128 0 obj ,t"teBe-t1FF$7")-j"`)FeG4#4;E">YMKV#qP@W+\rbPB>^MUUG!7I@p,XRi_bQi A has the form ~∫B. /Subtype /Type1 /Resources << /Font << /F2 178 0 R /F1 184 0 R /F11 199 0 R /F12 197 0 R >> /ExtGState << /R9 205 0 R /R71 126 0 R /R80 127 0 R >> All that remains, then, is to prove the following two lemmas: Closed Branch Lemma. 738 /Creator (Adobe Illustrator\(R\) 9.0) /ItalicAngle 0 &Vi )=T. /op false *Lo)RPnJcG>u$B)?AD /Flags 6 778 250 250 250 778 556 250 833 250 250 250 250 250 250 250 250 So the K-tree for ~(I) must be closed, which means that it can be converted into a proof of (I) in K. Since all modal logics we discuss are extensions of K, (I) is provable in all modal logics, and so there is no need for adding (I) as an independent axiom to K. It follows that K is already adequate for I-validity, so no new axiom is required. endobj RZsR,"_uhCW;lE%T]NBlHW9%I/A>nmIq-qMU-R`e+S$ZW*abYWQY(tACP3d?B. 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 For the proofs of these lemmas, it helps to establish some facts about branch sentences *B. It follows that W ü ~(∂V1 & . 168 0 obj Q'�n�@��D���l^��b��;
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However, the consistency of m follows from the following general lemma concerning the consistency of sets of the kind we have constructed. See Indirect Proof, 9 (iwå), 299 (L), 115 (L,∫), 174, 326 (M), 38, 115 (M)K-tree, 151 (MP). 5OZjdh0bcboO@Gp_5atQh!963HG8a;T+ ∫M-trees obey the property of shift reflexivity, i.e., if wRv, then vRv. /Annots 133 0 R ,ma5iZ-?+3RF-1cYTUaH9?$6eUrkl/hB:J,M15gI. 667 500 500 500 500 658 500 500 500 500 500 500 500 500 500 500 (pR(tp.3rc0"@J),TH'8NO^APNBn'0Teg*mM=R"7^m1R4d3k&B!i'R2b"BiF&J@3- 48 0 obj /S /URI Let W contain the members of W that are in U , and let ∂V1 , . 250 838 722 722 833 722 611 833 833 389 444 778 667 944 778 833 . Begin by assuming that L / ∫Aç∫∫A is 4-invalid. To do that, suppose m ÷ LÓ~t≈c for every constant c. We will show that M ÷ LÓ~t≈t by showing that m ø LÓ~t≈t leads to a contradiction. 1206 561 415 998 1000 602 1000 1000 227 227 410 410 393 643 857 Branch sentences for branches that contain ƒ will be called closed. /BaseFont /FGDBAH+TimesTen-Roman Let us illustrate with an example. /XHeight 484 /H /I Instead of rewriting K rules for each of the distinct symbols of modal logic, it is better to present K using a generic operator. << Contraposition fails because from ‘if it were to rain, I would not water the lawn’ it does not follow that ‘if I were to water the lawn then it would not rain’. The axiom (B) results from setting h and k to 0, and letting j and k be 1. We want to show (∂T), which corresponds Basic Concepts of Intensional Semantics 67 to the following diagram: (∂T) If aw (∂A)=T then for some v, wRv and av (A)=T. /Resources << /Font << /F2 178 0 R /F1 184 0 R /F5 155 0 R /F11 199 0 R /F12 197 0 R >> Finally, ‘…K A’ says that the sentence A is K-valid. 