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# Modal Logic PHI 134

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This 36 page Class Notes was uploaded by Marlee Kulas on Tuesday September 8, 2015. The Class Notes belongs to PHI 134 at University of California - Davis taught by George Mattey in Fall. Since its upload, it has received 20 views. For similar materials see /class/191923/phi-134-university-of-california-davis in PHIL-Philosophy at University of California - Davis.

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Module 10 B and Equivalent Systems G J Mattey May 1 2007 Contents 1 The Semantical System BI 2 2 The Derivational System BD 4 3 The Axiom System B 6 4 Applications of the BSystems 6 41 Alethic Modal Logic 6 42 Conditional Logic 7 43 Deontic Logic 7 44 Doxastic Logic 7 45 Epistemic Logic 7 46 Temporal Logic 7 461 The Brouwer Connection 8 We saw earlier that the Tsystems yield the entaillnent 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 Brouwersche system due to some similarities between its characteristic consequencerelations and an element of Brouwer s intuitionistic interpretation of mathematics to be discussed in the last section The system is obtained by adding a symmetry requirement for the accessibility relation to the semantical system The Bsystems are extensions of the Tsystems 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 Tsystems characteristic consequencerelations are independent of those of the Bsystems so there are systems with those consequence relations which are stronger than the K systems but weaker than 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 The axiom system B is built on T as a f i and r quot the i system BI is based on the semantical system T Thus the accessibility relation in a BIframe is re exive That relation is also symmetrical R is symmetrical if and only if HxHyny 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 Because accessibility in the semantical system BI is serial as a consequence of its re exivity the condition in the antecedent is always met for at least one world accessible to a given world W Each world has at least one world accessible to it and by symmetry they are mutually accessible So each world W stands in a symmetrical relation of accessibility to at least one world This fact is guaranteed as well by re exivity every world is accessible to itself in which case it trivially stands in an acessibility relation to itself If RWW then RWW Applied to frames this means that if a world Wi that belongs to a frame is accessible to W then W is accessible to W We can de ne a BIframe as a set WR such that HWW E W gt RWW and HWHWLW 6 WA W E W A RWWi gt RWL39W The symmetry of the accessibility relation in BI frames yields what will be called the characteristic consequence of the Bsystems 0 Hal Boa The proof of the entailment can be given using a metalogical derivation Sketch of a semantical proof that a 31 EIltgta 1 v1a W T Assumption 2 RWW1 gt RW1W Symmetry of R 3 RWW1 Assumption 4 RW1W 2 3 gt E 5 RW1WV1aWT 34AI 6 V1ltgtar W1 T 5 SRltgt 7 RWW1 gt V1ltgta W1 T 26 gt I 8 V1Iltgta W 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 KI or DI rather than TI to produce other semantical systems with the charateristic B consequence relation This reasoning can be illustrated with a modal truthtable a T Oar Oar T T EIltgtar T Note that the inclusion of a value for Oa at w1 is made because of re exivity In fact since wz is an arbitrary world wz could be w1 itself which is why this inclusion is not made in the metalogical derivation All TI 4entailments and hence all DI and KI entailments are BI entailments 71YnTI a gt Y1YnBI an Since the class of BI frames is a subset of the class of TI frames any entailment that holds in all TI frames also holds in all BI frames So the semantical system TI is contained in the semantical system BI BI is a stronger system than TI in that some BIentailments are not TI entailments because some TI frames are not BIframes So BI is an extension of TI and hence of DI and KI Speci cally 01 TI Boa Proof Let W in a frame Fr contain two worlds w1 and wz such that Rw1w1 Rwlwz and szwz R is therefore re exive and so Fr is a TI frame Now let v1a w1 T and v1a wz F If follows from SRltgt that V1ltgta wz F since a is false at all accessible worlds ie at wz itself Therefore V1EIltgta w1 F by SREI n 0 W1 4 W2 1 a T F Oar Oar T F EIltgtar F The semantical systems BI and S4I do not contain each other The accessibility relation R may be tran sitive 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 sup ports the characteristic consequence relation of the other The frame just given shows that the characteristic BI consequence does not hold in S4I The accessibility relation in that frame is trivially transitive since the condition that triggers the additional passthrough accessibility relation so to speak is not satisifed For the failure of the characteristic S4I consequence in a BI frame let the frame have three worlds and let Rw1w1 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 as as we 2 The Derivational System BD Just as the semantical system for BI contains the semantical rules inherited from KI DI and TI the deriva tional system inherits the derivational rules from these systems We will give an additional rule that gener ates 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 based on KD because the reiteration introduces a new operator altogether within the strict scope line rather than removing one1 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 On the other hand with the modal truthtables we could introduce an arrow pointing to the left So we write within the restricted scope line a sentence that is the re 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 B Our 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 BI 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 semantical reasoning used in the metalogical derivation above It does not mirror all the reasoning due to the limitation noted above of Fitchstyle systems 1This also occurs for similar reasons with the derivational rules needed to derive the axioms of K as described in Module 6 