Modal Predicate Logic G. J. Mattey June 11, 2001 1 The Dimensions of Modal Predicate Logic Modal Predicate Logic (MPL) is based on Predicate Logic PL. Its syntax is generated by adding modal operators to the syntax of PL. This will allow such sentences as ‘(∃x)Fx’, in which a modal operators occurs in the scope of aquan- tifier, and ‘(∃x)Fx’ where a quantifier lies in the scope of a modal operator. Semantical systems for MPL vary a great deal in their treatment of sentences of this kind. 1 In what follows, we will be treating Predicate Logic with an added identity predicate, PLI, and the resulting MPLI. But to minimize names for systems, we will refer to these systems as PL and MPL. There are many aspects of the semantics for Predicate Logic and for modal logic which allow for variation when they are combined. 2 In the first place, there are many systems of Modal Sentential Logic on which Modal Predicate Logic can be based. Secondly, the semantics for modal and predicate logics have complicating features not found in Sentential Logic. The two most important are these. Modal semantics has as its foundation frames containing sets of possible worlds at which sentences are assigned values by a valuation function. Semantics for Predicate Logic is based on a non-empty universe of discourse, which contains the individuals which are the referents of names and serve as the values of variables. When the two are combined there are several ways in which the possible worlds can be related to the universe of discourse. The simplest relation is that in which there is a single UD that applies across all possible worlds. This means that each world is populated by exactly the same objects. However, much of the appeal of modal logic lies in the idea that we can use it to represent states of affairs which are not actual. One way in which things might be different is that different objects exist in different possible worlds. To capture this, one might wish to allow that each possible world has its own universe of discourse. This greatly complicates the semantics, as it turns out. Another issue is how to treat the assignment of constants to members of the universe of discourse (at a world, if the UD varies across worlds). Some 1 for a systematic introduction to systems of Modal Predicate Logic, see James Garson, “Quantification in Modal Logic”, in Gabbay and Guenthner, eds., Handbook of Philosophical Logic, Volume II, pp, 249-307. 2 Some of these aspects will not be discussed here. 1
Transcript
Modal Predicate Logic
G. J. Mattey
June 11, 2001
1 The Dimensions of Modal Predicate Logic
Modal Predicate Logic (MPL) is based on Predicate Logic PL. Its syntax isgenerated by adding modal operators to the syntax of PL. This will allow suchsentences as ‘(∃x)�Fx’, in which a modal operators occurs in the scope of aquan-tifier, and ‘�(∃x)Fx’ where a quantifier lies in the scope of a modal operator.Semantical systems for MPL vary a great deal in their treatment of sentences ofthis kind.1 In what follows, we will be treating Predicate Logic with an addedidentity predicate, PLI, and the resulting MPLI. But to minimize names forsystems, we will refer to these systems as PL and MPL.
There are many aspects of the semantics for Predicate Logic and for modallogic which allow for variation when they are combined.2 In the first place,there are many systems of Modal Sentential Logic on which Modal PredicateLogic can be based. Secondly, the semantics for modal and predicate logics havecomplicating features not found in Sentential Logic. The two most importantare these. Modal semantics has as its foundation frames containing sets ofpossible worlds at which sentences are assigned values by a valuation function.Semantics for Predicate Logic is based on a non-empty universe of discourse,which contains the individuals which are the referents of names and serve as thevalues of variables.
When the two are combined there are several ways in which the possibleworlds can be related to the universe of discourse. The simplest relation is thatin which there is a single UD that applies across all possible worlds. This meansthat each world is populated by exactly the same objects. However, much of theappeal of modal logic lies in the idea that we can use it to represent states ofaffairs which are not actual. One way in which things might be different is thatdifferent objects exist in different possible worlds. To capture this, one mightwish to allow that each possible world has its own universe of discourse. Thisgreatly complicates the semantics, as it turns out.
Another issue is how to treat the assignment of constants to members ofthe universe of discourse (at a world, if the UD varies across worlds). Some
1for a systematic introduction to systems of Modal Predicate Logic, see James Garson,“Quantification in Modal Logic”, in Gabbay and Guenthner, eds., Handbook of PhilosophicalLogic, Volume II, pp, 249-307.
2Some of these aspects will not be discussed here.
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semantical systems require that constants be “rigid designators”, referring toexactly the same individuals, no matter what the world at which sentencescontaining them are evaluated. Other systems allow that a constant may refer todifferent individuals when sentences containing them are evaluated at differentworlds. Yet another system banishes constants from the syntax altogether.
If universes of discourse are allowed to vary from world to world, then even ifconstants are rigid designators, the UD might not contain the object to which itrefers. In that case, given the semantics for PLI, some sentences will be lackingin truth-values. This forces a change in the semantics, in that “truth-valuegaps” must be dealt with.
If the semantics has to be changed anyway, it might be profitable to look atan alternative to classical PLI as the basis for a satisfying Modal Predicate Logicwith Identity. A natural alternative is “free logic”, which allows for constantsthat do not refer to any individual in the relevant universe of discourse. Thisalternative is a natural fit for modal logic, because it provides a way of reasoningabout non-existence but possible individuals.
In this chapter, we will investigate two semantical approaches to ModalPredicate Logic with Identity. The first will be the simplest system Q1, dueto Kripke. This is the system in which there is only one universe of discourseand constants refer to the same object in that domain at all possible worlds.Then we will turn to the system Q1R, which allows for world-relative universesof discourse. Finally, we will look at a version of Q1R which is based on freelogic.
