of the conclusion One of the main tasks of symbolic logic is to represent the form of arguments in such a way that their validity or invalidity can be determined using standard ized techniques One can use truthtables for example to represent the validity or invalidity of arguments whose basic units are individual sentences In the logics that are generally taught in introductory logic courses the properties of validity or invalidity are not represented in the symbolic language itself There are symbols representing the truthfunctional operators and orl not etc but there is no symbol for thereforeli 1 In fact the operators of truthfunctional logic as well as those of predicate logic do not represent possibility or necessity at all Standard logic is nonmodal in this respect even though it might be used to establish modal properties of arguments Modal logics are precisely those logical systems which contain modal oper atorsi In the case of validity one might seek to build a logical language which contains an operator which is understood to express the modal property of va lidity That is it would contain a modal operator 2 The Lewis Systems Modern modal logic appeared in the early twentieth century not long after modern nonmodal logics had been popularized by Russell and Whiteheadls 1Many logic students mistakenly render the material conditional into English as there7 forel but this is a mist V Principz39a Mathematical2 A young philosophy instructor at UC Berkeley Cili Lewis used Principz39a as a text Lewis thought that Russell7s description of the truthfunctional conditional operator as material implication was misleading He built several axiomatic systems featuring a modal operator he called strict implication which he thought better represents the relation between premise and conclusion in an argument neplace modal operators for possibility and necessity were part of the Lewis systems The necessity operator can be understood as allowing the rep resention of the concept of necessary truthi Strict implication it turned out expresses the same thing as necessarily true material implication i 3 Semantics Lewis s systems were laid down in axiomatic formi The earliest work on the sys tems was to prove theorems of the given systems which follow from their speci c axiomsi Very soon thereafter the axiomatic systems were given interpretationsi The most prominent kind of interpretation was with matrices that resemble truthtablesi A useful matrix for modal logic typically contained more than two values Using matrices logicians were able to get important results about the systems They could determine which systems contain which other systems and whether a given axiom is independent of the other axiomsi While the matrix system was useful it provided no real insight into the mean ings of modal sentencesi It was udolf Carnap writing in the mid 1940s who rst provided an intuitively plausible semantics for one of the Lewis systems S5i ln Carnapls semantical system the truthvalues of nonmodal sentences are determined just as they are in truthfunctional logici A sentence whose main operator is a necessity operator is true if and only if the sentence it governs is a logical truthi Thus if a sentence is true on all rows of its truthtable then the sentence formed by pre xing a necessity operator to it is also true and in fact is true on all rows of all truthtablesi Carnap also provided a system for a predicatelogic version of S5 His se mantics is of interest because of the way it interprets the syntax of predicate logic Nonmodal semantics interprets terms as standing for objects in a uni verse or domain of discourse Predicates are interpreted as standing for sets of objects from the domain Beside this extensional type of interpretation Carnap developed an intensional interpretation suitable for the use of terms and predicates in modal contexts The intensional interpretation depends on the notion of a state descrip tion Carnap wrote that the state description represents Leibnizls possible worlds7 or Wittgensteinls possible states of affairs 7 4 A term might stand for dif ferent objects in different state descriptions so that its intension is a function 2The basic system in that book had been laid out by Frege in 1879 but had gone largely unnoticed 3System S contains system 3 just in case all the theorems of S are theorems of S 4Meaning and Necessity 9710 from state descriptions to objects A perdicate might have different extensions in different state descriptions so that its intension is a function from state descriptions to sets of objects In the late 1950s Carnapls semantics was generalized to the form in which it exists today5 The key notion in the semantics is that of a possible world77i ln sentential logic a possible world corresponds to a row of a truthtable ln predicate logic a possible world corresponds roughly to an interpretation Carnapls system in effect took necessity to be truth at all possible worlds This worked as a semantics for S5 but not for any of the weaker systems of Lewis and others The innovation was to add to the semantics a twoplace rela tion of accessiblity or alternativeness holding among the worlds themselves Then a necessity sentence could be taken to be true just in case the sentence governed by the necessity operator is true at all accessible possible worlds This generalization of the Carnapian semantics allowed Kripke and others to provide semantics for most of the known axiomatic systems of modal logic It also made it easy to generate new systems Most importantly perhaps it provided an intuitive way of understanding what the sentences of modal logic meani s ould be noted that for a long time modal logic was held in some dis repute due to the criticisms of WiviOi Quinei One of his objections was that valdity implication logical necessity etc are metalogical notions which have no place in logic itselfi Another was that the semantics for modal predicate logic requires the postulation of possible but nonexistent objectsi Quine be lieved that we should not commit ourselves to possibilia on the grounds that they do not have wellde ned identity conditions77i When generalized possible worlds semantics came on the scene philosophers welcomed it as a powerful analytic tool and brushed Quinels objections asidei It is probably not coincidental that about this time there was a powerful shift away from the austere metaphysics of Quine and his Harvard colleague Nelson Goodman not to mention the later Wittgenstein There remains vigorous debate about the metaphysical status of possible worlds and objects in themi At one extreme David Lewis advocated a modal realism according to which each possible world is just as real as the one we call actua 7 5 At another Michael Jubien has tried to treat modalities without appeal to possible worlds at alli7 4 Applications From the time generalized possible worlds semantics was invented and even be fore philosophical logicians began to recognize that it has applications beyond the logical notions of implication and logical truthi Jaakko Hintikka recognized that Lewis s necessity operator could be interpreted either as a belief opera 5The sematics is commonly attributed to Saul A Kripke but it was developed during the same period by Jaakko Hintikka and Stig Kanger 60n the Plumlity of Worlds 1986 7Contemporm y Metaphysics 1997 Chapter 8 Module 7 D and Equivalent Systems G J Mattey May 2 2007 Contents 1 The Semantical System D1 1 2 The Derivational System DD 4 3 The Axiom System D 6 4 Applications of the DSystems 6 In this and subsequent modules a number of systems stronger than the K systems will be developed by adding restrictions to the accessibility relation new derivational rules and new axioms Requiring that various conditions on accessibility hold means in general that more worlds are accessible to a given world which removes obstacles to entailment validity etc by limiting the range of potential counterexamples Adding new rules of inference increases the range of derivations that can be made And adding new axioms increases the stock of theorems that can be proved The weakest extensions of the K systems are the