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 To prove a I BD EIltgta 1 a Assumption 2 D Oar 1 SR B 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 a instead of a and strictly reiterate Ela Strict ReiterationEI B a Already derived Ia SRI 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 With this rule in hand we can prove that a I BD EIEIar which is the characteristic BD consequence in a syntax without the 0 operator To prove a I BD IIa Assumption 1 Double Negation 2 SREI B 2 3 El I 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 1 Assumption 2 Oar 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 Ewan This axiom is clearly valid in BI by the same reasoning that was used to validate the corresponding seman tical entailment 4 Applications of the BSystems Symmetry of accessibility makes for odd consequences For this reason the Bsystems 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 already 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 the Tsystems yield Assuming 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 oflogic For hypothetical modalities the symmetry restriction on accessibility may or may not be wanted de pending on the details of the condition accessibility is supposed to represent Consider Hughes and Cress well s example of conceivability discussed in the context of the S4systems We may conceive a situation which contains 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 indi viduals 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 We may regard strict implication as holding locally indicating only a connection of truthvalues at accessible worlds Then 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 Tsystems the Bsystems are too strong for deontic logic regardless of their characteristic consequencerelation But the characteristic 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 Tsystems 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 Tsystems 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 matter 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 Tsystems so the problem for deontic and doxastic logics posed by the containment of the Tsystems by the Bsystems does not arise Thus we have a tBIFSYta 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 Tsystems 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 or tBIKsthmar A logical saint might know what holds as a matter of logic but a contingent sentence a may or may not be compatible with what he knows and this does not seem to be guaranteed by its mere truth 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 Tsystems 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 Tsystems 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 47 The Brouwer Connection Intuitionist mathematicians such as Brouwer were opposed to any mathematical proofs which employ the rule we here call Negation Elimination If one assumes that the negation of some mathematical proposition is true and derives a contradiction it would follow from Negation Elimination that the proposition itself is true The intuitionists held that this would allow the proof of the existence of mathematical objects without any positive evidence that the object exists This kind of reasoning is used in metalogic as well It is often proved that a derivation exists by denying that there is a derivation and showing that a contradiction follows from this denial If we were to remove Negation Elimination from our primitive ruleset we would not be able to prove that 1 follows from a Thus for an intuitionist logic I a H a On the other hand the converse consequence a I a is allowed Consider Negation Introduction according to which if the assumption of 1 leads to a contradiction we may assert a Here we might be tempted to say that what is really proved by the rule is that a is impassible ie ltgta We can in fact get this result in KD 1 D Assumption 2 3 4 5 a Negation Introduction 6 ltgta ltgtI The sentenceschema ltgtar can be symbolized with the impossibility operator Then the characteristic B consequence a I B ltgtltgta becomes a I B a which lines up with what is allowed for negation by the intuitionists On the other hand ltgtltgta 3 a and thus a 3 a This fact is shown by constructing a modal truthtable in which ar is true and a is false at a world A n W1 3 W2 ltgtltgtar a a T F T ltgtltgta 001 001 F T T ltgtar ltgtar F F So the behavior of the impossibility operator in the Brouwersche systems has an af nity with that of the negation operator in intuitionistic logic3 33 Hughes and Cresswell A New Intraductian t0 Madal Logic pp 7071 Module 10 B and Equivalent Systems G J Mattey May 1 2007 Contents 1 The Semantical System BI 2 2 The Derivational System BD 4 3 The Axiom System B 6 4 Applications of the BSystems 6 41 Alethic Modal Logic 6 42 Conditional Logic 7 43 Deontic Logic 7 44 Doxastic Logic 7 45 Epistemic Logic 7 46 Temporal Logic 7 461 The Brouwer Connection 8 We saw earlier that the Tsystems yield the entaillnent 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 Brouwersche system due to some similarities between its characteristic consequencerelations and an element of Brouwer s intuitionistic interpretation of mathematics to be discussed in the last section The system is obtained by adding a symmetry requirement for the accessibility relation to the semantical system The Bsystems are extensions of the Tsystems 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 Tsystems characteristic consequencerelations are independent of those of the Bsystems so there are systems with those consequence relations which are stronger than the K systems but weaker than 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 The axiom system B is built on T as a f i and r quot the i system BI is based on the semantical system T Thus the accessibility relation in a BIframe is re exive That relation is also symmetrical R is symmetrical if and only if HxHyny 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 Because accessibility in the semantical system BI is serial as a consequence of its re exivity the condition in the antecedent is always met for at least one world accessible to a given world W Each