2 The Syntax of MPL
The syntax of MPL is built upon that of MSL. Sentence letters, SL operators,and modal operators are carried over from MSL to MPL. Predicate Logic addsseveral elements to this base. We begin with a set of terms, of which thereare two kinds: constants and variables. Constants are lower-case Roman lettersfrom ‘a’ to ‘v’, with or without positive integer subscripts. Variables are lower-case letters from ‘w’ to ‘z’, again with or without positive integer subscripts.
Next, there is a set of predicate letters, which are capital Roman letterswith a positive integer superscript, such as ‘F2’. The superscript indicates thenumber of terms that have to follow the predicate letter to create a formula ofMPL.3 The predicate letters may also carry a positive integer subscript, so ‘F2
3’is a predicate letter.
Finally, there are two quantifier symbols, ‘∃’ and ‘∀’, called the existentialand universal quantifiers, respectively. A pair consisting of a quantifier symboland a variable, enclosed in parentheses, is a quantifier. Thus ‘(∀x)’ and ‘(∃y)’are quantifiers.
Using these expressions, we can define a formula of MPL. We did not use thenotion of a formula in MSL; instead, we gave a definition for a sentence of MSL.In MPL, we shall allow expressions which are sentence-like but not sentences.
3Most of the time, the superscript will be suppressed for readability.
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These open sentences along with sentences make up the class of formulas. Weshall be in a position to define the notion of an open sentence shortly.
Where n is the superscript on a predicate P, we will say that P is an n-placepredicate letter. So ‘F2’ is a two-place predicate letter. A atomic formula ofMPL is an n-place predicate letter followed by n terms. Given this definition,‘F2ax’ is an atomic formula of MPL.
Non-atomic, or molecular formulas can be built from atomic formulas throughthe use of modal and non-modal operators. So ‘�F2ax’ is a formula of MSL.Another way of building molecular formulas is to prefix a quantifier to a for-mula containing the variable in the quantifier. Thus, ‘(∃ x)�F2ax’ is a formula.More precisely, it is a quantified formula, a formula whose main operator is aquantifier. We shall require that a quantifier may be attached only to those for-mulas in which its variable does not occur in the scope of any other quantifiercontaining that variable. In such a case, the variable is said to be free in theformula to be quantified. A variable which is not free is bound to the quantifiercontaining it, and in whose scope it lies. An open sentence is a formula with atleast one free variable. A sentence is a formula with no free variables.
Syntax of MPLIf α is a sentence letter of SL, then α is a formula of MPLIf α is a formula of MPL, then ∼ α is a formula of MPLIf α and β are formulas of MPL, then α & β is a formula of MPLIf α and β are formulas of MPL, then α ∨ β is a formula of MPLIf α and β are formulas of MPL, then α ⊃ β is a formula of MPLIf α and β are formulas of MPL, then α ≡ β is a formula of MPLIf α is a formula of MPL, then �α is a formula of MPLIf α is a formula of MPL, then ♦α is a formula of MPLIf α and β are formulas of MPL, then α ≺ β is a formula of MPLIf α and β are formulas of MPL, then α ◦ β is a formula of MPLIf Pn is a predicate of MPL and t1,. . . ,tn are terms of MPL,then Pnt1,. . . ,tn is a formula of MPLIf α(x) is a formula of MPL and x is not in the scope of any quantifiercontaining x, then (∃x)α(x) is a formula of MPLIf α(x) is a formula of MPL and x is not in the scope of any quantifiercontaining x, then (∀x)α(x) is a formula of MPLIf t1 and t2 are terms of MPL, then t1=t2 is a formula of MPLNothing else is a formula of MPL
3 Systems Q1
We begin our treatment of MPL with a family of semantical systems which arebased on the various systems of Modal Sentential Logic. We will refer to thesesystems collectively as Q1. If a specific system is based on K we will call it‘Q1-K ’, if it is based on D it will be called ‘Q1-D ’, etc. We will concentrate onthose features of the systems that are common to them all.
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3.1 The Semantical Systems Q1
3.1.1 Definitions
Q1-frames are the basic semantical structures. Such a frame is the result ofadding to a MSL frame 〈W,R〉 a universe of discourse D. So a frame Fr =〈W,R,D〉. The only restriction on D is that it be non-empty. It may containinfinitely many members. We shall assume that we can enumerate the membersof the universe of discourse, so that D={u1,. . . ,un,. . . }. The universe of dis-course consists of the “possible objects” to which we can refer using sentences ofMPL. An interpretation based on Fr just is Fr with the addition of a valuationfunction v.4 So an interpretation I = 〈W,R,D,v〉.
The valuation function v works as in the semantics for MSL for sentenceletters, truth-functional sentences, and modal sentences. But it is a good dealmore complicated than in the semantics for MSL. The reason is that there aremore syntactical expressions that require interpretation. These include con-stants, predicates, variables, and quantified sentences.
The valuation function in modal logic is two-place, its first place being asyntactical expression and the second being a possible world. When the expres-sion is a constant a, the value is a member of the universe of discourse D. Weshall call this value the designation of the constant. So we can say in generalfor an interpretation 〈W,R,D,v〉,
Designation for Constantsv(a, w) ∈ D
Since the universe of discourse contains at least one member, the following holdsfor any constant, any interpretation, and any world in that interpretation:
Non-Empty Designation(∃ui)(ui ∈ D & v(a,w) = ui)
Because we are assuming rigid designation, the value of a constant at a worldis the same as that at any other world.
Rigid Designation(∀wi)(∀wj)((wi∈ W & wj∈ W)⊃(v(a, wi)= v(a, wj))
The assumption of rigid designation greatly simplifies the semantics as well asfacilitating completeness results.5 But the simplification comes at a price. Wecannot use constants to stand for definite descriptions, since expressions like“the first human to walk on the moon” would be expected to have differenttruth-values at different possible worlds. Leibniz, and more recently DavidLewis, have also argued that no individuals exists at different possible worlds.If they are right, then Q1 is an inadequate logic.
