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The interaction axioms raise questions concerning asymmetries betweenthe past and the future. A standard intuition is that the past isfixed, while the future is still open. The first interaction axiom\((A\rightarrow GPA)\) conforms to this intuition in reporting thatwhat is the case \((A)\) will at all future times be in the past\((GPA)\). However \(A\rightarrow HFA\) may appear to haveunacceptably deterministic overtones, for it claims, apparently, thatwhat is true now \((A)\) has always been such that it will occur inthe future \((HFA)\). However, possible world semantics for temporallogic reveals that this worry results from a simple confusion and thatthe two interaction axioms are equally acceptable.
In propositional logic, a valuation of the atomic sentences (or row ofa truth table) assigns a truth value \((T\) or \(F)\) to eachpropositional variable \(p\). Then the truth values of the complexsentences are calculated with truth tables. In modal semantics, a set\(W\) of possible worlds is introduced. A valuation then gives a truthvalue to each propositional variable for each of the possibleworlds in \(W\). This means the value assigned to \(p\) for world\(w\) may differ from the value assigned to \(p\) for another world\(w'\).
Even in modal logic, one may wish to restrict the range of possibleworlds which are relevant in determining whether \(\Box A\) is true ata given world. For example, I might say that it is necessary that Ipay my bills, even though I know full well that there is a possibleworld where I fail to pay them. In ordinary speech, the claim that\(A\) is necessary does not require the truth of \(A\) in allpossible worlds, but rather only in a certain class of worlds which Ihave in mind (for example, worlds where I avoid penalties for failureto pay). In order to provide a generic treatment of necessity, we mustsay that \(\Box A\) is true in \(w\) iff \(A\) is true in all worldsthat are related to \(w\) in the right way. So for anoperator \(\Box\) interpreted as necessity, we introduce acorresponding relation \(R\) on the set of possible worlds \(W\),traditionally called the accessibility relation. The accessibilityrelation \(R\) holds between worlds \(w\) and \(w'\) iff \(w'\) ispossible given the facts of \(w\). Under this reading for \(R\), itshould be clear that frames for modal logic should be reflexive. Itfollows that modal logics should be founded on \(M\), the system thatresults from adding \((M)\) to \(\bK\). Depending on exactly how theaccessibility relation is understood, symmetry and transitivity mayalso be desired.
Work on modal logic in the 60s was primarily concerned with obtainingcompleteness results with respect to various conditions on theaccessibility relation. However as research progressed into the 70s,deeper connections were discovered concerning what modal axiomsexpress about frames. A central idea in this work is the notion offrame validity, which differs from the kind of validity which was laidout in Section 6 above. There an argument was considered valid for aset of conditions \(C\) on frames exactly when for every model\(\langle W, R, v\rangle\) whose frame obeys \(C\), and every world\(w\) in \(W\), the truth of the premises at \(w\) entails the truthof the conclusion at \(w\). In short, model validity amounts topreservation of truth on every model. Frame validity, on the otherhand, focuses more clearly on the frames of the model. A sentence issaid to be valid on a frame \(\langle W, R\rangle\) iff it istrue in every world in any model with frame \(\langle W, R\rangle\).Then an argument is ruled frame valid for a set of conditions\(C\) on frames iff it preserves frame validity, that is, for everyframe that obeys \(C\), if the premises are valid on that frame, thenso is the conclusion.
The concept of frame validity provides a basis for translating whatmodal axioms express into sentences of a second-order language wherequantification is allowed over one-place predicate letters \(P\).Replace metavariables \(A\) with open sentences \(Px\), translate\(\Box Px\) to \(\forall y(Rxy \rightarrow Py)\), and close freevariables \(x\) and predicate letters \(P\) with universalquantifiers. For example, the predicate logic translation of the axiomschema \(\Box A\rightarrow A\) comes to \(\forall P \forall x[\forally(Rxy\rightarrow Py) \rightarrow Px\)]. (The basis for thequantification over the predicate letters P is that frame validityquantifies over all valuations of the propositional variables p, butvaluations over p are functions from the set of possible worlds totruth values, and these can be likened to properties of worldsexpressed by p, namely the property that world w has when p is truethere.)