world has at least one world accessible to it and by symmetry they are mutually accessible So each world W stands in a symmetrical relation of accessibility to at least one world This fact is guaranteed as well by re exivity every world is accessible to itself in which case it trivially stands in an acessibility relation to itself If RWW then RWW Applied to frames this means that if a world Wi that belongs to a frame is accessible to W then W is accessible to W We can de ne a BIframe as a set WR such that HWW E W gt RWW and HWHWLW 6 WA W E W A RWWi gt RWL39W The symmetry of the accessibility relation in BI frames yields what will be called the characteristic consequence of the Bsystems 0 Hal Boa The proof of the entailment can be given using a metalogical derivation Sketch of a semantical proof that a 31 EIltgta 1 v1a W T Assumption 2 RWW1 gt RW1W Symmetry of R 3 RWW1 Assumption 4 RW1W 2 3 gt E 5 RW1WV1aWT 34AI 6 V1ltgtar W1 T 5 SRltgt 7 RWW1 gt V1ltgta W1 T 26 gt I 8 V1Iltgta W 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 KI or DI rather than TI to produce other semantical systems with the charateristic B consequence relation This reasoning can be illustrated with a modal truthtable a T Oar Oar T T EIltgtar T Note that the inclusion of a value for Oa at w1 is made because of re exivity In fact since wz is an arbitrary world wz could be w1 itself which is why this inclusion is not made in the metalogical derivation All TI 4entailments and hence all DI and KI entailments are BI entailments 71YnTI a gt Y1YnBI an Since the class of BI frames is a subset of the class of TI frames any entailment that holds in all TI frames also holds in all BI frames So the semantical system TI is contained in the semantical system BI BI is a stronger system than TI in that some BIentailments are not TI entailments because some TI frames are not BIframes So BI is an extension of TI and hence of DI and KI Speci cally 01 TI Boa Proof Let W in a frame Fr contain two worlds w1 and wz such that Rw1w1 Rwlwz and szwz R is therefore re exive and so Fr is a TI frame Now let v1a w1 T and v1a wz F If follows from SRltgt that V1ltgta wz F since a is false at all accessible worlds ie at wz itself Therefore V1EIltgta w1 F by SREI n 0 W1 4 W2 1 a T F Oar Oar T F EIltgtar F The semantical systems BI and S4I do not contain each other The accessibility relation R may be tran sitive 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 sup ports the characteristic consequence relation of the other The frame just given shows that the characteristic BI consequence does not hold in S4I The accessibility relation in that frame is trivially transitive since the condition that triggers the additional passthrough accessibility relation so to speak is not satisifed For the failure of the characteristic S4I consequence in a BI frame let the frame have three worlds and let Rw1w1 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 as as we 2 The Derivational System BD Just as the semantical system for BI contains the semantical rules inherited from KI DI and TI the deriva tional system inherits the derivational rules from these systems We will give an additional rule that gener ates 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 based on KD because the reiteration introduces a new operator altogether within the strict scope line rather than removing one1 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 On the other hand with the modal truthtables we could introduce an arrow pointing to the left So we write within the restricted scope line a sentence that is the re 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 B Our 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 BI 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 semantical reasoning used in the metalogical derivation above It does not mirror all the reasoning due to the limitation noted above of Fitchstyle systems 1This also occurs for similar reasons with the derivational rules needed to derive the axioms of K as described in Module 6 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 To prove a I BD EIltgta 1 a Assumption 2 D Oar 1 SR B 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 a instead of a and strictly reiterate Ela Strict ReiterationEI B a Already derived Ia SRI 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 With this rule in hand we can prove that a I BD EIEIar which is the characteristic BD consequence in a syntax without the 0 operator To prove a I BD IIa Assumption 1 Double Negation 2 SREI B 2 3 El I 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 1 Assumption 2 Oar 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 Ewan This axiom is clearly valid in BI by the same reasoning that was used to validate the corresponding seman tical entailment 4 Applications of the BSystems Symmetry of accessibility makes for odd consequences For this reason the Bsystems 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 already 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 the Tsystems yield Assuming 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 oflogic For hypothetical modalities the symmetry restriction on accessibility may or may not be wanted de pending on the details of the condition accessibility is supposed to represent Consider Hughes and Cress well s example of conceivability discussed in the context of the S4systems We may conceive a situation which contains 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 indi viduals 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 We may regard strict implication as holding locally indicating only a connection of truthvalues at accessible worlds Then 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 Tsystems the Bsystems are too strong for deontic logic regardless of their characteristic consequencerelation But the characteristic 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 Tsystems 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 Tsystems 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 matter 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 Tsystems so the problem for deontic and doxastic logics posed by the containment