The semantics for Q1 allows that the extensions of predicates can vary fromworld to world. As with non-modal PL, the extension of an n-place predicate
4The subscript reference to the interpretation I will no longer be used, since we will needto affix subscripts for different purposes.
5See Garson, pp. 261-265.
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P is a set of ordered n-tuples taken from D. For example, given a two-placepredicate P2, if D={u1,u2}, then v(P2,w) could be any of the following: ∅,{〈u1,u2〉}, {〈u1,u1〉, {〈u2,u2〉}, {〈u1,u1〉, 〈u2,u1〉,〈u2,u2〉}, etc. The set of allordered n-tuples drawn from a set is called the Cartesian product of that set,n times. The set of all ordered pairs drawn from the Cartesian product of D,D2, contains the extension of any two-place predicate as a subset. So we cansay for any n-place predicate Pn:
Extension of Predicatesv(Pn, w) ⊆ Dn.
We could easily define the notion of the intension of a predicate P in termsof the extensions of P at all the worlds in an interpretation. For example, theintension could be understood as a set of ordered pairs, the first member ofwhich is the extension of the predicate and the second member of which is apossible world along the following lines: 〈v(P, w), w〉. Some semantical systemsallow for the intensional interpretation of predicates, which are understood asproperties. This was first done in Carnap’s original semantics for MPL.6
At this point, we are ready for the truth-definition for atomic sentences ofMPL.
This simply says that the ordered set consisting of the values of the constants(in order) is in the extension of the predicate.
For example, let us suppose that D={u1,u2} and v(F,w)={〈u1,u1〉, 〈u2,u2〉}.If v(a, w)=u1, then v(Faa,w)=T, since 〈u1,u1〉 is in the extension of ‘P’ at w.But suppose further that v(b, w)=u2. Then v(Fab,w)=F, since 〈u1,u2〉 is notin the extension of ‘F’ at w. Note that it may be the case that the extensionof ‘F’ is different at a distinct world w1, in which case, ‘Faa’ and ‘Fab’ mighthave different truth-values at w1 from their truth-values at w.
The treatment of variables is not so straightforward. This is due to thefact that a variable might be taken to refer to any member of the universe ofdiscourse. To take this into account, we need a notion of a variable assignmentd (with or without a numerical or alphabetic subscript). A variable assignmentis a function whose arguments are variables and whose values are members ofthe universe of discourse: d(x)∈ D. These assigments are made independentlyof possible worlds, so we shall say that for any world w, d(x) = d(x, w). Forexample, using D={u1,u2}, we might have it that d(x,w)=u1, d(y,w)=u2,d(z,w)=u1, and so on for all the variables.
Now we can combine the notion of a variable assignment with that of avaluation function. The notation ‘vd’ indicates that a valuation-function vassumes a variable-assignment d. What this allows us to do is to evaluatesentences with variables. We shall say in general that for variable x, d(x) ∈D, that is, that the variable-assignment on an interpretation is taken from the
6See also Aldo Bressan, A General Interpreted Modal Logic.
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universe of discourse in that interpretation. Assignments made to constantsare not affected by assignments made to variables by variable-assignments. Ingeneral, v(a,w)=vd(a,w) for any f, a, w, and d. Similarly, for any predicate-letter P, v(P,w)=vd(P,w).
For example, consider ‘Fxa’ on the interpretation given above. We wantto say that if d(x)=u1, then vd(Fxa,w)=T.7 This is because the ordered pair〈d(x),v(a,w)〉, i.e., 〈u1,u1〉, is in the extension of ‘F’ at w. To simplify the truth-definitions that are to come, we shall adopt the convention that vd(x,w)=d(x).It does no harm to say that the value of a variable at a world given the assign-ment just is the value of the assignment, since worlds are irrelevant to variable-assignments in Q1. Given this notation, we can expand the truth-definition foratomic sentences to include all atomic formulas, including open sentences. (Welet ‘t’ stand for any term, constant or variable.)
We shall speak in terms of a variable-assignment satisfying an open sentence.The idea is that a open sentence expresses a condition which may or may notbe fulfilled, or satisfied, by members of the UD. So when a valuation-functionvd based on variable-assignment d gives an open sentence the value T, we shallsay that that d satisfies the open sentence.
At this point, there is still one more semantical notion required for the truth-definitions for quantified sentences. Consider the open sentence ‘Fxy’ on theinterpretation given above. There are four possible assignments to its variables:
(1) d1(x)=u1 and d1(y)=u1,(2) d2(x)=u1 and d2(y)=u2,(3) d3(x)=u2 and d3(y)=u1,(4) d4(x)=u2 and d4(y)=u2.
Strictly speaking, the four assignments we have just given are only partialvariable-assignments, since a variable-assignment assigns members of the UDto all variables. However, if the formula under evaluation does not contain anyvariables other than ‘x’ and ‘y’, the assignments to any other variables are ir-relevant given the truth-definition for an atomic formula. So even though thereare infinitely many assignments which begin in the same way as d1, d2, d3, andd4, we shall treat these four as if they were the only variable assignments onour interpretation when dealing with formulas containing only ‘x’ and ‘y’.
Now compare d1 and d2. They differ only with respect to the assignmentmade to ‘y’. We may call d2 a variant of d1 with respect to ‘y’ (or a y-variant ofd1). This can be noted by the equation d2=d1[u2/y]. The variable-assignmentd2 is the assignment d1 except that ‘y’ is given the value u2, where d1 gave ‘y’the value u1. In general, an x-variant of a variable-assignment d, d[ui/x], isan assignment just like d except perhaps for the member of the UD, ui, that is
7In The Logic Book, open sentences are not given truth-values. They are given truth-valueshere to simplify the exposition. The notation used here is adapted from The Logic Book andHughes and Cresswell, A New Introduction to Modal Logic.