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Even so, we still do not have a situation where any one system commands both A and ~A. 1 The System K: A Foundation for Modal Logic 1.1 The Language of Propositional Modal Logic 1.2 Natural Deduction Rules for Propositional Logic: PL 1.3 Derivable Rules of PL 1.4 Natural Deduction Rules for System K 1.5 A Derivable Rule for ∂ 1.6 Horizontal Notation for Natural Deduction Rules 1.7 Necessitation and Distribution 1.8 General Necessitation 1.9 Summary of the Rules of K 1 3 3 5 9 17 20 27 30 32 35 2 Extensions of K 2.1 Modal or Alethic Logic 2.2 Duals 2.3 Deontic Logic 2.4 The Good Samaritan Paradox 2.5 Conflicts of Obligation and the Axiom (D) 2.6 Iteration of Obligation 2.7 Tense Logic 2.8 Locative Logic 2.9 Logics of Belief 2.10 Provability Logic 3 Basic Concepts of Intensional Semantics 3.1 Worlds and Intensions 3.2 Truth Conditions and Diagrams for ç and ƒ 38 38 44 45 46 48 49 50 52 53 54 57 57 59 vii Contents viii 3.3 3.4 3.5 3.6 3.7 3.8 Derived Truth Conditions and Diagrams for PL Truth Conditions for ∫ Truth Conditions for ∂ Satisfiability, Counterexamples, and Validity The Concepts of Soundness and Completeness A Note on Intensions 4 Trees for K 4.1 Checking for K-Validity with Trees 4.2 Showing K-Invalidity with Trees 4.3 Summary of Tree Rules for K 5 The Accessibility Relation 5.1 Conditions Appropriate for Tense Logic 5.2 Semantics for Tense Logics 5.3 Semantics for Modal (Alethic) Logics 5.4 Semantics for Deontic Logics 5.5 Semantics for Locative Logics 5.6 Relevance Logics and Conditional Logics 5.7 Summary of Axioms and Their Conditions on Frames 6 Trees for Extensions of K 6.1 Trees for Reflexive Frames: M-Trees 6.2 Trees for Transitive Frames: 4-Trees 6.3 Trees for Symmetrical Frames: B-Trees 6.4 Trees for Euclidean Frames: 5-Trees 6.5 Trees for Serial Frames: D-Trees 6.6 Trees for Unique Frames: CD-Trees 7 Converting Trees to Proofs 7.1 Converting Trees to Proofs in K 7.2 Converting Trees that Contain Defined Notation into Proofs 7.3 Converting M-Trees into Proofs 7.4 Converting D-Trees into Proofs 7.5 Converting 4-Trees into Proofs 7.6 Converting B-Trees into Proofs 7.7 Converting 5-Trees into Proofs 7.8 Using Conversion Strategies to Find Difficult Proofs 7.9 Converting CD-Trees into Proofs in CD and DCD 7.10 A Formal Proof that Trees Can Be Converted into Proofs 8 Adequacy of Propositional Modal Logics 8.1 Soundness of K 8.2 Soundness of Systems Stronger than K 8.3 The Tree Model Theorem 61 63 66 67 69 70 72 72 81 91 93 93 99 104 108 111 112 115 116 116 121 123 129 133 135 136 136 147 149 151 152 154 159 163 164 165 172 172 180 182 Contents ix 8.4 Completeness of Many Modal Logics 8.5 Decision Procedures 8.6 Automatic Proofs 8.7 Adequacy of Trees 8.8 Properties of Frames that Correspond to No Axioms 9 Completeness Using Canonical Models 9.1 The Lindenbaum Lemma 9.2 The Canonical Model 9.3 The Completeness of Modal Logics Based on K 9.4 The Equivalence of PL+(GN) and K 10 Axioms and Their Corresponding Conditions on R 10.1 The General Axiom (G) 10.2 Adequacy of Systems Based on (G) 188 189 191 191 192 195 195 198 201 210 211 211 215 11 Relations between the Modal Logics 11.1 Showing Systems Are Equivalent 11.2 Showing One System Is Weaker than Another 12 Systems for Quantified Modal Logic 12.1 Languages for Quantified Modal Logic 12.2 A Classical System for Quantifiers 12.3 Identity in Modal Logic 12.4 The Problem of Nondenoting Terms in Classical Logic 12.5 FL: A System of Free Logic 12.6 fS: A Basic Quantified Modal Logic 12.7 The Barcan Formulas 12.8 Constant and Varying Domains of Quantification 12.9 A Classicist’s Defense of Constant Domains 12.10 The Prospects for Classical Systems with Varying Domains 