of the Tsystems by the Bsystems does not arise Thus we have a tBIFSYta 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 Tsystems 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 or tBIKsthmar A logical saint might know what holds as a matter of logic but a contingent sentence a may or may not be compatible with what he knows and this does not seem to be guaranteed by its mere truth 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 Tsystems 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 Tsystems 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 47 The Brouwer Connection Intuitionist mathematicians such as Brouwer were opposed to any mathematical proofs which employ the rule we here call Negation Elimination If one assumes that the negation of some mathematical proposition is true and derives a contradiction it would follow from Negation Elimination that the proposition itself is true The intuitionists held that this would allow the proof of the existence of mathematical objects without any positive evidence that the object exists This kind of reasoning is used in metalogic as well It is often proved that a derivation exists by denying that there is a derivation and showing that a contradiction follows from this denial If we were to remove Negation Elimination from our primitive ruleset we would not be able to prove that 1 follows from a Thus for an intuitionist logic I a H a On the other hand the converse consequence a I a is allowed Consider Negation Introduction according to which if the assumption of 1 leads to a contradiction we may assert a Here we might be tempted to say that what is really proved by the rule is that a is impassible ie ltgta We can in fact get this result in KD 1 D Assumption 2 3 4 5 a Negation Introduction 6 ltgta ltgtI The sentenceschema ltgtar can be symbolized with the impossibility operator Then the characteristic B consequence a I B ltgtltgta becomes a I B a which lines up with what is allowed for negation by the intuitionists On the other hand ltgtltgta 3 a and thus a 3 a This fact is shown by constructing a modal truthtable in which ar is true and a is false at a world A n W1 3 W2 ltgtltgtar a a T F T ltgtltgta 001 001 F T T ltgtar ltgtar F F So the behavior of the impossibility operator in the Brouwersche systems has an af nity with that of the negation operator in intuitionistic logic3 33 Hughes and Cresswell A New Intraductian t0 Madal Logic pp 7071 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 As was stated in the last module it seems desirable that if a subject believes that a at a time then a is compatible with what the subject believes We have this result in DI Ema IDI Pmar Moreover there are no DI frames containing deadends so if Ema is true at a world it is because there are accessible worlds at which 1 holds and it holds at all of them This is a much more realistic view of belief than one that yields a belief when nothing is compatible with what one believes The same considerations hold for knowledge insofar as it requires belief It is also obvious that a temporal logic of the future andor the past can only be adequately represented in systems at least as strong as the Dsystems It is plausible to say that if it always will be the case that a then it will at some time be the case that a If there is a future at all then it contains some times Less obvious is the requirement in the semantics that for each time there is a future time which is what would be represented by a serial accessibility relation where an accessible world is a future world Note that if the number of times is nite the serial relation would lead to some kind of loop where an accessible world is either present or past In that case either the present or the past lies in the future If there are in nitely many times then the seriality condition requires that there be a future time for any time which is as it should be The same remarks hold mutatis mutandis for what has at all times been the case Philosophy 134 Module 1 Informal Introduction to Modal Logic G J Mattey March 27 2007 Contents 1 Motivation for Modal Logic 1 2 The Lewis Systems 2 3 Matrix Semantics 2 4 PossibleWorlds Semantics 3 5 Applications 5 6 Natural Deduction Systems 5 7 Plan of the Text 5 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 passible 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 a valid argument in turn depends essentially on the madal concepts of possibility and necessity These concepts are called modal because they indicate 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 standardized 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 a operators and or not etc but there is no symbol in the language for therefore 1 One might see the symbol 39 but this is an expression of the metalanguage In fact the operators of sentential and 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 operators 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 validity That is it would contain a madal apemtar 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 Whitehead s Principia Mathematical2 A young philosophy instructor at UC Berkeley CI Lewis used Principia as a text Lewis thought that Russell s description of the truth functional conditional operator as material implication was misleading In response he built several axiomatic systems with various modal operators including one for impossibility and another for consistency The most interesting of the operators was one he called strict implication which Lewis thought better represents the relation between premise and conclusion in an argument than does material implication After several false starts Lewis produced a system that he later called S3 3 Oneplace modal operators for possibility and necessity eventually became components of the Lewis systems The necessity operator can be understood as allowing the represention of the concept of logically necessary truth Strict implication in S3 is equivalent to a necessarily true material implication The strict implication operator and its relation to material implication will be described in more detail in Module 6 Some time after the introduction of the