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assigned to x. As a consequence, we can say that a valuation function based ond[ui/x] assigns ui to x:
Variant Assignmentvd[ui/x](x, w1)=ui
(We will allow that a variable-assignment is a variant of itself, so there needbe no difference in the values of the variable x between an assignment and itsx-variant.)
Some formulas will call for the use of variants of variants, variants of variantsof variants, etc., depending on the number of free variables they contain. If aformula contains three free variables, ‘x’, ‘y’, and ‘z’, for example, we mighthave to consider the x-variants of d, the y-variants of the x-variants of d, andthe z-variants of the y-variants of the x-variants. (The order of the quantifierswhich bind the variables will determine the exact sequence of variants required.)To take an arbitrary example, we might have d[u2/x][u1/y][u2/z]. This can beabbreviated as d[u2/x, u1/y, u2/z].
We are finally able to give the truth-definitions for quantified formulas. Thebasic idea is that an open sentence α(x), with x free, may be true (at a world)on some, all, not all, or no variable-assignments. If it is true at a world w onsome variable assignments, then (∃x)α(x) is true at w. If it is true at w on novariable-assignment, then the existential sentence is false at w. If α(x) is trueat w on all variable-assignments, then (∀x)α(x) is true at w, and if it is not thecase that it is true on all variable-assignments, it is false at w.
The actual definition has to be a bit more complicated. In order to dealwith multiple quantifiers, we have to define truth of an open sentence relativeto a valuation based on a variable-assignement, vd. So what we need is to sayis that an existential formula (∃x)α(x) is assigned T given vd just in case theembedded open sentence α(x) is assigned T on all x-variants of d. Similarly,for a universal formula (∀x)α(x): the embedded sentence α(x) must be assignedthe value T on all variants of d with respect to x. (Note that this is why werequired that all variable-assignments be variants of themselves, since what dassigns to x is just as important as what any of its non-identical variants assignswhen assessing a universal formula.)
Truth-Definition for ∃ (Q1)vd((∃x)α(x), w)=T iff vd[u/x](α(x), w)=T for some x-variant d[u/x] of d
Truth-Definition for ∀ (Q1)vd((∀x)α(x), w)=T iff vd[u/x](α(x), w)=T for all x-variants d[u/x] of d
For example, given the interpretation and variable-assignments above, wecan evaluate the sentence ‘(∀x)(∃y)Fxy’ at world w with respect to the variable-assignment d2. Recall that D={u1,u2} and v(F,w)={〈u1,u1〉, {〈u2,u2〉}. Nowconsider the atomic formula ‘Fxy’. On variable-assignment d2, 〈d2(x),d2(y)〉=〈vd2(x,w),vd2(y,w)〉=〈u1,u2〉. Since 〈u1,u2〉 is not in vd2(F,w), vd2(Fxy,w)=F.
However, d1 is a y-variant of d2. That is, d1=d2[u1/y]. The assignmentd1 is the same as d2 except that it assigns u1 to ‘y’ where d2 assigns u2 to
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‘y’. Now we have it that 〈d1(x),d1(y)〉=〈u1,u1〉. Since 〈u1,u1〉 ∈ vd1(F,w),vd1(Fxy,w)=T. So there is a y-variant of d2 which gives ‘Fxy’ the value T, andso vd2((∃y)Fxy,w)=T.
To evaluate ‘(∀x)(∃y)Fxy’, we need to look at how all x-variants of d2 assignvalues to ‘(∃y)Fxy’. There are exactly two x-variants of d2, that is variable-assignments that differ from d2 at most with respect to the value they assign to‘x’. Since d2 assigns u2 to ‘y’, the x-variants are d4 and d2 itself. We alreadyknow that d2 assigns ‘(∃y)Fxy’ the value T, so the only question is what valued4 assigns that formula. Now there is a y-variant of d4, namely d4 itself, thatsatisfies ‘Fxy’, so d4 does satisfy ‘(∃y)Fxy’. This means that all x-variants ofd2 satisfy that formula, in which case vd2((∀x)(∃y)Fxy,w)=T.
Exercise. Determine whether the other three variable-assignments satisfy‘(∀x)(∃y)Fxy’.
The truth-definition for the universal quantifer makes a universally quan-tified formula true (at a world w and valuation function v) given a variableassignment d just in case the embedded open sentence is true on all variants ofd relative to the variable in the quantifier.
vd((∀x)α(x), w)=T iff vd[u/x](α(x), w)=T for all x-variants d[u/x] of d
Recall that an x-variant of d is a variable assignment just like d except that itmay assign a different member ui of the UD to x.
Now suppose that vd((∀x)α(x), w)=T. Then vd[u/x](α(x), w)=T for allx-variants d[u/x] of d. Suppose further that u1 is an arbitrary member of D.Then vd[u1/x] is an x-variant of d, and vd[u1/x](α(x), w)=T. Therefore, if u1
∈ D, then vd[u1/x](α(x), w)=T. Since the choice of u1 is arbitrary, for all u inD, if u ∈ D, then vd[u/x](α(x), w)=T. For the converse, suppose that for anymember u of D, vd[u/x](α(x), w)=T, and suppose further that vd[u1/x] is anarbitrary x-variant of d. Then u1 is a member of D, so vd[u1/x](α(x), w)=T.Therefore, if vd[u1/x] is an x-variant of d, then vd[u1/x](α(x), w)=T. Since thisholds for any x-variant of D, vd((∀x)α(x), w)=T.