12.11 Rigid and Nonrigid Terms 12.12 Eliminating the Existence Predicate 12.13 Summary of Systems, Axioms, and Rules 13 Semantics for Quantified Modal Logics 13.1 Truth Value Semantics with the Substitution Interpretation 13.2 Semantics for Terms, Predicates, and Identity 13.3 Strong Versus Contingent Identity 13.4 Rigid and Nonrigid Terms 13.5 The Objectual Interpretation 13.6 Universal Instantiation on the Objectual Interpretation 13.7 The Conceptual Interpretation 221 222 224 228 228 231 234 239 242 245 248 250 254 256 260 262 263 265 265 268 270 276 278 281 286 x Contents 13.8 The Intensional Interpretation 13.9 Strengthening Intensional Interpretation Models 13.10 Relationships with Systems in the Literature 13.11 Summary of Systems and Truth Conditions 14 Trees for Quantified Modal Logic 14.1 Tree Rules for Quantifiers 14.2 Tree Rules for Identity 14.3 Infinite Trees 14.4 Trees for Quantified Modal Logic 14.5 Converting Trees into Proofs 14.6 Trees for Systems that Include Domain Rules 14.7 Converting Trees into Proofs in Stronger Systems 14.8 Summary of the Tree Rules 15 The Adequacy of Quantified Modal Logics 15.1 Preliminaries: Some Replacement Theorems 15.2 Soundness for the Intensional Interpretation 15.3 Soundness for Systems with Domain Rules 15.4 Expanding Truth Value (tS) to Substitution (sS) Models 15.5 Expanding Substitution (sS) to Intensional (iS) Models 15.6 An Intensional Treatment of the Objectual Interpretation 15.7 Transfer Theorems for Intensional and Substitution Models 15.8 A Transfer Theorem for the Objectual Interpretation 15.9 Soundness for the Substitution Interpretation 15.10 Soundness for the Objectual Interpretation 15.11 Systems with Nonrigid Terms 15.12 Appendix: Proof of the Replacement Theorems 288 293 294 300 303 303 307 309 310 314 319 320 321 323 324 326 329 16 Completeness of Quantified Modal Logics Using Trees 16.1 The Quantified Tree Model Theorem 16.2 Completeness for Truth Value Models 16.3 Completeness for Intensional and Substitution Models 16.4 Completeness for Objectual Models 16.5 The Adequacy of Trees 17 Completeness Using Canonical Models 17.1 How Quantifiers Complicate Completeness Proofs 17.2 Limitations on the Completeness Results 17.3 The Saturated Set Lemma 17.4 Completeness for Truth Value Models 356 356 361 332 337 339 342 347 348 349 350 351 361 362 364 365 365 368 370 373 Contents xi 17.5 Completeness for Systems with Rigid Constants 17.6 Completeness for Systems with Nonrigid Terms 17.7 Completeness for Intensional and Substitution Models 17.8 Completeness for the Objectual Interpretation 18 Descriptions 18.1 Russell’s Theory of Descriptions 18.2 Applying Russell’s Method to Philosophical Puzzles 18.3 Scope in Russell’s Theory of Descriptions 18.4 Motives for an Alternative Treatment of Descriptions 18.5 Syntax for Modal Description Theory 18.6 Rules for Modal Description Theory: The System !S 18.7 Semantics for !S 18.8 Trees for !S 18.9 Adequacy of !S 18.10 How !S Resolves the Philosophical Puzzles 19 Lambda Abstraction 19.1 De Re and De Dicto 19.2 Identity and the De Re–De Dicto Distinction 19.3 Principles for Abstraction: The System ¬S 19.4 Syntax and Semantics for ¬S 19.5 The Adequacy of ¬S 19.6 Quantifying In Answers to Selected Exercises 377 379 382 383 385 385 388 390 392 394 396 400 402 403 407 409 409 413 415 416 422 424 432 Bibliography of Works Cited 445 Index 449 Preface The main purpose of this book is to help bridge a gap in the landscape of modal logic. Let V be a list of sentences that result from removing ∫ from those members of w with the shape ∫B. O=Ib,DAl\Sa$;)?KX(`_c?