original systems of modal logic one of which he came to call S3 Lewis formulated several related systems of modal logic Systems S4 and S5 were stronger than S3 in that more modal sentences could be proved in them Systems S1 and S2 are weaker in that some sentences that could be proved in S3 could not be proved in S1 or S2 In this text we will be examining in detail only the strongest Lewis systems S4 and S5 These systems belong to a family of normal systems which have the same basic semantics 3 Matrix Semantics Lewis s systems were laid down in axiomatic form The axioms were taken to express basic facts about modality given an intuitive way of understanding what they meant The earliest work on the systems was to prove theorems of the given systems which follow from their speci c axioms Very soon thereafter the axiomatic systems were given interpretations The most prominent kind of interpretation was with matrices that resemble truthtables A useful matrix for modal logic typically contained more than two values The following is an example of a matrix for the strict implication symbol 3 The numbers 1 and 1Many logic students render the material conditional into English as therefore but it will be seen in the next module that this is a mist e 2The basic system in that book had been laid out by Frege in 1879 but had gone largely unnoticed 3The system of A Survey afSymbalic Lagic 1918 Chapter V contained a devastating error in the formulation of one of the axioms As a result this chapter appears only in the rst edition The stable version of S3 is given in Strict Implicationian Emendation Jaumal afPhilasaphy Vol 17 1920 pp 300302 2 are designated values which play the same role as truth in a truthtable while the numbers 3 and 4 correspond to truthtable falsehood 4 Three rows of interest are highlighted A D bbbbWWWWNNNNHHHHgt wab thb wab wap uy NNNNhNhNWWNwaBN L A brief examination of the matrix reveals differences between strict and material implication In the second and tenth rows B has the designated value but A 3 B does not In the tenth and twelfth rows A does not have the designated value andA 3 B does not With material implication if B has the designated value truth or A does not have it falsehood then A D B has the designated value truth Note however that there is no row of the table where A has a designated value B has a nondesignated value and A 3 B has a designated value Strict implication respects a necessary condition of any implication that it may not take us from truth to falsehood 4 PossibleWorlds Semantics Using matrices logicians were able to get other important results about the systems They could determine which systems contain which other systems and whether a given axiom is independent of the other axioms5 While the matrix technique was useful it provided no real insight into the meanings of modal sentences It was Rudolf Carnap writing in the mid 1940s who rst provided an intuitively plausible semantics for one of the Lewis systems SS6 In Carnap s semantical system the truthvalues of nonmodal sentences are determined just as they are in truthfunctional logic A sentence whose main operator is a necessity operator is true if and only if the sentence it governs is a logical truth 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 truthtables 4C 1 Lewis and C H Langford Symbalic Lagic 1932 Appendix II This matrix is entitled Group V 5System S contains system S just in case all the theorems of S are theorems of S S is then said to be stronger than S and S weaker than S An axiom is independent of the other axioms in an axiom set just in case it cannot be proved as a theorem from the other axioms 6 Modalities and Quanti cation Jaurnal UfSymbalic Lagic Vol 11 1946 pp 3364 See also Meaning and Necessity 1947 Carnap also provided a system for a predicatelogic version of SS His semantics is of interest because of the way it interprets the syntax of predicate logic Nonmodal semantics interprets terms as standing for ob jects in a universe 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 description Carnap wrote that the state description represents Leibniz s possible worlds or Wittgenstein s possible states of a airs 7 A term might stand for diITerent objects in diITerent state descriptions so that its intension is a function from state descriptions to objects A predicate might have diITerent extensions in diITerent state descriptions so that its intension is a function from state descriptions to sets of objects In the late 1950s Carnap s semantics was generalized to the form in which it exists today8 The key notion in the semantics is that of a possible world In sentential logic a possible world corresponds to a row of a truthtable In predicate logic a possible world corresponds roughly to an interpretation which spells out what objects there are and what properties they have Carnap s system in eITect took necessity to be truth at all possible worlds This worked as a semantics for SS but not for any of the weaker systems of Lewis and others The innovation was to add to the semantics a twoplace relation 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 mean It should be noted that for a long time modal logic was held in some disrepute due to the criticisms of WVO Quine9 One of his objections was that valdity implication logical necessity etc are metalogical notions which have no place in logic itself Another was that the semantics for modal predicate logic requires the postulation of possible but nonexistent objects Quine believed that we should not commit ourselves to possibilia on the grounds that they do not have wellde ned identity conditions When generalized possible worlds semantics came on the scene philosophers welcomed it as a powerful analytical tool and brushed Quine s objections aside It is probably not coincidental that about this time there was a powerful shift away from the austere ontologies of Quine and his Harvard colleague Nelson Goodman not to mention the later Wittgenstein to ontologies which allow the existence of possible but not actual objects There remains vigorous debate about the metaphysical status of possible worlds and objects in them At one