So to say that vd[u/x](α(x), w)=T for all x-variants d[u/x] of d is just tosay that for any member ui of the UD, vd[ui/x](α(x), w)=T. More formally,
Alternative Truth-Definition for ∀ (Q1)vd((∀x)α(x), w)=T iff (∀ui)(ui ∈ D ⊃ vd[ui/x](α(x), w)=T).
Note that this definition respects the fact that ui may be the member of the UDassigned to x by d itself. Similarly, we may give an alternative truth-definitionfor existentially quantified formulas.
Alternative Truth-Definition for ∃ (Q1)vd((∃x)α(x), w)=T iff (∃ui)(ui ∈ D& vd[ui/x](α(x), w)=T).
These alternative definitions will be useful later.Atomic formulas with identity predicate ‘=’ get special treatment in the
semantics. We want to say that they are true just in case the terms flankingthe identity sign are assigned the same member of the UD.
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Truth-Definition for = (Q1)vd(t1=t2, w)=T iff vd(t1,w)= vd(t2,w)
For example, given the semantics above, we can evaluate vd2(a=x,w). Recallthat v(a,w)=u1, in which case, vd2(a,w)=u1. Moreover, d2(x)=u1 and sovd2(x,w)=u1. As a result, vd2(a,w)=vd2(x,w). So the sentence ‘a=x’ is trueat w given vd2 .
With the definition of truth for sentences on a variable-assignment in hand,we can now define truth for sentences, that is, formulas with no free variables.A sentence will be said to be true at a world w on an interpretation I just incase it is true at w on all variable-assignments in I:
Truth-Definition for Sentences of MPLv(α,w)=T iff vd(α,w)=T for all d in I.
As with Modal Sentential Logic, a valid sentence is one which has the value Tat all worlds on all interpretations.
Summary of Systems Q1-xI=〈W, R, D, v〉
v defined for �, ♦ as for K Modal restrictions on R for system xv defined for ∀, ∃, = specially for Q1
Non-Empty DesignationRigid Designation
3.1.2 Quantifiers and Modalities in Q1
The Q1 systems are very strong, in the sense that the interplay between modal-ity and quantification is somewhat transparent. Specifically, the following equiv-alences hold:
�(∀x)α(x) and (∀x)�α(x),
♦(∃x)α(x) and (∃x)♦α(x).
The valid schemata (∀x)�α(x) ⊃ �(∀x)α(x) and ♦(∃x)α(x) ⊃ (∃x)♦α(x) areknown in the literature as variations of the Barcan formula, as the latter was anaxiom schema of the first published axiomatization of modal predicate logic.8
Any sentence that is an instance of one schema is an instance of the other. Thevalid schemata �(∀x)α(x) ⊃ (∀x)�α(x) and (∃x)♦α(x) ⊃ ♦(∃x)α(x) are theforms of the converse Barcan formula. Again, any instance of one is an instanceof the other.
Exercise. Using Duality and Quantifier Negation (from the derivational systemfor PLI ), prove the claims just made that any instance of one schema is aninstance of the other.
8Ruth Barcan, “A Functional Calculus of the First Order Based on Strict Implication”,Journal of Symbolic Logic 11 (1946) 1-16. The existential sentence was use in the originalpaper, as the ‘♦’ was generally used as a primitive operator at that time. Barcan’s systemwas based on Lewis’s S2.
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Given our emphasis on the consequence-relation in modal logic, we will belooking at what will be called the Barcan consequences and the Converse Barcanconsequences: :
Informally, the Barcan consequence with the universal quantifier and the neces-sity operator states that if everything necessarily satisfies the condition specifiedby α(x), then necessarily, everything satisfies that condition. So for example, ifeverything is such that necessarily it is either round or not round, then neces-sarily, everything is round or not-round.
Suppose that everything is such that necessarily it is either round or not-round, and consider an arbitrary accessible possible world. Everything is eitherround or not-round at all possible worlds, so everything is either round or not-round at that arbitrary world. Therefore, since the choice of worlds is arbitrary,at every world, everything is round or not-round. Hence, necessarily, everythingis round or not-round.
We shall now give a formal proof of an instance of the Barcan consequence,‘(∀x)�Fx’, quite straightforwardly.
Exercise. Prove this instance of the converse Barcan consequence: {�(∀x)Fx}�Q1−x (∀x)�Fx’.
We shall now turn to the converse Barcan consequence. This time, we willshow that an instance of the ∃♦ version is an entailment in the semantics for anyQ1 system. Informally, what we have is that if something possibily meets the
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conditions specified by α(x), then possibly something satisfies the condition.Suppose that some existing thing is possibly round. Then there is an acces-
sible possible world at which it is round. Therefore, something at that worldis round. And since there is an accessible possible world at which something isround, it is possible that some existing thing is round. The formal proof for aninstance of the converse of the Barcan formula is as follows.
In system Q1-K, we have the following consequence-relation:
{(∃x)�α(x)} �Q1−x �(∃x)α(x).
Informally, it states that if there is an individual that satisfies the conditionspecified by α(x) in every accessible world, then in every accessible world thereis an individual which satisfies the condition specified by α(x). Given the an-tecedent, we can always presume that the same individual will serve to satisfythe condition in each world. So, for example, if there is an object which neces-sarily is human, then there is an object which is necessarily human.
Exercise. Show that {(∃x)�Fx} �Q1−x �(∃x)Fx.
One might wonder whether any combination of quantifiers can be freely re-versed, as with the Barcan consequence and its converse. The answer is negative.The combinations �/∃ and ♦/∀ are not reversible in one direction. So in thepresent case, we have:
{�(∃x)α(x)} 2Q1−x (∃x)�α(x).