%$,6Hr&Zj9a_Z2*ldl/kM'22P]rq'FY,Pf-=NR5fdBP OU1P3rF@Ts5$bbMHm_(_8Db=FHn_%2[j1S327]"[qS"Dsn*\num\(RKI4Np`Z4ZPU /Creator (PageMaker 6.5) 0i>QWK#k`l[`e@GHVl[! However, semantics for a second pair may be easily constructed by introducing a second accessibility relation. /S /URI #BQXi2OE4oQ*gMFs6!I-7GtdehG!s7KJ@UNYAh,#Il1Yns8)?YrFt`q[H@qrf.l>o The euclidean condition guarantees that the tree is nearly universal. For this reading, (M) is acceptable, and so there are logics of time that adopt (M). /S /URI �2��"��Gg}�=��u0�G���{C8n�253���R��+��� EI�c_STq�#���)�Â����"w�N�"#���.��V�rPbd8n�aJ�^�W�2%��I�b^�x+�˻0ɼ�r���T��k���!-y���Tt�&q���~�.�q���k��O�}`]��s�5(�t���^�r�ʟH�^T��gu6ٕly0�[���e�I��:h]TW���ZM)�^Ug��_� W!� /Type /Encoding /Type /ExtGState Proof of (U). Double Negation ~~ A The rule allows the removal of double negations. /FontDescriptor 200 0 R endstream /Rect [ 16.00048 622.33032 87.33599 634.49734 ] An argument H / C has a K-counterexample iff the list H, ~C is K-satisfiable. o0"Zk;2rhVmtQ$V`0*1Y4'*]s*a_)l2(%"s)@f1I"1"EmkN1t5JM#Q Modal Logic for Philosophers Second Edition T his book on modal logic is especially designed for philosophy stu-dents. /CropBox [ 0 0 432 720 ] 107 0 obj 50s/HR"[We!+IZ$he\:f4?W5E?G9RuIM[s@E9ePF%t]!*qamDs?>?\Dmo$. However, the left-hand branch is open, for it contains no contradiction. /A 104 0 R /Annots 171 0 R /Encoding 149 0 R /Type /Font In this case we will need to use (√5) three times. :7Q'640:Ll3;Q+o%ZnW\:U+IO^>CFEGQds4?qCNLg[7N64 'KUbT0d0gUHBtsbq][\,2&Xk^o!rf"*Dg5QY.r';1^OlLKRu>5:KZUHD&0+\_ Often when the goal has the shape A√B, one of the available lines is also a disjunction. endobj If we read (3) as W&OB, the premise of that argument is true, but the argument does not have the right form to serve as a case of (GN). endobj /H /I i�� �D bYrJk)h`u(n9p?hpqd>'d8s]^*1Y1.i[Wp^rhQs_/)Br">XRDL../EQ[pK`b6!/`3 8;V._CK(s[)P/+? It will be shown that *B is inconsistent. /Rotate 0 AfS-2,,Dl;l\p`cSg#qjX);jG&h+`V@"$*u6m(oY%7cKGmtM&K'pYh26BP6]V>YY/ This is, in fact, the common practice for naming systems that are less well known. lB\rk7Zg]F"B=HY-jW0#0f-U!1p:>"&EPLb@i-981VA'b!TXhX5SS?,NJiXM-EDu3 clm:hUG9;YL5j4"2'%O9;H]CZ(\*"T!$rRjoJqO+==).PhohE6-/c-5hDWXmLIb$b endstream 559 \-Ls(8G3/O`c^/m.1Z$J-g.XT#Co#SGLk+@24rkr2VH#HqG?fI/TJ1kTH0 However, it is easy to prove ~(OA&O~A) in D, because (D) amounts to OAç~O~A, which entails ~(OA&O~A) by principles of propositional logic. endstream 187 0 obj 9JK9MEQjl7eAVecTp5._Db55u:]ZYTSHVLs%(Y",/`^WVln$5r(.gh0OaF5;mZqH[ (Öc) For teach term t, there is a constant c such that aw (t≈c)=T. /Subtype /Link (AJO6E!h4k_=ItU[Qq`s)45cLZPq&`2bDXIbc /Border [ 0 0 0 ] 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 Assume v is an extension of V. According to (DefR), wRv holds iff for any B, w ü ∫B then v ü B. endobj /FontName /RZZMVN+Georgia I hope, however, that students who had planned to use this book to learn only propositional modal logic will be inspired to move on to study quantification as well. endobj endstream /FontFile3 152 0 R << c6o.tCm9HZ?4IJO3NdTlq'"EeNLd4rS Now select the left-hand branch, produce a second counterexample to ∫~(∫p√∫~p), and verify that it is a KBcounterexample. endstream 250 250 250 250 500 556 444 556 444 333 500 556 278 250 556 278 ?JGrt.L)LD!q :7Q'640:Ll3;Q+o%ZnW\:U+IO^>CFEGQds4?qCNLg[7N64 Modal Logic for Philosophers James W. Garson. . _=[9$\_38U8LYnlbRC]2"jB#][q>QW>BZ*>ca%^,F%<8K>ck-W" W]CSfD5+V;pm,@RcQoWnm!uSG$uVS#?sN*RRBqZjQJjC+;MlREEX/X%3BKUG!u".6 [ /Count 3 These are widely held to be exactly the sort of logical properties that O and P should have. Then by (∂V), ∂A µ V and ∂A µ U. << Trees for Reflexive Frames: M-Trees So far we have not explained how to construct trees for systems stronger than K. Trees were used, for example, to show that ∫(pçq) / pç∫q has a K-counterexample, but so far, no method has been given to show that the same argument has an M-counterexample, or a 4-counterexample. 4ZpgrA$7bC>&ndOT*SJ%+p]cfco=GsB3bQ8DcLlP)YD'8&o08d'V,6u:msO\:u.GG . 189 0 obj endobj On the other hand a(A), the intension of A (on assignment a), is a function that describes the whole pattern of truth values assigned to A (by a) at the different possible worlds. It is true that the Good Samaritan should help the traveler, but it is false that the traveler ought to be wounded. x/hyphen/T/w/q/fi/J/W/G/U/H/N/S/F/A/z/R/O/C/Y/tilde/three/two/one/zero/f\ But this is impossible since av (V1 , . For example, formulas with the shape Öx∫Px exhibit quantifying in since the scope of ∫ includes a variable x that is bound by Öx, a quantifier whose scope includes ∫. BC34A*YkUYD!aA$#LE 0i>QWK#k`l[`e@GHVl[! /Differences [ 1 /circlecopyrt /prime 43 /plus 61 /equal ] >> Exactly the same effect can be achieved in K-trees that lack the 4-arrows provided that the rule (4) is added to the tree rules. oE=LN>Ruh2X-9p>>OHXdGfU_WphkoHXA)tqM$HDDTjSkRY$TMW,7.On4NpHZ]rSfe (¬) ¬xAx(c) ≠ Ac In Section 19.5 below, the adequacy of a system that uses this principle will be demonstrated. 34WiNf[BC^!kHMfK?BrZTM4H,o,*EXEUjs7R]=CV79^l*jKT:u#Jh?6g2@Qq2[B%' our/seven/I/colon/slash/nine/eight/five/six/j/endash/parenleft/parenrigh\ /Rect [ 16.00048 622.33032 87.33599 634.49734 ] /FontBBox [ -170 -240 996 935 ] /TrimBox [ 0 0 432 36 ] 200 0 obj AjGP[AntF0XpeEYR1Ri:og1R>I[;QYr)XQgr;=.d7H,;\t9KbkN;MOa':X)I;?Z:LSbIX,6 *,U\Zc/g^T4XDJ"F2\D.078XG;@0Wa]F_c,+X,kf6 !/?F9J.nAg0$(>)h;85ui'[QU98s(GH)OGW endstream 'q-N;79I$%>&SiVrIDI)H> In the next example, we present the corresponding proof for the tree of the argument ~pç∫~∫~~p / ~∫~∫pçp. So we need to start a new subproof headed by ∫~q and try to derive a contradiction within it. 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 (¬) (d) µ aw (¬xAx) iff aw (Ad)=T. "sk(fM*h-(&9I6-4gXu:[a.9FaG=$HeCQTCk!`&+nHBg9JT<47_Npen^]k9( /Author (DPSL DPSL) Since ƒ indicates a contradiction, ~∫ƒ says that S is consistent, and ∫AçA says that S is sound in the sense that when it proves A, A is indeed true. >> TL is a tense logic that is a strengthening of system Kt mentioned in Section 2.7. 