extreme David Lewis advocated a modal realism according to which each possible world is just as real as the one we call actual 10 At another Michael Jubien has tried to treat modalities without appeal to possible worlds at all11 7Meaning and Necessity 910 8The sematics is commonly attributed to Saul A Kripke but it was developed during the same period by Jaakko Hintikka and Stig Kanger 9See for example Three Grades of Modal Involvement Praceedings 0f the XIth Internatianal Cangress 0f Philasaphy Brussels 1953 Vol 14 reprinted in Quine s The Ways afParadax 1966 100 the Plurality afWarlds 1986 11Carttempamry Metaphysics 1997 Chapter 8 Module 14 Free Predicate Logic G J Mattey June 1 2007 Contents 1 NonDesignating Terms 1 2 Syntax of Free Predicate Logic 2 3 Semantics for Free Predicate Logic 2 Inner and Outer Domains 2 32 The Interpretation of Expressions 2 321 Terms and Predicates 2 322 The Existence Predicate 3 323 Quanti ed Sentences 3 324 Identity 4 33 Formal Semantics for Free Predicate Logic 4 4 Derivations in Free Predicate Logic 5 1 NonDesignating Terms One of the reasons to investigate modal logic is that we ordinarily talk about objects that could exist although they do not Philosophers call such wouldbe existents possibilia One way to talk about possibilia is to use de nite descriptions that do not refer We might use the expression the ying horse to describe a nonexistent object in Greek mythology Or we might use the name Pegasus to that same end If we think of constants of PL as surrogates for names in natural language we seem to be faced with a problem In the standard semantical system PI on a given interpretation every constant refers to an object in that interpretation s domain Therefore if we assert a sentence containing the name such as Pegasus is a horse that ies we would symbolize it as perhaps Hla A F 1a from which it can be inferred that 3xH1x A F136 If the existential quanti er is to represent only those objects that exist in the actual world then we will either have to give up the use of a constant to stand for the name Pegasus say that Pegasus is actual or stop using the existential quanti er to represent what is actual One way to evade this trilemma is to modify underlying Predicate Logic so that we are free to use constants to refer to objects that are not actual Free Predicate Logic allows the expression of true sentences about individuals that do not exist The de ning feature of Free Predicate Logic is that its constants need not designate an existing object Thus we can make assertions about nonexistent objects such as Pegasus without being committed to their existence merely by associating their name with a constant 2 Syntax of Free Predicate Logic The syntax of Free Predicate Logic is an extension of the syntax for Predicate Logic A single oneplace constant predicate letter is added It will be written in nonitalicized Roman type to distinguish it from predicates whose extension varies from interpretation to interpretation Intuitively this predicate applies only to what exists Precise meaning will be given to this notion when the semantics for Free Predicate Logic is developed Existence Predicate E1 is a predicate letter of Free Predicate Logic 3 Semantics for Free Predicate Logic Lablanc and Thomason have speci ed a semantics for nonmodal Free Predicate Logic1 Here we adapt their semantics to mesh with the semantical system PI described in the previous module 31 Inner and Outer Domains The heart of the LeblancThomason semantics is the bifurcation of the unitary domain of PI into two exclu sive domains one called inner and the other called outer The inner domain contains the objects which are taken to exist while the outer domain contains those objects that are taken not to exist We will assume here that the inner domain is not empty though the outer domain may be empty2 If D is the inner domain in an interpretation we will call D the outer domain Thus a Free Predicate Logic interpretation is an ordered triple D D V rather than an ordered pair consisting of a single domain and a valuation function 32 The Interpretation of Expressions Because there are two domains in the semantics for Free Predicate Logic the rules for interpreting its expressions are more complex than those for the interpretation of expressions of standard Predicate Logic In this section we will discuss the interpretation of terms and predicates of the special existence predicate and of quanti ed sentences 321 Terms and Predicates In an interpretation of Free Predicate Logic the extensions of predicates and the designations of terms constants and parameters are taken from the union D U D of the two domains That is they are taken from 1Hughes Lablanc and Richmond H Thomason Completeness Theorems for Some PresuppositionFree Logics F undamenta Math 62 125164 See also Ermanno Bencivenga Free Logics in Gabbay and Guenther eds Handbaak 0f Philasaphical Lagic Vol 111 pp 373426 2Bencivenga notes that the inner domain may be empty on the LablancThomason semantics the combination of all the objects both existent and nonexistent The extension of a predicate letter P is a set of ordered ntuples drawn from the union of the two domains that is a subset of D U D Because the objects in the extensions of predicates are not limited to the inner domain there can be true predications of objects in the inner domainiobjects that do not exist For example Fa can be true on an interpretation when a designates an object in the outer domain and that object is in the extension of F even though it does not exist which is to say that it is not a member of the inner domain 322 The Existence Predicate To say solely on the basis of Pegasus being a ying horse that ying horses exist requires that Pegasus exist In standard Predicate Logic this might be symbolized by 3xx a where a designates Pegasus In Free Predicate Logic it would be symbolized using the special existence predicate as Ela 3 The existence predicate is introduced specially because of the role it plays in the derivational system for FPL and because of the role it might play in more sophisticated systems of free modal logic The extension of the existence predicate