If there were such a consequence-relation, it would allow us to infer from thepremise that it is necessary that there exists something satisfying the conditionspecified by α(x) to the conclusion that there exists something that necessarilymeets the condition specified by α(x).
The premise expresses necessity de dicto, while the putative conclusion ex-
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presses necessity de re. The failure of this consequence-relation shows that inQ1-K, de re necessity is stronger than de dicto necessity. If, in each accessibleworld, there is an individual which is α, then there is a single such individualwhich is α at all the worlds. Suppose that at every world there exists at leastone human that is shorter than two meters tall. Does it follow that there is ahuman that is shorter than two meters tall at all the worlds? We might imaginea person who was malnourished as a child and thus is shorter than two metersbut in another world was nourished properly and exceeds two meters in height.
A formal counter-example due to Kripke is an interpretation with two worlds,w1 and w2, such that each world is accessible to itself and the other. Theuniverse of discourse D consists of two distinct objects, u1 and u2. Now sup-pose that for a value-assignment d, vd[u1/x](Fx, w1)=T, vd[u2/x](Fx, w2)=T,vd[u2/x](Fx, w1)=F, and vd[u1/x](Fx, w2)=F. Then vd((∃x)Fx, w1)=T andvd((∃x)Fx, w2)=T.
In each case, there is at least one x-variant of d which satisfies ‘Fx’ at aparticular world. So ‘(∃x)Fx’ is true under vd at all worlds accessible to w1.Hence, vd(�(∃x)Fx, w1)=T. However, since vd[u2/x](Fx, w1)=F and Rw1w1,vd[u2/x](�Fx, w1)=F. Similarly, since vd[u1/x](Fx, w2)=F and Rw1w2, it fol-lows that vd[u1/x](�Fx, w1)=F. So for both x-variants of d, ‘�Fx’ is false.This yields the conclusion that vd((∃x)�Fx, w1)=F. This result holds in anyQ1 system, since R is an equivalence relation, and so the consequence relationdoes not hold in Q1-S5 and therefore not in any weaker system either.
Parallel results hold for the combination of universal quantifier and thepossibility-operator:
{♦(∀x)α(x)} �Q1−K (∀x)♦α(x).
{(∀x)♦α(x)} 2Q1−K ♦(∀x)α(x).
The demonstration of the consequence-relation is straightforward, and the samecounter-example as just given may be applied here. The similarity of the resultsis due to fact that ♦(∀x)α(x) is equivalent to ∼ �(∃x) ∼ α(x), and so forth.
3.1.3 Identity in Q1
We will now turn to some results for identity in the Q1 systems. The firstresult is that �Q1−x (∀x)x = x. This is a simple consequence of the fact thatvd[u1/x]=vd[u1/x] for any x-variant of any variable-assignment d. Closure yieldsthe result that �Q1−x �(∀x)x = x. And the converse of the Barcan consequencegives us the further result that �Q1−x (∀x)�x = x.
Exercise. Show that �Q1−x (∀x)�x=x directly using the Q1 semantics.
The effect of Rigid Designation is that all identities are necessary:
�Q1−x ai = aj ⊃ �ai = aj.
�Q1−x (∀x)(∀y)(x = y ⊃ �x = y).
It is fairly obvious why the first schema is valid given Rigid Designation. If twoconstants are assigned the same value at a world by v, then they are assigned
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the same value at all worlds, so that ai = aj is true at all worlds. The argumentfor the quantified sentence-schema is just an application of this reasoning tovariable-assignments. The following is a meta-logical derivation showing thatan instance of the first schema is valid.
To show validity of an instance of the quantified sentence-schema, assumethat u1 and u2 both belong to D and that d is a variable-assignment. Then sup-pose that vd[u1/x,u2/y](x=y, w)=T. That is, we have a y-variant of an x-variantof d that makes ‘x=y’ true at w. By the truth-definition for identity, we havevd[u1/x,u2/y](x, w)=vd[u1/x,u2/y](y, w). The variable-assignment gives the samemember of the UD to both ‘x’ and ‘y’. Now let w1 be an arbitrary world ac-cessible to w. Because variable-assignments are made independently of possibleworlds, ‘x’ and ‘y’ get the same values at w1 as they did at w. So vd[u1/x,u2/y](x,w1)=vd[u1/x,u2/y](y, w1). This means that vd[u1/x,u2/y](x=y, w1)=T, againby the truth-definition for identity. Since w1 is arbitrary, vd[u1/x,u2/y](�x=y,w)=T. Now we have it that if vd[u1/x,u2/y](x=y, w)=T, then vd[u1/x,u2/y](�x=y,w)=T. By the truth-definition for the material conditional, vd[u1/x,u2/y](x=y⊃ �x=y, w)=T. Also, since the choice of u2 is arbitrary, vd[u1/x]((∀x)(x=y ⊃�x=y), w)=T. For the same reason, vd((∀x)(x=y ⊃ �x=y), w)=T.
A result that is more eye-catching is this:
�Q1−x �(∃x)x = a.
Whatever is named by a constant a necessarily exists. This is a result of the factthat every constant names a member of the UD, and that every member of theUD exists at all possible worlds. In non-modal Predicate Logic with Identitywe have it that �PLI (∃x)x = a, so we can get its necessitation by Closure. Aninstance of this schema can also be shown to be valid as follows.
Whatever is named by a constant a necessarily exists. This is a result of thefact that every constant names a member of the UD, and that every member ofthe UD exists at all possible worlds. An instance of this schema can be shownto be valid as follows.
This result should be compared with the result for non-modal PredicateLogic, which has as a valid sentence-form (∃x)x=a. This only says that eachconstant refers to an object in the UD. The present result is stronger, statingthat each constant refers to an object existing in the UD for each possible world.But this just follows from the fact that there is a single domain of individualsfor all the worlds.