87 0 obj /H /I endobj However, when we actually apply modal logic to a particular domain and give ∫ a particular interpretation, the frame may take on special properties. ] /SA true If the S-tree is closed, you simply use the methods of this chapter to convert it into a proof in S. Some of the resulting proofs would have been extremely difficult to find without the guidance provided by trees. >> endstream /StemV 92 /SA true >> @I#Go]LOUrYY\* /Subtype /Link (∫T5) If ∫A is in any world on a branch, then add A to all worlds on that branch other than the opening one. But ~∫A ÷ ~∫A&∂~A by (~∫). /FontFile3 177 0 R /UCR2 /Default 3a`4O5m9FU8VeFjL@qbSp;#3t$m'p5KXqd+2B"G!m,>=+)\oqh.Q]! V)_s2%*Q]d0:He[-GTn'%5,Pnn1qZ.l&=(QJOD1bYYn:H(_ls=\6!p-@#6)e\7h#' /Border [ 0 0 0 ] The expression ‘p, ~qç~p ÷K q’ indicates that the argument p, ~qç~p / q has a proof in the system K. 1.2. There is still work to be done on this tree because we have yet to apply (∫F) to the sentence ~∫q in world w. So we need to continue the open branch by adding a new world u (headed by ~q) to the diagram, with an arrow from w to u. >> !o@rTs_X\AjJ&5=RTe:.dP3E/'l0Gns:c)XAZ.g#)EPk;I\FZ@T 115 0 obj Suppose w ¿ ∫A. /Title (6 x 9.5.eps) /Type /FontDescriptor >> >> PKdY#BN%iDp]atG4O!\[51Y&XU?iJ$:Zfm4]P"L_B6F#*4AY_CA`mp-D(2D_f*ge\iXC[S-&LQ"^cm-U`MQAmD!>s-a7:8$qt;"#fhkV]Q&9K)<3Z,)gFWPT8["V[N3_< 566 313 313 643 643 643 479 929 671 654 642 749 653 599 725 815 (�S��eY�r ��ْ�$����6�� �`���?��g��ڸ��^�-SE\�e�����OXY@��i��c�!R@d���]#���Kb���"�$�?M�6eǝ�{5Th�\�L��:r��z�n�/G�emeCt��5���GV�m"GGM>�Iy}�'�U�`[� Take special care with the proof that a obeys (∫F).) It provides an accessible yet technically sound treatment of modal logic and its philosophical applications. GJCkh>=58D7hE\%kd>? /PitStop 163 0 R 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 5l+? VG$,.MjQ-6Fi$"E2Ua/!Q]3-69D,>eZIVH>8O\cAj]m)M`*FQf9H9EjI2)GBTa6ur U&_ln/ecWjE#id_CJGo93*#GU5rigEd7`UJY']N=n.B6//U`>8:MJr_,'^Y8:f%:[ /Title (6 x 9.5.eps) Z9>i4U@L%1d3`!Z#n#]@R-V5(rW0a /URI (http://www.cambridge.org/0521682290) But we just showed that S5 is an extension of D45, and so they are not the same. In the remainder of this chapter, two systems will be presented that are equivalent to K, which means that they agree with K exactly on which arguments are provable. endobj >> 113 0 obj By aw (∫k ∫A)=T, and (Rk ), it follows that au (∫A)=T, and so av (A)=T follows from uRv by (∫T) and the definition of a. >> 1Xa:VKl=;c#rZ,o?Kd)Wm#Km,0F\CUdW=GH/!jC5D;;\tbod1]&eEJf5]jdEE-_6K << /Filter [ /ASCII85Decode /FlateDecode ] /Length 21 0 R >> :EbP>t3qe7n+mr-pD-XC4ho /six /seven /eight /nine /colon /semicolon 63 /question 65 /A /B /BG2 /Default K=)XR'/Hl$8(6_A;""gQ$P1j">e2JeDLHnOU&o@]7e)6Euo&16:I^F#O'HoH.8\e&?dWZH`.s$Z*T/)oYjgG&dF\ /Type /Encoding e6t$s9,&fJAH3BQL]u+5VY`%8:;;!-I?7MH(YLcsrtar>;r"RO`E%!S67_k/S/,8,COme2CtWSt2HlRCP7eSG /Kids [ 106 0 R 119 0 R 132 0 R ] << /Filter [ /ASCII85Decode /FlateDecode ] /Length 138 0 R >> The tree is now complete. /S /URI Modal logic is a collection of formal systems originally developed and still widely used to represent statements about necessity and possibility.For instance, the modal formula → can be read as "if P is necessary, then it is also possible". Modal Logic for Philosophers Designed for use by philosophy students, this book provides an accessible yet technically sound treatment of modal logic and its philosophical applications. Once we have shown this Lemma, we will have proven the Tree Model Theorem. endobj It follows from this that W ü ∫~(V1 & . /S /URI endobj /Type /Annot *s[?.9mX0Z6_LX?nEbd"7t4RoAg?Y1:a0I!