Dsystems The semantical system DI will be examined rst followed by the derivational system DD The axiom system D will be given a brief treatment Following this there will be a discussion of the suitablity of the Dsystems for the various applications discussed in the last module 1 The Semantical System D1 The semantical system DI is just like the system KI except for the requirement that R be serial That is each world must have at least one world accessible to it We may express the serial character of a relation R as follows R is serial if and only if Hx2yny We can de ne a DI frame as a set W R such that HWW E W gt ZWiXWi E W A wai1 1The 6 symbol indicates membership in a set An interpretation I is based on a frame and the frame s set of worlds W is a member of I So we can expand the description of a serial accessibility relation in a frame to that of a serial accessibility relation in an interpretation nww e w A w e 1 gt Ewiwi e w A w e I A wai Because the conjunction W E W A W E I is somewhat cumbersome we introduce a twoplace B predicate that indicates that a world W belongs to an interpretation I which just means that W E W A W E I This conforms to our practice in previous modules of referring to worlds as being in interpretations So we can say that a DI interpretation meets the following condition HWBWI gt EWiXBWiI A RWWi If a world W belongs to an interpretation I then there is an accessible world Wi possibly the same as W in the interpretation as well It is easily seen that all KI entailments are DI entailments If 71 yn PK 1 then 71 yn PD 1 This is because the class of DI frames is a subset of the class of KI frames all DI frames are KI frames So any entailment that holds in all KI frames also holds in all DI frames In this sense the semantical system KI is cantained in the semantical system DI This fact means that the semantical results we have proved for KI carry over to DI and other systems formed by placing restrictions on the accessibility relation Speci cally Modal Bivalence Modal Truth Functionality and Closure were proved without in any way taking into account the nature of the accessibility relation in a frame The semantical system DI is a stranger system than KI in that some DI entailments are not KI entailments In this sense DI is an extensian of KI Speci cally Dar 0 Oar but Dar KI Oar Proof If V1EIa WT then for all worlds Wi accessible to W V1a WiT By the seriality of R there is a world Wi such that RWWi so there is an accessible world Wi such that V1a WiT So V1ltgta WT The nonentailment in KI was proved in the last module and it hinges on the fact that a Kframe may contain deadend worlds to which no world is accessible The requirement that R be serial prohibits the presence of deadends in the set W of worlds in a frame There is guaranteed to be a world Wi accessible to a given world W which guarantees that V1ltgta WT This holds regardless of the choice of W and I so long as the frame on which I is based is serial The proof of the entailment can be given using a metalogical derivation DOOQCI LIIlkwwt H O 11 Sketch of a semantical proof that Ba DI 0a v1la W T THWXEWQRWW EWiXRWWi RWW1 HWRWW gt V1a Wi T RWW1 gt V1a W1 T V1a W1 T RWW1 V1a W1 T EWRWWL39 A V101 Wi T V1ltgtoz W T mm W T Assumption Seriality of R 2 H E Assumption 1 SREI 5 H E 4 6 gt E 4 7 A I 8 E I 9 SRltgt 3 410 2 E The result can be represented graphically using two steps First we establish that given the truth of Bar at a world W a is true at all accessible worlds Wi W Wi Ela T a T Then we consider a world W1 satis es the seriality requirement by being accessible to W We then carry over our result that a is true at all accessible worlds Wi to world W1 W W1 Dar T a T Oar T We can now establish another desirable result of the system There are valid sentences of the form Oar Speci cally if a is valid in DI then so is Oar If IDa then DIltgtar Proof Suppose IDI 1 Then at all worlds W on all interpretations I based on any DI frame v1a WiT For every world W there is an accessible world Wi So there is an accessible Wi such that v1a WiT Hence V1ltgta WT Because this result holds for all worlds on all interpretations based on any DI frame IDIltgtar which was to be proved 2 The Derivational System DD Just as the semantical system DI builds on the semantical rules for KI the derivational system DD adds a rule to the derivational rules for KD with the aim of allowing the derivation of Oar from Ba As before the derivational rule will closely follow the semantical rule In the semantical system if a necessitysentence Bar is given the value T at world W not only is a true at any arbitrary accessible world if there is one but it is true at at least one accessible world So when we write down a restricted scope line indicating an accessible world and get a result a there we can end the restricted scope line and write down Oar We shall call this a weak rule of 0 Introduction because we will later introduce a stronger rule Weak 0 Introduction Provided that a is not in the scope of any assumption within the strict scope line We assert without proof that the derivational system DD resulting from adding this rule to the deriva tional rules of KD is complete with respect to the semantical system DI We also assert without proof that the derivational system is sound This claim can be motivated by the way in which a derivation mirrors the semantical reasoning used in the metalogical derivation above To prove Dar I DD 0a 1 Bar Assumption 2 D a 1 SREI Oar 2 W O I Some texts give as rule the derivation of Oar directly from Bar This can be treated in DD as a derived rule as is clear from the preceding derivation If the alternative rule is taken as primitive then our rule of Weak 0 Introduction would be a derived rule It is easily seen why it would be Weak 0 Introduction as a derived rule D Ba EII Oar Alternative Primitive Rule If Weak 0 Introduction is adopted as a primitive rule then system D requires a primitive possibility operator in its syntax If only the El operator is taken as primitive and sentences with 0 as their main operator are de ned in terms of sentences with the El operator a new EI rule is needed The most natural rule would be modeled on the Impossibility Introduction rule That is if a occurs inside a single restricted scope line and not in the scope of any undischarged assumptions then the scope line may be ended and Iar written I Introduction D My Ia I 1 Provided that a is not in the scope of any assumption within the restricted scope line Exercise Give a justi cation for this rule using the semantical rules for DI Either rule can be derived from the other given Duality as a replacement rule Here is how the EI Introduction rule can be derived using the Weak 0 Introduction rule Assume that a occurs within a restricted scope line Then the scope line can terminated in favor of ltgta With Duality this is shown to be equivalent to Ela I Introduction as a derived rule D My ltgtar W O I Ela Duality Exercise Show that with Duality as a derived rule and EI Introduction the rule of Weak 0 Introduction can be derived It might be noticed that neither Weak 0 Introduction nor EI Introduction requires that any sentence of the form Ela occur outside the restricted scope line In this way the two rules deviate from the semantical proof that Dar ID Oar This does not threaten to make the rule unsound however If a sentence of the form Ela does occur outside the restricted scope line then the semantical reasoning is followed perfectly If it does not there are no untoward consequences If we can derive a entirely with the strict scope line then I DDa More generally It is easy to see that in general if I D a then I D Oar which parallels the result from the semantical system Here is an example To Prove I DDltgtP A P 1 D P A P Assumption P 1 A E P 1 A E P A P 13 I ltgtPA P 14WltgtI L11wa Now the last step of this derivation could just as well have been by El Introduction EIP A P So we could have derived this result rst which would then have allowed Bar to occur outside the restricted scope line Then the semantical reasoning used above would be re ected in the derivation of ltgtP A P 3 The Axiom System