A similar result is the validity of the sentence-form (∃x)�x=a. This saysthat some existing thing is identical at all accessible worlds to what a desig-nates. Again, this is a straightforward result of Non-Empty Designation, RigidDesignation, and unitary universes of discourse. Since a designates somethingin D, and the same thing at all worlds, there is something in each accessibleworld which is identical to something which exists at a given world.
3.2 The Derivational Systems Q1
The most straightforward way to obtain an integration of the inference rules ofModal Sentential Logic and non-modal Predicate Logic is to take the union ofthe two sets of rules. This yields derivational system Q1-x for MSL system x.So to the rules for K, we can add the PLI rules to get the system Q1-K, etc.
The derivation rules for MSL prohibit the use of SL rules across restricedscope line. The reason for this was that truth-values at a given world could notbe assumed to hold at accessible worlds. This kind situation also rises in MPLderivations, so that we shall prohibit the use of quantifier rules across restrictedscope lines.
To see why there must be a restriction for the Q1 systems, suppose that ‘Fa’occurs outside a restricted scope line and we wish to write down ‘(∃x)Fx’ inside
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the scope line. This is tantamount to assuming that ‘Fa’ is true at a world andinferring that ‘(∃x)Fx’ is true at an accessible world. Because ‘Fa’ is held tobe true at a world, what ‘a’ designates is taken to be in the extension of ‘F’ atthat world. But it might be the case that ‘F’ has an empty extension at theaccessible world. Then we would not be entitled to infer ‘(∃x)Fx’ at that world.Similar counter-examples can be generated for the other quantifier rules.
Although the quantifier rules may not be used across restricted scope lines,there is no restriction on their use within restricted scope lines. In particular,it does not matter whether the relevant constant or variable occurs in a “modalcontext”, i.e., within the scope of a modal operator. So, for example, from‘(∀x)�Fx’, ‘�Fa’ can be inferred, and from ‘�Fa’, ‘(∃x)�Fx’ can be inferred.The reason these inferences are permitted is the fact that all assignments madeto terms are taken from the same UD, the fact that variable-assignments arenot sensitive to worlds, and Rigid Designation.
The rule of = Elimination is another that might be applied across a restrictedscope line. Because variable-assignments are not made relative to worlds andbecause of Rigid Designation, there is no reason in principle to restrict this kindof use of the rule for the Q1 systems. We shall, however, make the restrictionin order to keep the derivational rules uniform and to allow their use for weakersystems where the application of = Elimination across restricted scope lineswould not be permitted.
A further issue is whether to allow the use of = Elimination in cases wherethe substitution would be made into a modal context. For example, if we have‘a=b’ and ‘�Fa’, should we be able to infer ‘�Fb’? With the unitary UD andRigid Designation, this would be a sound move in the Q1 derivational systems.9
But again, it would not be sound in weaker systems. We shall restrict the useof = Elimination even further to accommodate weaker systems, allowing thesubstitution to be made only into atomic formulas.10
We shall need a new rule to reflect Rigid Designation. Specifically, if thereis an identity sentence ai=aj , we can write �ai=aj . We also have a rule thatallows the move from ∼ai=aj to ∼ �ai=aj .
� = Introduction
ai=aj�ai=aj � = I
∼ai=aj� ∼ai=aj � = I
That the first is a sound rule can be seen from the fact that by Rigid Designation,v(ai, wi)=v(ai, wj) for all worlds wi and wj . So if v(ai=aj , w)=T, then v(ai,w)=v(aj , w), and so v(ai, wi)=v(aj , wi) at all accessible worlds wi, whichyields the result that v(�ai=aj , w)=T.
The negated-identity rule can be justified as sound as well. Suppose v(∼ai=aj ,w)=T. Then v(ai, w) = ui and v(aj , w) = uj for distinct ui and uj in theUD at w. Because the UD at w is the same as that as the UD at wi for all
9It can also be derived using the rules for Q1, as will be shown below.10This modification is based on a suggestion by James Garson, op. cit..
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accessible worlds wi, ai and aj refer to distinct individuals in the UD at wi, sov(ai=aj , wi)=F, in which case v(� ∼ai=aj , w)=T.
Derivational Systems Q1-xModal operator rules as with modal system x
No PL rules may be used across restricted scope linesQuantifier Rules as for PL, with no restrictions in modal contexts
= Elimination may be used only on atomic formulas� = Introduction
It will be assumed without proof that these rules are sound and complete relativeto the semantics for Q1 systems. The soundness of some uses of these rules will,however, be demonstrated.
We shall now provide derivations of some important consequences and the-orems in the systems. The derivation of the an instance converse Barcan conse-quence follows in outline the semantical proof already given, but it differs fromit in that a constant is involved.
1 �(∀x)Fx Assumption2 (∀x)Fx 1 SR3 Fa 2 ∀ E4 �Fa 2-3 � I5 (∀x)�Fx 4 ∀ I
The only rule in this derivation that is not a simple instance of a modal orquantifier rule is step 5, where ∀ Introduction is used. To show that this is asound use of the rule, we turn to the semantical argument for this consequence.
Suppose vd(�(∀x)Fx,w)=T for an aribtrary world w on an arbitrary inter-pretation. Then at all accessible worlds wi, vd((∀x)Fx, wi)=T. This holds justin case every x-variant of d satisfies ‘Fx’, which in turn holds just in case everymember of the UD, ui, is in the extension of ‘F’. So if we pick an arbitraryconstant ‘a’, we can say that whatever object ui in the UD is referred to by ‘a’,〈ui〉 is in the extension of ‘F’. Therefore, vd(Fa, wi)=T.