in an interpretation is the set of onetuples of the members of the inner domain of the frame on which the interpretation is based V1Eu u E D This set is the 1st Cartesian product of D which is D1 323 Quanti ed Sentences The interpretation of quanti ed sentences is like the interpretation of the existence predicate sensitive to the distinction to the inner and the outer domains What makes a quanti ed sentence true depends entirely on which objects are in the inner domain Thus we say that a universally quanti ed sentence Vxax u is true just in case arll is true for all the objects in the inner domain which could be assigned to u by the interpretation An existentially quanti ed sentence 3xarx u is true if and only if there is some object in the inner domain such that if u designates it the sentence arll is true In the case of constants we lose our ability to generalize existentially Consider an interpretation I on which v1a E D v1a E v1F and there is no object d in D such that d E v1F That is the value of a is in the outer domain and hence not in the inner domain and the onetuple consisting of that value is in the extension of F while no onetuple consisting of an object from the inner domain is in the extension of F As a result the sentence F a is true while 3xFx is false The latter sentence is false because there is no variant of VI which assigns a member d of the inner domain D to M such that FM is true on that variant In ordinary English we might assert Pegasus is a ying horse without thereby asserting Flying horses exist The same considerations apply to existential generalizations of sentences containing parameters On some interpretations a parameter u might be assigned an object in the outer domain Thus the sentence aru might be true while the existential generalization of that sentence 3xarx u is false The truth or falsehood of the latter sentence does not depend at all on what u designates All that matters is that there is some variant of the interpretation which assigns to u a member of the inner domain such that arll is true Sentence with the existence predicate applied to a term are relevant to the truthvalue of existential sentences IfI assert Lorena Ochoa is a dominant female professional golfer and Lorena Ochoa exists then I am entitled to assert There exists a dominant female professional golfer Symbolically we can say that if F a and Ea are true on an interpretation 1 then a designates a member of the inner domain D which allows existential generalization If vad then there is a variant of v which assigns d to a parameter u so that FM is true in which case 3xFx is true on that interpretation 3We shall from this point forward drop the placeindex 1 The existence predicate is needed to generate a sound version of another rule of inference Universal Instantiation when instantiation is made to constants To say that everything is F given the semantics is only to say that every existing thing is F so Fa need not be true if a does not designate an existing thing On the other hand F a may be true given that the extensions of predicates can be drawn from the outer domain 324 Identity The treatment of identity sentences in the semantics for Free Predicate Logic is the same as for the seman tics for standard Predicate Logic Identity in PI is treated semantically by the simple rule that an identity sentence is true on an interpretation just in case its two terms designate the same individual As a result we can validly assert the existence of everything to which a constant may refer It is trivial that IP1 a a and that P1 ll ll Now consider the case of u a On every PI interpretation I there is a single domain D such that V1a E D and V1ll E D No matter what value V1 assigns to u there is a variant of V1 which assigns to u the same member 1 of the domain as V1 assigns to a Hence V1dua V1a in which case V1Clllll v1dua From this we can infer that v1duu a T from which it follows that V13xx a T Since the choice of interpretations is arbitrary we have it that Ip1 3xx a The situation is different in Free Predicate Logic where a constant can fail to designate an object in the inner domain We may have a formula 14 a where on an interpretation I a designates a member of the outer domain In that case there is no variant of a valuation function assigning a member 1 of the inner domain to u such that V1dMM is the same as V1a so the identity formula is false on all those variants In that case 3xx a is false If 3xx a is false then so is Ea since a fails to designate an object in the inner domain 33 Formal Semantics for Free Predicate Logic Below we give a formal treatment of the semantics for Free Predicate Logic Not listed are semantical rules that carry over directly from the semantics for Predicate Logic such as the rules for truthfunctional compounds Although the rules for the quanti ers are stated identically it must be kept in mind that the domain D in the case of Free Predicate Logic is the inner domain which shares no members with the outer domain Interpretations in Free Predicate Logic IDD v D DnD Special Semantical Rules for Free Predicate Logic H v1aEDUD N V1llE DUD 9 VP DUD 5 V1E D1 U V1VXaXllT where u is free for X in aX if and only if for all 1 E D V1Clllarll T v1VXaXuF if and only if for some 1 E D v1duau F 0 v13XarXuT where u is free for X in aX if and only if for some 1 E D V1Clllarll T V13XarXuF if and only if for all d E D v1duau F gt1 v1titjT if and only if V1tiV1tj v1titjF if and only if v1ti v1tj Here it should be noted that D in the rules for the quanti ers denotes the inner domain only so that the quanti er rules stated here are substantively different from those for standard Predicate Logic We will carry the notions of semantical entailment semantical equivalence validity and semantical consistency directly over from PI There is nothing in F PI that requires a modi cation in any of them They may be referred to as Free Quanti catianal Semantical Entailment etc As was intimated in the earlier discussion Free Predicate Logic is weaker than Predicate Logic In general VXaXu FPL araX and aaX FPL 3XaXu In both cases 3 might designate in an interpretation a member of the outer domain