Chapter 9 Modal Language, Syntax, and Semantics - David W. Agler

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A. Valuating wffs at a world in a Model: Consider the following model and determine whether ornot the following propositions are true or false:M = whereW ={w 1 , w 2 },R ={, }, andI(P, w 1 )=T, I(P, w 2 )=F, I(S, w 1 )=T, I(S, w 2 )=T1. v M (¬P, w 1 )2. v M (P∧S, w 2 )3. v M (P→S, w 1 )4. v M (◊P, w 1 )5. v M (◊P, w 2 )6. v M (□S, w 1 )7. v M (□S, w 2 )8. v M (◊¬P, w 1 )9. v M (¬□P, w 1 )10. v M (◊□P, w 1 )B. Testing Your Understanding: Using the above model, what additional information is neededto determine the truth value of the following proposition: ◊¬P. What does this say about modallogic semantics and the nature of possibility and necessity?10
9.5 Modal Systems: S, D, T, B, S4, S5In the previous section, a generic definition of a model was provided along with a description ofhow to assign truth values to wffs given a fleshed-out model. In this section, we consider thesemantics for various modal systems. There are several obstacles for constructing a semantics forML. First, as the modal operators ◊ and □ are not truth-functional, it is not possible to representthe semantics of ML using truth tables or truth-tree diagrams. Recall that a propositionaloperator is truth-functional if and only if the truth value of the complex proposition P, composedof propositions R and S, is determined entirely by the truth values of R and S. That is, since thepropositional operator “∧” is truth-functional, we can determine the truth-value of “A∧B” giventhe truth values of “A” and “B”. In contrast, “possibly” and “necessarily” (and theircorresponding operators) are not truth-functional. That is, the truth values of “possibly P” and“necessarily P” are not determined by the truth value of P. If “possibly’ and “necessarily” weretruth-functional, then it would be possible to determine the truth value of sentences like “It ispossible that John is planning a picnic” from the truth value of “John is planning a picnic.”Unfortunately, this cannot be determined for if we suppose that John is not planning a picnic, thetruth or falsity of “It is possible that John is planning a picnic” will depend upon additionalfactors we don’t have access to: is John even alive to do the planning? is John capable ofplanning a picnic?The second obstacle is more problematic, philosophical, and irksome. In propositionaland predicate logic, barring a few isolated cases, there is a strong consensus about the formulathat should count as tautologies in PL and RL. Proposition of the form “P∨¬P”, “P→P”,“P→(Q→P)” are true independent of the specific propositions we substitute for “P” and “Q”.For a language involving ◊P and □P, what propositions should count as tautologies and whatarguments should count as valid? While there are some formula that clearly should count astautologies, e.g. □(P∨¬P) and ¬◊(P∧¬P), there are several formula where it is uncertainwhether they should count as tautologies, e.g., ◊□P→P.Rather than solving this philosophical problem, we sidestep it by developing a variety ofdifferent modal systems. These systems are named K, D, T, B, S4, and S5. While each modalsystem shares a common set of symbols and syntax, and uses the generic notion of a model, eachof these systems is distinguished by their semantics. Specifically, K, D, T, B, S4, and S5 aredistinguished according to what formal properties belong to the accessibility relation R. Theseformal properties (and their corresponding systems) are as follows:Modal SystemKDTBS4S5Property of Accessibility RelationNoneSerialReflexiveReflexive and SymmetricReflexive and TransitiveReflexive, Symmetric, and TransitiveThus, in system S4 (i.e., in a S4-model), the accessibility relation R is both reflexive andtransitive. That is, for any set of worlds W in a S4-model, every member of W accesses itself and11
Page 4 and 5: 9.3 Language & SyntaxThe language a
Page 6 and 7: John is standing between Liz and Sa
Page 9: To clarify (6): ‘It is possibly t
Page 13 and 14: Exercise Set #2A. Semantics1.2.3.4.
Page 15 and 16: However, if the above conditional i
Page 17 and 18: Chapter 10Modal Logic TreesIn chapt
Page 19 and 20: 10.4 Analysis with Modal Logic Tree

Chapter 9 Modal Language, Syntax, and Semantics - David W. Agler

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