D The axiom system D is obtained by adding to the axioms of K the further axiom schema I D Dar 3 Oar This axiom is clearly valid in DI by the extending the reasoning that showed that Dar IDI Oar Assume that Bar is true at an arbitrary world W in an arbitrary interpretation I based on a DI frame Then a is true at all accessible worlds WI There is such an accessible world by the seriality of R so a is true at that world which makes Oar true at W Thus by SRD Dar 3 Oar is true at W which was to be proved 4 Applications of the DSystems In the last chapter we considered the use of the K systems to represent various modalities of which we have informal conceptions In every case it appeared that a stronger system is needed We shall now look at the adequacy of the Dsystems to represent the various modalities treated in this text The Dsystems yield a result which is amenable to the notion of logical necessity What is true of logical necessity by virtue of the laws of logic or by its logical form should at least be possibly true With respect to hypothetical necessity the Dsystems do not allow worlds at which everything is trivially necessary because nothing is impossible If we take the accessibility relation as specifying a condition that might or might not hold relative to a world we must then say that the condition can be satis ed in that it is satis ed at an accessible world It seems plausible on the face of it that given almost any condition this would be a desirable result There is a change in the Dsystems which is of some small signi cance for the strictimplication inter pretation of the shhook In the semantical system KI all sentences of the form a 3 are true at deadends This result can be obtained in two ways First El is always true at a deadend and if El is true at a world then a 3 is also true of that world Second ltgtar is true at a dead end and any strictimplication sentence with an impossible antecedent at a world is true at that world Ridding the semantics of deadends does away with this peculiar way of generating necessities and impossibilites But the paradoxes of strict implication remain in the Dsystems and indeed in any systems based on the K systems If for more orthodox reasons Bar is true at a world then so is 3 a and so forth The deontic interpretation seems to require the restriction on accessibility laid down by semantical sys tem D If something is obligatory relative to a world it should be treated as being permissible relative to that world as well2 We have this result because Oar IDI Par This is about as strong as we want a system of deontic logic to be Indeed the name of the axiom system D is an abbreviation for deontic It should be noted that it is possible meaningfully to combine the deontic modal operators with other modal operators which would produce a system with more expressive power Thus we may wish to assert that what is obligatory is possible or that ought implies can so thath should entail Oar Another option is to combine the denotic operators with temporal operators We shall not explore these combined modalities in this text ZPennissibility which may be de ned in terms of obligation or viceversa must therefore be understood in the same way in the semantics Some would claim for example that there are acts which are legally obligatory but not morally permissible Module 18 Systems with a Unitary Domain of Possible Objects G J Mattey June 6 2007 Contents 1 The FQIx Systems 1 11 The Semantical FQII x Systems 1 12 The Derivational F QIDx Systems 4 2 The QIx Systems 5 21 The Semantical QIIx Systems 5 22 The Derivational QIDx Systems 6 Systems with a unitary domain of possible objects are strong systems that can be built on the framework of Sentential Logic and Free Predicate Logic we have developed to this point They sanction both the Barcan Consequences and the Converse Barcan Consequences and hence allow the reversal of the pairs V El and 3 0 The other two pairs work in only one direction a fact which will be discussed below A family of systems the F Q x systems is built on the platform of Free Predicate Logic and will be considered rst here A stronger family of systems the QIx systems is based on standard Predicate Logic and will be the last topic to be treated in this series of modules 1 The F Q1 x Systems In this section we will give semantical rules for systems F QIIx and derivational rules for systems FQlDX 11 The Semantical F QIIx Systems The key semantical feature of the F QIIx systems is that there is only one domain for all the objects at all the possible worlds As noted in the last module the effect of a unitary domain is the consequence of both the symmetry of the accessibility and the Included In or Includes requirements on the domains at worlds This is enough to yield the F Q x systems Here we will simplify our treatment and begin with the assumption of a single domain serving all worlds Since the underlying logic is Free Predicate Logic we will have to reintroduce the distinction between an inner and an outer domainia distinction which in the QIRIx systems had been transposed to the dif ference between the domain of the world and the remaining objects in the domain of the frame Since the unitary domain exhausts the objects existing at all the possible worlds the outer domain can only consist of what are from the standpoint of the frame impossible objects So if we want a to refer to a round square we would still be able to assert that it is round which might be symbolized as Fa The semantical rules for F QIIx are the same as those for FPI only relativized to worlds Interpretations in FQIIx IDD v D DnD W Special Semantical Rules for F QIIx H v1aEDUD N V1llE DUD 9 HwiHWjWi E W A Wj E W gt V107 Wi V107 W j VP DUD v1E D1 05quot V1VXaXll WT where u is free for X in aX if and only if for all 1 E D V1dllall W T V1VXaXll WF if and only if for some d E D V1dllarll W F v13XarXu WT where u is free for X in aX if and only if for some 1 E D V1dllarll W T v13XarXu WF if and only if for all d E D V1dllall W F 1 OD v1titj WT if and only if V1ti WV1tjW v1titj WF if and only if v1ti v1tj W It should be obvious that the Barcan and Converse Barcan Consequences hold in this semantical system since any reasoning involving the inclusion relation between domains of worlds holds for a system in which there is a single world Here is a diagram that illustrates how a variation of the Barcan Consequence would be proved in F Q SS Since there is no longer any need to appeal to the domains of worlds so the diagram is more simple than that for QIRCSS W1 W2 IVxaXu T VXaX u T au Hd u T Iall Hd u T VXIaX u T The fact that both the Barcan and Convese Barcan Consequences hold in the semantical system means that sentences beginning with V EI are equivalent to those beginning with El V and likewise for the pairs 3 0 and 0 3 This leaves two more pairs each of which allows an entailment in only one direction Thus we have 3XUaXll FQ1174 03XarX ll ltgtVXaXu IFQ1H VXltgtaXu The rst result will be illustrated with a diagram n 0 W1 W2 ltgtVXaX u T VXaXu T aVu HdV T ltgtav u Hdv T VXOaX u T The second result is left as an exercise for the reader The following nonentailments were proved in the previous model using the underlying system KI with an interpretation with a single domain so they apply to the present systems as well We shall now illustrate why they fail even if the underlying semantical system is SSI 33XaXuFQ1FS5 3XDaXu VXltgt IXll i FQJLSS 03XaXll n 0 W1 3 W2 F 1 F 2 I3xFx T 3xFx 3xFx T T F14 114 F14 214 T T F14 214 F14 114 F F EIF14 114 F EIF14 214 F 3xEIFx F Exercise Explain why the counterexample works for the other nonentailment and illustrate your reasoning with a diagram Summary of F QIIxSystems IW R D D v Semantical rules for EI 0 3 analogous to those for KI Semantical rules for V 3 as for FPI Conditions on R for system X Rigid Designation 12 The Derivational F Q1Dx Systems For the derivational systems F Q Dx we simply combine the rules for QIRCDx and QIRBx If we were to drop the indices then we would have to decide how to handle situations like the following Attempt to prove EIVxFx IQ1 RCD K VxIFx 1 IVxFx Assumption 2 IVxFx 1 Reiteration 