Another complication is that some logicians believe that modalityrequires abandoning classical quantifier rules in favor of the weakerrules of free logic (Garson 2001). The main points of disagreementconcerning the quantifier rules can be traced back to decisions abouthow to handle the domain of quantification. The simplest alternative,the fixed-domain (sometimes called the possibilist) approach, assumesa single domain of quantification that contains all the possibleobjects. On the other hand, the world-relative (or actualist)interpretation, assumes that the domain of quantification changes fromworld to world, and contains only the objects that actually exist in agiven world.
thus ensuring the translation is counted false at the present time.Cresswell (1991) makes the interesting observation that world-relativequantification has limited expressive power relative to fixed-domainquantification. World-relative quantification can be defined withfixed-domain quantifiers and \(E\), but there is no way to fullyexpress fixed-domain quantifiers with world-relative ones. Althoughthis argues in favor of the classical approach to quantified modallogic, the translation tactic also amounts to something of aconcession in favor of free logic, for the world-relative quantifiersso defined obey exactly the free logic rules.
A problem with the translation strategy used by defenders offixed-domain quantification is that rendering the English into logicis less direct, since \(E\) must be added to all translations of allsentences whose quantifier expressions have domains that are contextdependent. A more serious objection to fixed-domain quantification isthat it strips the quantifier of a role which Quine recommended forit, namely to record robust ontological commitment. On this view, thedomain of \(\exists x\) must contain only entities that areontologically respectable, and possible objects are too abstract toqualify. Actualists of this stripe will want to develop the logic of aquantifier \(\exists x\) which reflects commitment to what is actualin a given world rather than to what is merely possible.
However, recent work suggests that the fixed domain option may not beas actualist as originally thought; see Menzel 2020 and the entry onthe possibilism-actualismdebate. And some actualists might respond that they need not becommitted to the actuality of possible worlds so long as it isunderstood that quantifiers used in their theory of language lackstrong ontological import. Furthermore, Hayaki (2006) argues thatquantifying over abstract entities is actually incompatible with anyserious form of actualism. In any case, it is open to actualists (andnon-actualists as well) to investigate the logic of quantifiers withmore robust domains, for example domains excluding possible worlds andother such abstract entities, and containing only the spatio-temporalparticulars found in a given world. For quantifiers of this kind,world-relative domains are appropriate.
Such considerations motivate interest in systems that acknowledge thecontext dependence of quantification by introducing world-relativedomains. Here each possible world has its own domain of quantification(the set of objects that actually exist in that world), and thedomains vary from one world to the next. When this decision is made, adifficulty arises for classical quantification theory. Notice that thesentence \(\exists x(x=t)\) is a theorem of classical logic, and so\(\Box \exists x(x=t)\) is a theorem of \(\bK\) by the NecessitationRule. Let the term \(t\) stand for Saul Kripke. Then this theorem saysthat it is necessary that Saul Kripke exists, so that he is in thedomain of every possible world. The whole motivation for theworld-relative approach was to reflect the idea that objects in oneworld may fail to exist in another. If standard quantifier rulers areused, however, every term \(t\) must refer to something that exists inall the possible worlds. This seems incompatible with our ordinarypractice of using terms to refer to things that only existcontingently.
One response to this difficulty is simply to eliminate terms. Kripke(1963) gives an example of a system that uses the world-relativeinterpretation and preserves the classical rules. However, the costsare severe. First, his language is artificially impoverished, andsecond, the rules for the propositional modal logic must beweakened.
(Here it is assumed that \(A(x)\) is any well-formed formula ofpredicate logic and that \(A(y)\) and \(A(t)\) result from replacing\(y\) and \(t\) properly for each occurrence of \(x\) in \(A(x)\).)Note that the instantiation axiom is restricted by mention of \(Et\)in the antecedent. The rule of Free Universial Generalization ismodified in the same way. In \(\mathbfFL\), proofs of formulas like\(\exists x\Box(x=t)\), \(\forall y\Box \exists x(x=y)\), \((CBF)\),and \((BF)\), which seem incompatible with the world-relativeinterpretation, are blocked. 2ff7e9595c
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