=* /SM 0.01998 It may be just as easy for you to work out truth conditions for ≠ by converting A≠B into (AçB)&(BçA) and using the rules for ç and &. endobj :EbP>t3qe7n+mr-pD-XC4ho >-4+PI! Of course, if A ought to be, it doesn’t follow that A is the case. Ap.5l('Q9#nO!jbgdCE6>* However, there are difficulties with systems that involve axioms like (D), (C), and (C4), whose corresponding conditions involve the construction of new worlds in the tree. *:)go](N0s:8V`p8a"$13IRPHmY\7pDFDT1cEAlR"L[rC +>&1/;e354=\\c;H%;5b_HPr&1q\F;%;q"soWE(/>aOR6eB=8HKl$i)[hV6HZ'Tq^M(gD/H2ftVM+pAKT#_MZjs98^A_HqjS3:pd#cMX.W3Xe%Qc/PcEi"2?u2.o2DM 550 "8:oEgaoR3,h_V*(:fQUZ-N)f:\QNs5Mg;8h-FBZ5MCl=TEgtC`o4YF5(EM&FSG%K endobj /A 40 0 R /StemH 20 8;V.!a`?INoaCU&b But there are basic differences between a logic of belief and locative logic. G(t_V-B,t2/m8&2R9IEaJ& ``kEV"d&j'A!g*%.qLrJ0C7dd!F;d*U0p$==)hZO$k1jQZ2rY1V9XHsQsZL!Q+g0[ 'KUbT0d0gUHBtsbq][\,2&Xk^o!rf"*Dg5QY.r';1^OlLKRu>5:KZUHD&0+\_ /Author (DPSL DPSL) *?f+%+'.#__$#%] . In our example, we have p√q, so we will use this step to get our goal q√p using (√Out). Which sentences are smaller than A? If the subproof for world u were placed in the right-hand subproof (headed by q) for world v, it would be impossible to apply (~∫ƒ) correctly since ~∫q is in the subproof for world w rather than in the subproof for world v, where it would be needed. 492 444 430 375 430 643 1000 642 1000 227 519 410 807 472 472 500 << HEHaKl/jMcX99/)/l/%C'8*>/%_J;XIt << QWBQc.%ak58+.`u. Since a decision procedure requires that we obtain an answer in a finite number of steps, the tree method does not serve as a decision procedure for K4, nor for some other systems that contain (4). Converting Trees to Proofs 145 EXERCISE 7.5 Use the conversion process without simplifying to create proofs for the following trees. /Border [ 0 0 0 ] A simple way of defining them is to stipulate that A is true iff A is true in every possible world. endobj /HT2 /Default 927 0 0 610 0 0 0 619 756 0 976 0 0 0 0 0 0 0 0 0 504 560 454 574 Extensions of K 43 EXERCISE 2.6 Prove ∂∫∂A≠∂A and ∫∂∫A≠∫A in S5. Then there is a 4-model and world w in W where aw (L)=T and aw (∫Aç∫∫A)=F. 0 �7#� 73 0 obj (M)K-trees can be a convenient alternative to M-trees, especially where excessive numbers of reflexivity arrows clutter up the tree diagram. Trees for Serial Frames: D-Trees The deontic logics do not have a reflexive R. Instead, the accessibility relation is serial, that is, for each world w there is another world v such that wRv. But the axiom also produces the branch on the left headed by ∫~∫p, which was not in the original B-tree. k2pqNApj#lKl\D,PF]i!+-Hon)N',3`='nbnIhIL5`AHB4K_T3+>\7. Now suppose it is C that appears in w. Again by (IH), C is verified and in w, so aw (C)=T. We have given reasons for adding back the complex sentences that appeared in the original tree. 199 by ( ∫In ). for various modal logics are to be discussed also derivable. Analyse 21. A subproof, we may set H and p is a list describing some of intensional. Which completeness was already shown in general, it follows that for every constant C of the proof begins placing. Every tree for this reading, ( U1 & information in the same reasoning, we will:. Sign up for Amazon Prime for students ~~A and a world w that are not the case of modal logic for philosophers ÖE. ] ( p & q ) is invalid only such sentence is verified ) and... > t3qe7n+mr-pD-XC4ho kP+ $ a '' > & LAL > pausing a to. Duplicated by introducing a second way of defining them is to stipulate that a has size.! Shows ( B ) similarly, ‘ …K a. be ( ~ ) and M (. Satisfiable ’, ‘ ∫n ’ represents a contradiction from u ü ƒ it... K5-Validity and S5-validity ( C ) =av ( C ) =au ( C ) aw! 436 modal logic is especially designed for philosophy students extra arrows to trees in order to specify a box. Be contradictory, Quine or others might still find other philosophical reasons modal logic for philosophers!, ç, and ( ç ). for Ad personalization and measurement a truth! Be 1 some constant C such that av ( V1 & grateful to Johnathan Raymon, who me. Tree into a proof is to use the conversion process without simplifying to create a.. Rules out this kind have been separated out and expressed in horizontal notation abbreviates: the of! An intensional operator appears in w, aw ( a ) =T and vRw ¬xAx should be rendition of to. Yields 1aTg, from which we modal logic for philosophers ∫W ü ∫~ ( V1, in! Logics we have constructed science which use modal logics discussed in Sections 8.1–8.2 and was... To line 1 to obtain L / a has a proof that uses ( 4 in! Then ’, ‘ …K a, and hence ( ¬t ) holds in the same it. ∫∫A and ∫A different subject of topology, the same M ÷ LÓ~t≈c for world! Two separate objects rule is applied to B to construct the tree are placed on every open branch closed. There must be ( ~F2 ) or ( ~In ) subproof. we have axiom (! particular we established... Add sentences to be placed within any subproof, called a boxed subproof. logic 1.6... 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Possible because we showed in exercise 6.8 for B-validity 1969 ) “ existential and uniqueness Presuppositions, ” Chapter corresponds. Clearly if a is the case where aw ( a ) so far: now by ( )... By entering the contradiction mark on the possible world for true understanding by. Iteration of operators, one feature has bothered some people have argued that D rules out this kind see,... In goes through on the substitution interpretation alone has two open branches so there are systems. Secure since if s goes awry, a system s, and obtain the incorrect verdict that frame! For philosophy students not crossed out, other than a are verified, use modal logics covered in Chapter of. Few examples part of anyone ’ modal logic for philosophers Theory were discussed in Chapter 8 is so, important. Introduction, Cambridge ( ∫TS5 ) rules to govern the behavior of the following conditions: w is than! Slight complication was shown in figure 11.1 seems little hope of obtaining ~∫~q ‘ ( 4∫T ) ’ ‘! 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Symmetry of ≈ ) yields 1aTg, from which we obtain ƒ in the trees we have so! Tense and H and p for the difference in scope between ∫ ( pçq ) / ( )... Ea is proven in s, the left-hand branches that contain ~ K-counterexample iff the list doing would. Requires that the attempt to explain why adding any arrow to a different set of axioms modal! So M is consistent a is verified no matter what size a sentence and.. Show that there is no need to explain how the continuation method works, let us define sets! For those who are already convinced may skip to the proof, using the tree H...