3 D VxFx 2 SREI 4 F X 3 V E F Q 5 F K 6 BR V F Q 6 EIFZ 2 37 El 1 7 VxEIFx 8 Misapplication of V I FPI We would then need some rule which tells us when we can take the nal step If we allowed unrestricted generalization we would permit invalid inferences such as the following Attempt to prove VxltgtFx l QlRCDiK ltgtVXFX 1 VxltgtFx Assumption 2 VxltgtFx 1 Reiteration 3 F g 2 V E 4 D F g Strict Assumption 5 V F E 4 Reiteration 6 VxFx 5 Missapplication of V I 7 ltgtVxFx 2 46 W 0 I 8 ltgtVxFx 6 BR FQ 2 The Q1 x Systems In this section we will give semantical rules for systems QIIx and derivational rules for systems QlDX These rules will be simplest of all 21 The Semantical QIIx Systems The di erence between the systems F Q 36 and QIIx lies in the absence of a distinction between inner and outer domains in the semantics There is a single domain D in every frame We shall require that every term parameter or constant designates an object in D Interpretations in QIIx I W RDV D at a W at a Special Semantical Rules for QIIx 1 V1a E D N v1u E D 9 HwiHWjWi E W A Wj E W gt V107 Wi V107 W j vam g D 5 15 V1E D1 0 V1VXarXll WT where u is free for X in aX if and only if for all 1 E D V1duaru W T V1VXaXll WF if and only if for some d E D v1duau W F 7 v13XarXu WT where u is free for X in aX if and only if for some 1 E D V1duaru W T V13XaXu WF if and only if for all d E D v1duau W F 8 v1titj WT if and only if V1ti Wv1tjW v1tit WF if and only if v1ti v1t w This change in the semantics has important consequences for sentences containing constants It restores the PI validity in systems as weak as QIIK of3XX a Because of Closure we have VDashQ11KEI3XX a We also have VDashQ11K3XIX a Summary of IQIIxSystems IW R Dv Semantical rules for EI 0 3 analogous to those for KI Semantical rules for V 3 with parameters as for FPI Semantical rules for V 3 with constants as for PI Conditions on R for system X Rigid Designation 22 The Derivational Q1Dx Systems We will take over all the derivational rules for the F QIRDx systems except that to re ect a unitary domain we can now relax the rules for the instantiation of univerally and existentially quanti ed formulas and use the rules for PD though limited to constants Universal Elimination QID Vxax u Already Derived aau V I Existential Introduction QID arau Already Derived 3xax u 3 I No longer must we have Ea to instantiate from Vxarx u to aau Nor is it needed for the generalization from arau to 3xarxu This change does not a ect the rules involving parameters When the rules of Universal Instantiation and Existential Generalization are applied to formulas by Virtue of their constants there is need for a barrier Using the rst of these relaxed rules we can prove as theorems what before were shown to be valid To prove I Q10K I3xx a 1 1 D a a I 2 3xx a 1 3 IQDx 3 D3xx a 12 El I To prove I Q10K 3xlx a 1 D a a I 2 la a 1 El I 3 3xux a 2 31Q1D B and Equivalent Systems G J Mattey May 10 2005 We saw earlier that the T systems yield the entailment of Oar from a and that this seemed a desirable result for some of the applications we have been discussing The Bsystems take this a step further with its characteristic consequence of EIltgtar from a The system is sometimes called the Brouwerian or Brouwerische system due to some similarities between its characteristic consequencerelations and some elements of Brouwer s intuitionistic interpretation of mathematics1 The system is obtained by adding a symmetry requirrnent for the accessibility relation to the semantical system T I The Bsystems are extensions of the T systems and therefore are extensions of the K and Dsystems as well Like the S4systems the B systems are not based directly on the K systems The T systems characteristic consequencerelations are independent of those of the Bsystems As with the other families of systems we will discuss the semantical system rst then the derivational system then brie y the axiom system and nally applications 1 The Semantical System BI As with the axiom system the semantical system for B is built on T as a founda tion so the accessibility relation is re exive To this is added the requirement of symmetry R is symmetrical i r HxlTyny gt Ryx If a world is accessible to another then the other is accessible to it Note that as with transitivity this restriction is conditional But because accessibility in the se mantical system B is serial because it is re exive the condition in the antecedent is always met Each world has at least one world accessible to it and by symmetry they are mutually accessible The limiting case is where w is accessible to itself in which case symmetry is trivial wa gt wa 1See Hughes and Cresswell A New Introduction to Modal Logic pp 7071 Applied to frames this means that if a world w is accessible to w then w is accessible to w We can de ne an BIframe as a set W R such that Hww E W gt wa and HwHww E W Awi E W A wai gt Rwiw The symmetry of the accessibility relation in BIframes yields what will be called the characteristic consequence of the Bsystems 0 HM Boa The proof of the entailment can be given using a metalogical derivation Sketch ofa semantical proofthat a 341 EIltgtar l v1arw T Assumption 2 wa1 gt Rwlw Symmetry ofR 3 wa1 Assumption 4 Rwlw 2 3 gt E 5 RwlwAv1arwT 34AI 6 v1ltgtarw1 T 5 SRo 7 wa1 gt v1ltgtarw1 T 26 HI 8 v1lltgtarw T 7 SREI Note that this derivation does not depend on any restrictions on R other than symmetry So symmetry could be added to the semantical systems K or D rather than T I to produce other semantical systems with the charateristic B consequence relation All T I entailments and hence all DI and KIentailments are BIentailments 7177YnT1 a gt 71777n31 a Since the class of BIframes is a subset of the class of T I frames any entailment that holds in all T I frames also holds in all BIframes So the semantical system T I is contained in the semantical system B B is a stronger system than T I in that some BIentailments are not T I entail ments because some T I frames are not BIframes So BI is an eXtension of T I and hence of D and K Speci cally 01T1 Boa Proof Let W in a frame Fr contain two worlds w1 and wz such that Rwlwl Rwlwz and szwz R is therefore re exive and so Fr is a T I frame Now let V1a w1 T and v1a wz F If follows from SRO that V1ltgta wz F since a is false at all accessible worlds ie at w itself Therefore V1EIltgta w1 F by SREI The semantical systems BI and S4 do not contain one another The acces sibility relation R may be transitive and re exive without being symmetric and re exive and it may be symmetric and re exive without being transitive and re exive More generally transitivity does not imply symmetry nor does symmetry imply transitivity The independence of the two semantical systems can be seen also from the fact that neither system supports the characteristic consequence relation of the other Consider an S4Iframe with three worlds in which Rw1w1 szwz RW3W3 Rwlwz RW2W3 and RW1W3 In this frame accessibility is transitive and re exive Now let a be true at w1 and false at W3 Then Oar is false at wz in which case EIltgtar is false at w1 For the failure of the characteristic S4 consequence in a BIframe let the frame have three worlds and let Rwlwl szwz RW3W3 Rwlwz RW2W3 szwl RW3W2 In this frame accessibility is symmetric and re exive Now let a be true at w1 and wz Since a is true at all worlds accessible to w1 Bar is true at w1 Now let a be false at W3 The Bar is false at wz in which case EIEIa is false at w1 2 The Derivational System BD Just as the semantical system for BI contains the semantical rules inherited from K D and T I the derivational system inherits the derivational rules from these systems We will give an additional rule that generates the elfects of symmetry of accessibility It will be a special version of Strict Reiteration SR B which allows that when 1 occurs Oar may be reiterated across a single restricted scope line This rule is unique among those we have considered until now because the reiteration introduces a new operator altogether within the strict scope line rather than removing one This is an expedient needed because of the incongruity between symmetry and the mechanics of