rather than the inner domain We do have the weaker results that VXaXu Ea FPL aaX and aaX Ea FPL 3XaXu Though proofs will not be given here it seems clear from the semantical rules just given that the prop erties of Free Quanti catianal Bivalence and Free Quanti catianal TruthFunctianality hold for FPI 4 Derivations in Free Predicate Logic A derivational system for Free Predicate Logic F PD can be built by modifying the elimination rule for the universal quanti er and the introduction rule for the existential quanti er We begin with Universal Elimination In the derivational system PD we were able to instantiate a universal sentence to any term whatsoever But in the semantical sysem for Free Predicate Logic this procedure is not valid On an interpretation where a term t is not assigned to a member of the inner domain the universal sentence VXaXu may be true while the instantiation atu is false This result applies to all terms both constants and parameters To prevent such a result we will restrict the use of Universal Elimination in a manner similar to the restriction on Universal Introduction in PD4 We shall allow instantiation only to a parameter that anks a restricted scope line in which the sentence to be instantiated occurs We can replace the rst invalid inference with one that conforms to the semantics for Free Predicate Logic 1 VxFx Assumption 2 F u 1 V E PD 1 VxFx Assumption F u 1 V E FPD 2 4The following rules of inference are adaptations of the rules given in Karel Lambert and Bas van Fraassen Derivatian and Caunterexample An Intraductian t0 Philasaphical Lagic 1972 In e ect this isolation of the instantiation re ects the semantical reasoning that the derivation is supposed to capture We suppose for an interpretation that VxFx is true Then we assume that some object d is in the inner domain and we let u stand for that member Then F u is true But this is only under the assumption that the parameter M denotes an existing thing We state the rule as follows Universal Elimination for Parameters FPL V Vxax u Already Derived av u V E Parameters In the system PD there is nothing be to be done with the result in our example except to generalize universally once again Proof that VxFx rm VyFy 1 VxFx Assumption 2 VxFx 1 Reiteration 3 F u 2 V E Parameters 4 VyF y 23 V I PD To instantiate to constants we must use a diITerent technique since constants may not ag restricted subderivations Here we will use the existence predicate So we will say that if some condition holds for everything in the inner domain and a designates something in that domain and hence Ea is true then that condition holds for what a designates So we will require that a universal sentence be instantiated when for constant a the sentence Ea occurs at the same scope line Universal Elimination for Constants FPL Vxax u Already derived Ea Already derived ara u V E Constants We must similarly restrict Existential Generalization From a semantical point of view we want to be able to generalize only when a term is assigned to a member of the inner domain As we saw with Universal Generalization this can be represented by agged restricted subderivations A sentence containing a pa rameter may be generalized upon only when it occurs within a restricted scope line with that parameter as its ag Existential Introduction for Parameters V avu Already Derived 3xarx u 3 I Parameters The effect of this rule as stated may be stronger than we would like There are only two rules which allow us to end a quanti cational restricted subderivation Universal Introduction and Existential Elimination The former rule requires that the agging parameter be replaced with a variable and an appropriate quanti er added But the rule 3 I Parameters itself carries out this replacement So the only way the result of Exis tential Introduction can be brought out is through the use of Existential Elimination as with the following derivation Proof that 3xFx l FPL 3yFy 1 3xFx Assumption 2 F u Assumption 3 3yF y 2 3 I Parameters 4 3yFy 1 23 3 E PD We have no way however to generate the following derivation since there is no way to remove the barrier5 Attempt to prove VxFx l FPL 3xFx 1 VxFx Assumption 2 VxFx 1 Reiteration 3 F u 2 V E Parameters 4 3xFx 3 3 I Parameters 5 3xFx 7 To this end we need a new rule which allows us to end the barrier and bring out a sentence which does not contain the parameter which ags the barrier so long as the sentence is not in the scope of any assump tion This makes sense semantically because the barrier is supposed to re ect an assumption about what the agging parameter refers to That assumption is moot if the sentence does not contain the parameter Barrier Removal FPL 11 a 5This is an intended feature of the derivational system of Lambert and van Fraassen since they want their system to accomodate a semantics that allows for the domain to be empty In our semantics the domain may not be empty so the above inference is valid and should be permitted by the rules Provided that u does not occur in a a does not lie in the scope of any undischarged assumption Use of this rule allows us to get the desired result Proof that VxFx I FPL 3xFx 1 VxFx Assumption 2 VxFx 1 Reiteration 3 F u 2 V E Parameters 4 3xFx 3 3 I Parameters 5 3xFx 4 BR To generalize on a constant a we require the occurrence of the sentence Ea occurs at the same scope line Existential Introduction for Constants arau Already Derived Ea 3xarxu 3 I Constants Here are some examples of derivations that hold in PD but are not permitted by the rules of FPD These are derivational correlates of examples discussed in the semantical section above In each case the problem lies in the attempt to use constants as the basis of the application of quanti er rules We begin with Existential Generalization Attempt to prove Fa I FPD 3xFx 1 F a Assumption 2 3xFx 1 3 I PD Step 2 is not permitted because the a is a constant and not a parameter The following attempt to generalize from a parameter fails as well since the parameter does not fall within a restricted scope line containing u Attempt to prove FM I FPD 3xFx 1 F u Assumption 2 3xFx 1 3 I PD

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