Fitchstyle natural deductions When we introduce a strict scope line to the right it indicates a world accessible to the current world To indicate that the current world is accessible to another world we would have to write a line to the left which is not feasible in the Fitch system So instead we write within the restricted scope line a sentence that is the e ect of the current world being accessible to that world Because a is true at the current world Oar is true at any world to which the current world is accessible So that is what we write down to the right of a strict scope line2 Strict Reiteration B 1 Already derived ltgta SR B The rule is sound If a is true at a world then Oar is true at an arbitrary accessible world since by symmetry the current world is accessible from it We assert without proof that the derivational system BD resulting from adding this rule to the derivational rules of TD is complete with respect to the semantical system B We also assert without proof that the derivational system is sound This claim can be motivated by the way in which a derivation mirrors some of the se mantical reasoning used in the metalogical derivation above It does not mirror all the reasoning due to the limitation noted above of Fitchstyle systems To prove a I BD EIltgtar l a Assumption 2 B Our 1 SR B 3 Iltgta 2 El I For systems with El as primitive and 0 as derived we can amend the strict reiteration rule using Duality We begin with a instead of a and strictly reiterate Ela 2This is similar to the alternative rule for S4 SR4F which allows Ea to be written down across one restricted scope line when Ea occurs SR4F simulates the elfect of a third world at which a is written down Strict Reiteration EI B a Already derived Ia SREI B This is easily seen to be a derived rule given the original system with SR B Strict Reiteration for I B as a derived rule a Already derived ltgtar SR B Ela Duality For systems with 0 alone as primitive we can use the rule of Strict Reiteration B in conjunction with ltgt Introduction to derive the dual of Dog ie ltgtltgtar from a To prove a I BD ltgtltgta l Assumption 2 0a 1 SR B 3 ltgtar 2 Double Negation 4 ltgtltgtar 23 ltgt I 3 The Axiom System B The aXiom system B is obtained by adding to the aXiom schemata of T the further aXiom schema I B a D EIltgta This axiom is clearly valid in B by the same reasoning that was used to validate the corresponding semantical entailment 4 Applications of the B Systems Symmetry of accessibility makes for odd consequences For this reason the B systems only seem suitable for alethic applications which call for even stronger systems that contain them 41 Alethic Modal Logic The Bsystems take us closer to representing the logical modalities We have al ready remarked that if possibility is viewed logically what is the case at a world should possibly be the case at that world as well This what system T yields As suming a uniform set of laws of logic it seems reasonable to take this a step further and say that when something is the case at a world it is logically necessary that it be logically possible That is when something a is the case it follows from the laws of logic that a is consistent with the laws of logic For hypothetical modalities the symmetry restriction on accessibility may or may not be wanted depending on the details of the condition accessibility is sup posed to represent Consider Hughes and Cresswell s example of conceivability discussed in the context of the S4systems We may conceive a situation which con tains individuals who cannot conceive of our situation So from their standpoint our world is not a possible one On the other hand we might want to represent conceivable worlds as containing individuals perhaps even ourselves who can conceive of us in which case we would want accessibility to be symmetrical If the condition is a set of laws of nature the considerations noted in the last module may apply On the one hand it may be that exactly the same laws hold at all accessible worlds in which case symmetry seems appropriate On the other hand there may be some phenomena in an accessible world that are governed by additional laws of nature Since condition holding for the accessible world is expanded it may not be thought to apply appropriately to the home world which is what symmetry would require 42 Conditional Logic As far as a logic of implication is concerned we have the result that 31 a 3 000 If we regard strict implication as representing a notion of logical implication the considerations about logical modalities from the last section apply here The validity of a 3 EIltgtar may be thought desirable If a is true then it is a matter of logic that a is possibly true and it is also a matter of logic that the possible truth of a is necessary If we regard strict implication as local the considerations cited in the last two paragraphs show that we may or may not desire the result that at all accessible worlds where 1 holds it is necessarily possible that 1 holds 43 Deontic Logic Because they contain the respective T systems the Bsystems are too strong for deontic logic regardless of their characteristic consequencerelation But the char acteristic consequencerelation for the Bsystems is itself undesirable for a logic of obligation If we deny that everything that is the case at a world is permissible as is required by the T systems we would be even less inclined to admit a principle that states that everything that is the case is permissible as a matter of obligation 44 Doxastic Logic The same considerations apply to doxastic logic We have denied when discussing the T systems that what is the case is as a matter of logic compatible with what a logical saint believes at a time There is all the more reason to deny that as a mat ter of logic a logical saint believes of whatever is the case that it is compatible with what the person believes 45 Epistemic Logic Epistemic logics generally are built on the T systems so the problem for deontic and doxastic logics posed by the containment of the T systems by the Bsystems does not arise Thus we have a tBInym But any item of knowledge seems to be as well a belief and we have already seen that the Bsystems consequence relation is unsuitable for belief For this reason what the Bsystems add to the T systems is not suitable for a logic of knowledge It implausibly requires that any truth is not only compatible with what the subject knows at the time but is known to be compatible a tBIKmFma 46 Temporal Logic A temporal logic of what is past and what is future would have to regard the past and future in a very odd way if it were to admit the characteristic consequence relation of the Bsystems Such a logic would be based on the T systems which have already been seen to be undesirable since something s being the case now does not entail that it will be the case at some future time Of course we could introduce new operators signifying what is the case now and forever and what is the case now or at some future time Then the T systems would be appropriate What is now and forever the case is now the case and what is now the case is the case now or at some time in the future Still we would not want to say that if a is now the case that it now and forever is the case that it now is or will at some time be the case that a In the Bsystems the original operators representing the future and the past minus the reference to the present would function symmetrically A time in the future now is such that now is in its future etc This would undermine entirely the directional character of our notions of future and past Module 11 S5 and Equivalent Systems G J Mattey May 4 2007 Contents 1 The Semantical System S51 2 11 Accessibility as Re exive and Euclidean 2 12 Accessibility as Re exive Transitive and Symmetrical 6 13 Accessibility as an Equivalence Relation 7 14 Accessibility as a Universal Relation 8 15 Semantics without the Accessibility Relation 9 16 Semantics without Possible Worlds 9 17 Reduction ofModalities in SSI 10 2 The Derivational System S5D 11 3 The Axiom System S5 16 4 Applications of the S5 Systems 16 41 Alethic Modal Logic 16 42 Conditional Logic 17 43 Deontic Logic 17 44 Doxastic Logic 18 45 Epistemic Logic 18 46 Temporal Logic 18 The SS systems are the strongest normal systems That is they are the strongest modal systems in which the semantical rules behave uniformly at all the worlds in a frame In one formulation of the semantics in every frame all worlds must be accessible to all worlds No stronger constraint can be placed on the accessibility relation1 The SS systems are extensions of the Bsystems and the S4systems and therefore are extensions of the K D and Tsystems as well As with the other families of systems we will discuss the semantical system rst then the derivational system then brie y the axiom system and nally applications 1Axiom systems known as S6 S7 S8 and S9 are neither contained in nor contained in S5 but are formed by adding axioms to S2 or systems based on S2 1 The Semantical System SSI SSI is unusual in that it allows for a number of different formulations each yielding the same entailments In this section we will examine four ways of formulating the semantics 11 Accessibility as Re exive and Euclidean One way to build SSI is to begin with the semantical systems TI in which accessibility is re exive and add the further requirement that accessibility be euclidean R is euclidean i HXHyHzny A RXZ gt Ryz If a world is accessible to any two not necessarily distinct worlds then those two worlds are accessible to each other The de nition of a euclidean relation speci es that one is accessible to the other But each of the two accessible worlds could serve as the value of y or of z so they are mutually accessible Applied to frames this means that if a world W is accessible to W and a world W is accessible to W then Wj is accessible to Wi We can de ne an SSIframe as a set W R such that HWW E W gt RWW and HWHWiHWjW 6 WA w E W A Wj E W Awai A waj gt Rwiwj The euclidean character of the accessibility relation in SSIframes yields what will be called the charac teristic consequence of the SS systems Oar F551 Boar The proof of the entailment can be given using a metalogical derivation Semantical proof that Our 551 EIltgta 1 V1ltgta W T 2 7RWW2 A RWW1 gt RW2W1 3 ZWiXRWWi A V1a Wi T 4 RWW1 A V1a W1 T 5 7 RWW2 6 RWW1 7 RWW2 A RWW1 8 RW2W1 9 VI01W1 T 10 RW2W1 A V1aW1 T 11 EwiXRWZWi A V1a Wi T 12 VIltgta W2 T 13 RWW2 gt V1ltgta W2 T 14 V1Iltgta W T 15 V1EIltgta W T Assumption R is euclidean SRltgt Assumption Assumption 4 A E 5 6 A I 2 7 gt E 4 A E 8 9 A I 10 E I 11 SRltgt 412 gt I 13 SREI 3 414 2 E Note that this derivation does not depend on any restrictions on R other than its being euclidean So being euclidean could be added to the semantical systems KI or DI rather than TI to produce other semantical systems with the charateristic 5 consequence relation The reasoning can be represented graphically as follows A W2 A W Oar T Oar T EIltgtar T Note that only the requirement that R be euclidean and not that it be re exive plays a role in this proof Since R is re exive in an SSI frame SSI contains TI But the characteristic SSI entailment does not hold in TI 001 in Goa Proof Let W in a frame Fr contain three worlds w1 w2 and W3 such that Rw1w1 szwz RW3W3 Rwlwz and RW1W3 R is therefore re exive and so Fr is a TI frame Now let V1a wz T and V1a W3 F If follows from SRltgt that V1ltgtar w1 T and V1ltgtar W3 F From the second result it follows from SREI that V1EIltgta w1 F n n n gt W1 4 W2 W3 1 a T F Oar Oar T F EIltgtar F It can also be proved that 851 contains both S41 and BI and that it is stronger than each of the two systems To prove the former claim we show that if a relation is both re exive and euclidean it is also symmetrical and transitive and hence that any S41 frame and any BI frame is an SSI frame 2 Sketch of a proof that if R is re exive and euclidean then R is symmetrical 1 2 3 4 5 6 7 8 9 HXRXX HXHyHZRXy RXZ gt Ryl 7 Rab Raa Rab A Raa gt Rba Rab A Raa Rba Rab gt Rba HXXHYXRXY gt RYX Assumption Assumption Assumption 1 H E 2 H E X 3 3 5 A I 5 6 gt E 37 gt I 8 HI X 2 2If n quanti er introductions or eliminations are used in a single step the rule will be cited as having been used n times as in 2 HEX3 Sketch of a proof that if R is re exive and euclidean then R is transitive DOOQCDLIIJkWNt H O 11 HXRXX 7HxHYHZRXy RXZ gt Ryl Rab A Rbc iRaa Rab A Raa gt Rba Rab Rab A Raa Rba Rbc Rba A Rbc gt Rac Rba A Rbc Rac Rab A Rbc gt Rac HXHyHZRXy A Ryl gt RXZ Assumption Assumption Assumption IHE 2HEX3 3AE 46AI 57 gtE 3AE 2HEX3 89AI 1011 HE 312 gtI 13HIX3 It can be proved that the characteristic SSI entailment fails in both S41 and BI Our 5541 Boar Proof Let W in a frame Fr contain three worlds W1 W2 and W3 such that RW1W1 szwz RW3W3 RW1W2 and RW1W3 R is therefore re exive and trivially transitive so Fr is a S41 frame3 Now let v1a W2 T and v1a W3 F It follows from SRltgt that V1ltgtar W1 T From the same semantical rule it follows that v1ltgta W3 F Therefore by SREI V1EIltgta W1 F 001 31 Boa n n n gt W1 gt wz W3 1 a T F Oar Oar T F EIltgtar F Proof Let W in a frame Fr contain two worlds W1 W2 such that RW1W1 szwz RW1W2 and szwl R is therefore re exive and symmetric so Fr is a BIframe Now let v1a W1 F and v1a W2 T It 3R is trivially transitive because no two distinct worlds meet the antecedent of the conditional de ning transitivity follows from SRltgt that V1ltgtar w1 T From the same semantical rule it follows that V1ltgtar w1 F since a is false at both w1 Therefore by SREI V1EIltgta w1 F 12 Accessibility as Re exive Transitive and Symmetrical n 0 W1 SW2 0 F T EIltgtar F A second way to generate the semantical system is to require that an SSIframe is both transitive and sym metrical as well as being re exive In this way 851 is built on both S41 and BI We can de ne an SSIframe as a set W R such that HWW E W gt RWW and HWHWiHWW 6 WA W E W A Wj E W Awai A wai gt Rwiwj and HWHWL39W 6 WA w E W A wai gt Rwiw We have already seen that if a relation is euclidean then it is transitive and symmetrical We can also prove the converse Sketch of a proof that if R is transitive and symmetrical then it is euclidean H OOOQCI LIIkaN HXHYHZRXY Ryl gt RXZ 7HXXHyXRXy a Ryx Rab A Rac Rba A Rac gt Rbc Rab gt Rba Rba A Rac Rbc Rab A Rac gt Rbc HXHYHZRXY A RXZ gt Ryl Assumption Assumption Assumption 1HEX3 ZHEXZ 3AE 46 gt E 3AE 7 8 A I 49 gt E 310 gtl 11HIX3 From this result it follows that the rst condition on SSI frames that they be euclidean is a consequence of the two conditions just stated Since being euclidean is equivalent to being transitive and symmetrical the semantical system would produce the same results if it is structured in either of the two ways An example is a proof of the characteristic entailment Sketch of a semantical proof that Our 551 EIltgta 1 V1ltgta W T Assumption 2 7szw A RWW1 gt RW2W1 R is transitive 3 waz gt szw R is symmetrical 4 EwiRWWi A V1a Wi T SRltgtC 5 RWW1 A V1a W1 T Assumption 6 7 RWW2 Assumption 7 szw 3 6 gt E 8 RWW1 5 A E 9 szwAwal 7 8 A I 10 szwl 3 9 gt E 11 V1aw1T 5AE 12 szwl A v1aw1 T 1011 AI 13 Ewiszwi A V1a wi T 12 El 14 V1ltgta W2 T 13 SRltgt 15 waz gt V1ltgta W2 T 514 gtI 16 v1lltgtaw T 15 SREI 17 V1Iltgta w T 4 516 2 E 13 Accessibility as an Equivalence Relation A relation that is re exive transitive and symmetrical is called an equivalence relatian So we can de ne the accessibility relation for SSI simply as being an equivalence relation We can de ne an SSIframe as a set W R such that R is an equivalence relation Examples of equivalence relations are being identical t0 being the same size as being semantically equivalent t0 In the case of semantical equivalence we can say that every sentence in a language say SL is semantically equivalent to itself If a is semantically equivalent to and is semantically equivalent to y then a is semantically equivalent to 7 And if a is semantically equivalent to then is semantically equivalent to a Note that in SI not every sentence is equivalent to every other sentence Instead sentences which stand in the equivalence relation to one another constitute an equivalence class of sentences For example the class of all valid sentences is an equivalence class as is the class of all negations of valid sentences4 In the case of being the same size as for each size there is an equivalence class of things of that size Applied to frames this means that W can be divided into equivalence classes such that the members of each class stand in the equivalence relation to themselves and all the other members of the class It will be shown that no member Wi of one equivalence class is accessible to or from any member W J of any other equivalence class If it were the case that Wi is accessible to W j then Wi would stand in an equivalence relation to W j and by the properties of the equivalence relation it would stand in an equivalence relation to all the other members of the class Moreover because Wi stands in an equivalence relation to all the members of its awn equivalence class all of them would stand in the equivalence relation to all the members of the other equivalence class Then there would only be one equivalence class which is contrary to the assumption that there are two equivalence classes We can reprove the characteristic entailment of 51 in terms of the equivalence relation Suppose that V1ltgta W T Then there is some world Wi such that wai where V1a Wi T Because R is an equivalence class all worlds W that are accessible to W are such that W is accessible to them Therefore at all Wj accessible to W V1ltgtar Wj T in which case V1EIltgta W T 14 Accessibility as a Universal Relation If we take accessibility to be an equivalence relation the set of worlds W might be partitioned into a number of isolated equivalence classes That this kind of frame is permitted does not affect any of the semantical results we have obtained From the standpoint of a given equivalence class the truthvalue assigments at worlds belonging to the other classes are completely irrelevant If for example Oar is true at a world W then the truthvalue of EIltgtar will be determined by the value of Oar at all worlds accessible to W which is to say all worlds in the equivalence class The validity in 51 of Oar D EIltgtar depends on its being true in all worlds on all interpretations given any SSIframe And it is true at each world in each equivalence class so it is true at all the worlds in a frame So there is no result that holds for a partitioned set of worlds that does not hold for an unpartitioned set where all the worlds in a frame stand in the equivalence relation This suggests that we can get the same semantical results for 51 if we make accessibility a universal relation ie a relation holding between each world in W In general R is universal if and only if HXHyny We can de ne an SSIframe as a set WR such that HWHWL39W 6 WA Wi E W gt RWWL39 This simpli es considerably the semantical proofs in 51 For example suppose that on an SSIframe Fr of this type on an arbitrary interpretation I based on Fr and at an arbitrary world W in Fr that v1ltgta W T Then at some accessible world Wi a is true Because R is universal Wi is accessible to all worlds so that Oar is true at all worlds And because Oar is true at all worlds it is true at all worlds accessible to W in which case EIltgtar is true at W 4The negations of valid sentences are often called contradictions or inconsistent or logically false sentences So the class of all contradictions is an equivalence class 15 Semantics without the Accessibility Relation As the preceding proof indicates when accessibility is a universal relation it plays no role in determining semantical results We could just as well have reasoned as follows Suppose that at an arbitrary world W V1ltgtaW T Then at some world Wi V1a Wi T So v1ltgta Wj T at all worlds Wj in which case v1lltgta W T So yet another version of the semantical system SIS is one in which there is no relation of accessibility Then a frame would be only an ordered single consisting of a set of worlds W and an interpretation would be a pair W V If this simplifying move is to be made the semantical rules must be changed to omit the reference to accessibility The following are the semantical rules for the two main modal operators EI and ltgt SREI5 V1EIa WT if and only if V1a WiT at all worlds Wi in I V1EIa WF if and only if V1a WiF at some world W in I SRltgt5 V1ltgtar WT if and only if V1a WiT at some world W in I V1ltgtar WF if and only if V1a WiF at all worlds Wi in I It is easily proved that these semantical rules are equivalent to the semantical rules with the accessibility relation R if R is a universal relation If R is universal then reference to R in SREI can be eliminated 1 HWHWiRWWi R is universal 2 HwiRWWi gt V1a Wi T SREI 3 HwiRWWi 1 H E 4 RWW1 3 H E 5 RWW1 gt V1a W1 T 2 H E 6 V1aW1T 454E 7 HWV1a W T 6 HI Exercise Prove the converse 16 Semantics without Possible Worlds A nal simpli cation generates what is roughly the semantical system of Carnap 19465 It is possible to dispense with possible worlds altogether in favor of in effect possible interpretations An interpretation is de ned as in SI the semantical system for Sentence Logic It consists of a one place valuation function which assigns the value T or the value F but not both to each sentence letter The semantical rules for the nonmodal operators are just as they are with SI 5 Modalities and Quanti cation Jaumal afSymbalic Logic 11 1946 pp 3364 and Meaning and Necessity 1948 The chief dilferences are that Carnap does not use the modern notion of an interpretation in which a valuation function assigns truth values to sentences In place of interpretations he uses statedescriptions which in Sentential Logic would be a set of sentence letters What corresponds to the assignment of T is membership in the set and what corresponds to the assignment of F is non membership in the set Hintikka based his early modal semantics on this method but Kripke s formulation in terms of valuation functions ultimately became the standard Carnap s idea for the modal operator EI was that it should re ect the semantical concept of L truth His informal de nition of Ltruth in a system S is truth in such a way as can be established on the basis of the semantical rules of the system S alone without any reference to extralinguistic facts 6 The formal de nition of L truth in terms of the semantical system SI presented here is truth in every interpretation ie validity The semantical rule which is supposed to re ect the notion of L truth is unusual The two possible outcomes of the use of the rule are not truth and falsehood within a single interpretation as with SI Instead the alternatives are truth in all interpretations or falsehood in all interpretations v1laT if and only if l39lIv1arT V1IaF if and only if EIV1arF The corresponding rule for possibility then would be as follows VIltgtaT if and only if EIV1aT V1ltgtaF if and only if HIV1aF A consequence of these rules is that the truthvalue of any modal sentence is the same on all inter pretations All necessarily true sentences on a given interpretation are true on all interpretations and all possibly true sentences on a given interpretation are true on all interpretations Suppose that on some in terpretation I V1IaT Then by the semantical rule 1 HV1arT Using the semantical rule in the other direction we can assert that Bar has the value T on an arbitrary interpretation and so it has the value T on all interpretations Exercise Prove that that the corresponding claim holds for possibility The following is a proof of the characteristic SSI entailment In giving the proof we will appeal to the following lemma which states the result just given for the case of the 0 V1ltgtaT gt HIV1ltgtaT Now it is easy to prove the entailment Sketch of a semantical proof that Our 551 EIltgta 1 V1ltgta T Assumption 2 HIV1ltgtar T Lemma 3 V1Iltgta T 2 Rule for El Carnap had good reason to give the semantical rules he did as his goal was to represent logical necessity But given the structure of the rules he adopted to represent logical necessity there is no hint of any kind of relativity of the truthvalues of modal sentences It is small wonder that Carnap did not nd in this semantical system a key to possibleworlds semantics with the accessibility relation 17 Reduction of Modalities in 51 We saw that in the S4systems there remain a number of irreducible strings of modalities That is there are sentences which are pre xed with a string of two or more modal operators of both kinds which are not equivalent to sentences with fewer operators In the SS systems this result does not hold Every sentence pre xed with a string of two or more modal operators is equivalent to a sentence with only one operator the innermost one This is the ultimate reduction of modalities The only further reduction would be to allow 5Meaning and Necessity Second Edition p 10 a sentence with a single modal operator as its main operator to be equivalent to a sentence with no moral operator This would collapse the modal system into the nonmodal system The modal sentences ltgtZlltgtar and Dog are reducible to Oar and the modal sentences EIOEIa and Omar are reducible to the Bar We have already seen how EIltgtar reduces to Oar The result from left to right holds in the TI and the result from right to left has been shown in this module to hold in 51 in its various guises Exercise Prove the other three reductions in the variants of the semantical system 2 The Derivational System S5D As was the case with the symmetrical accessibility relation in the semantical system BI the euclidean re lation is not readily mapped onto the structure of Fitchstyle derivations So as with ED we shall resort to a device that simulates the effect of a euclidean accessibility relation Inspection of the characteristic 5 entailment Oar 551 Dog shows that if Oar is taken to be true at W then it must also be true at all accessible worlds and hence at an arbitrary accessible world So a sound rule would be one that allows us to use Strict Reiteration to bring a sentence of the form Oar across a restricted scope line intact Strict Reiteration for El 5 Oar Already derived D Oar SREI5 We shall assume that the derivational system resulting from the addition of SR5 to the rules for TD is complete as well as being sound With SR5 proof of the characteristic derivationalrelation is straightforward To prove 0a I S5D EIltgta 1 Oar Assumption 2 D Oar 1 SREI5 3 ma 2 El 1 For systems with El as primitive and 0 as derived we can amend the strict reiteration rule using Duality We begin with Ela instead of a and strictly reiterate Ela Strict Reiteration for I 5 Ela Already derived Ia SRI 5 This is easily seen to be a derived rule given the original system with SR 5 Strict Reiteration for I 5 as a derived rule Ela Already derived ltgtar Duality El ltgta SR 5 Ela Duality We can show that the SRB and SR4 rules are derived rules given SRS and Strong 0 Introduction which re ect the euclidean and re exive nature of accessibility respectively For SRB this is easy to see When we are given a sentence a on a line we can use Strong 0 Introduction to get Oar and then apply the SRS rule Strict Reiteration B as a derived rule 1 1 Already derived 2 Oar Strong 0 I 3 u 4 5 6 0a SRS The proof that SR4 is a derived rule is much more complicated This is mirrored by the greater com plexity of the derivation of transitivity of a relation from its being re exive and euclidean In particular two di erent uses of H Introduction had to be made in the former derivation Strict Reiteration for El 4 as a derived rule Elar Given that the semantical system for 851 can be generated by making accessibility an equivalence rela tion it should be the case that SREI5 is a derived rule given El E SRB and SREI4 Respectively they re ect re exivity symmetry and transitivity Indeed it can be shown that SREI5 is a derived rule given just the two Strict Reiteration rules This is most easily demonstrated using Duality and the derived SD rule Double Negation Already derived Strong 0 I SREI5 Assumption Duality SREI5 Duality ltgt I Reiteration E SREI Strict Reiteration for El 5 as a derived rule Oar Already derived El ltgtar Assumption Elar Duality ltgtltgtar SRB ltgtltgta Double Negation ltgtEla Duality EIEIar Duality D D ar SREI4 Ia El 1 IIa El 1 Oar E Given the simpli cation introduced by the noaccessibility semantical system 551 it might be wondered whether a simpler derivational system could be based on it The simpli ed semantics has the property that the truthvalue of a modal sentence is invariant across worlds7 If Oar is true at a world W then it is true at all worlds in W since all that it required is that a be true at some world in W Similar reasoning holds for EIarar 3 andao To re ect this situation we can still use restricted scope lines to represent other worlds but now the only restriction is that a reiterated sentence must be modal To generate a simpli ed derivational system 85D we can consolidate the all the Strict Reiteration rules into a single rule If a is a modal sentence it may be strictly reiterated across any number of restricted scope lines We can call this rule SRM or Strict Reiteration for Modalities Strict Reiteration for Modalities 1 Already derived a SRM Provided a is a sentence whose main operator is modal 7Recall that a modal sentence is one whose main operator is modal 14 This blanket rule gives us the eITect of the basic Strict Reiteration rule for El which is also the rule for KD since Ela can be strictly reiterated by SRM and El Elimination applied to yield a which is what SREI requires Strict Reiteration for 1 as a derived rule Ela Already derived D Ela SRM a El E The proof of Strict Reiteration for EI 4 as a derived rule is straightforward There are two uses of SRM followed by a use of El Elimination Strict Reiteration for I 4 as a derived rule Ela Already derived D Ela SRM D Ela SRM a El E For SRB if 1 occurs in a step Weak 0 Introduction may be applied and the resulting Oar may be strictly reiterated by SRM Strict Reiteration B as a derived rule 1 Already derived Oar Strongltgt I D Oar SRM This shows that SRM yields all the lesspowerful Strict Reiteration rules as derived rules It remains to be shown that the original Strict Reiteration rules yield SRM But we have already shown Fitch s version of Strict Reiteration for El 4 allows the strict reiteration of Dar intact across a strict scope line And the Strict Reiteration for El 5 similarly allows the strict reiteration of Oar intact Exercise Show why sentences whose main operator is 3 may be reiterated intact across strict scope lines given the rules for KD through 85D 3 The Axiom System SS The axiom system 85 is obtained by adding to the axiom schemata of T the further axiom schema IS 5 Oar 3 D00 This axiom is valid in 51 by the same reasoning that was used to validate the corresponding semantical entailment 4 Applications of the SSSystems The SS systems are extremely strongitoo much so for most applications They seem only to represent logical modalities 41 Alethic Modal Logic The original application of the S5systems was made by Carnap who understood the El to represent logical necessity His way of understanding the meaning of logical necessity is close to our notion of what holds by virtue of laws of logic The concept of logical necessity seems to be commonly understood in such a way that it applies to a proposition 1 if and only if the truth of p is based on purely logical reasons and not dependent upon the contingency of facts in other words if the assumption of notp would lead to a logical contradiction independent of facts Meaning and Necessity p 174 Carnap defended his choice of 51 as the semantical system for logical necessity and possibility by claiming that the fact of Ltruth is itself based on purely logical reasons Here is a quotation from Meaning and Necessity Carnap uses different notation than ours It should be kept in mind that his S2 is our 51 and his N is our EI Let A be an abbreviation for an Ltrue sentence in S2 for example Hs V Hs Then NA is true according to semantical rule 391 And moreover it is L true because its truth is established by the semantical rules which determine the truth and thereby the Ltruth of A together with the semantical rule for N say 391 Thus generally if N is true then NN is true hence any sentence of the form Np D NNp is true This constitutes an af rmative answer to the controversial question mentioned in the beginning Meaning and Necessity Second edition p 174
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