In Q1 systems, every object that is in the UD at w is also in the UD atan accessible world wi. Moreover, by Non-Empty Designation, ‘a’ picks out amember of the UD at w (by Rigid Designation, the same one it picks out atwi). As a result, we can say that whatever object in the UD at w ‘a’ mightrefer to, ‘�Fa’ is true at w. This holds just in case every x-variant of d satisfies‘�Fx’, and so vd((∀x)�Fx, w) = T. Thus what supports the converse Barcanconsequence in the semantics is the fact that the UD of a given world is asubset of the UD of any accessible world, and (if constants are involved in thereasoning) Rigid Designation.
Now we turn to an instance of the Barcan consequence. The derivation isas simple as with its converse.
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1 (∀x)�Fx Assumption2 �Fa 1 ∀ E3 Fa 2 SR4 (∀x)Fx 3 ∀ I5 �(∀x)Fx 3-4 � I
Of note in this derivation are two moves: the instantiation of ‘x’ to a constantoccurring in the scope of a necessity-operator, and the use of ∀ Introductioninside a strict sub-derivation. These steps are permitted by the rules, and aperusal of the semantical proof of the consequence shows why they should be.
Suppose vd((∀x)�Fx, w) = T. Then all x-variants of d satisfy ‘�Fx’. This isequivalent to saying that any assignment of a member of the UD to an arbitraryconstant ‘a’ makes ‘�Fa’ true. So for every accessible world wi, vd(Fa, wi) =T. Now every object in the UD of wi is in the UD of w, since the UDs areidentical. Also, by Rigid Designation, ‘a’ picks out the same object at eachworld. Therefore, ‘Fa’ holds no matter what object in the UD of wi ‘a’ refersto, which is equivalent to saying that ‘Fx’ holds at wi for all x-variants of d.So vd((∀x)Fx, wi) = T. It follows that vd(�(∀x)Fx, w) = T, which was to beproved. So the Barcan consequence is supported in the semantics by the factthat the UD of an accessible world is a subset of the UD of any given world,and Rigid Designation (if constants are involved in the reasoning).
A derivation of interest is that of an instance of the Barcan consequence,{♦(∃x)Fx} `Q1−x (∃x)♦Fx. The derivation presents a problem if we wish toproceed straightforwardly using ♦ Elimination.
1 ♦(∃x)Fx Assp2 (∃x)Fx Modal Assp3 Fa Assp
The problem is that we would like to be able to discharge the assumptionsin the reverse order, inferring first ♦Fa, then (∃x)♦Fx. But this is forbiddenby the rules. We would have to discharge the existential Assumption made atstep 3 before discharging the Modal Assumption at step 2, but this cannot bedone while preserving the constant ‘a’. The derivation may be made indirectly,however, with the help of a Quantifier Negation rule at one point where thederivation would become tedious.
Exercises. Give derivations to show that the following consequences hold.{(∃x)♦Fx}`Q1−x ♦(∃x)Fx, {(∃x)�Fx}`Q1−x �(∃x)Fx.
We shall now turn to derivations involving identity formulas. The first resultsto be established are that {a=b} `Q1−x �a=b and `Q1−x (∀x)(∀y)(x=y ⊃�x=y). The former is established using a two-step derivation.
Another result from the semantical section to be established in the deriva-tional system is that `Q1−x (∃x)�x = a. This result is very easily established.
1 a=a = I2 �a=a 1 � = I3 (∃x)�x=a 2 ∃ = I
Now we shall establish some further results. First, recall that the use of =Elimination was severely restricted. A substitution could be made only intoatomic sentences and was not allowed to cross restricted scope lines. This re-striction can be overcome with � = Introduction. To illustrate this, we shallshow that {a=b, �Fa}`Q1−x �Fb. This will show both how substitution canoccur inside a modal non-atomic sentence and how restricted scope lines can becrossed.
This concludes our treatment of identity in the Q1 systems. Before turningto the next family of systems, it will be useful to look at a further consequencethat can be established in systems for which the B Strict Reiteration rule holds.We shall show that an instance of the Barcan consequence is derivable withoutthe use of ∀ Introduction across the scope of the necessity operator. This showsthat even if the rules were weakened to prohibit such use (as will be the case inthe next section), the Barcan consequence still holds.
Note that in step 4, ∀ Elimination was used across the scope of a modal operator.This is what allows for the derivation of the converse Barcan consequence. Soin systems with the SR(B) rule, if the converse Barcan consequence holds, theBarcan consequence holds. On the semantic side, it is the symmetry of theaccessibility relation that generates this result. The SR(B) rule was intendedto simulate the effect of symmetry in the derivational system.
Exercise. Show that the Barcan consequence holds if we assume that theaccessibility relation is symmetrical and that the UD of a world is a subset ofthe UD of any accessible world.
4 Systems QPL
In this section, we develop systems that are related to the Q1 systems butlack the Barcan consequence.11 As was just seen, such systems must not allow
11The semantical systems QPL were given by Hughes and Cresswell, An Introduction toModal Logic, Chapter Ten.
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that accessibility is symmetrical, so there are no QPL systems based on eitherB or S5.12 Without the Barcan consequence, we can use the logic to representpossible individuals that are not actual (at the world under consideration). Thiswould seem to be desirable for most applications, so systems QPL are worthlooking into. But it will be seen that like the Q1 systems, the QPL systems donot allow for anything existing at a world not to exist at any accessible world.So they may be too weak for the most part. It will also emerge that thesesystems require a rather drastic change in the base predicate logic—a changethat has some rather implausible ramifications.
4.1 The Semantical Systems QPL
4.1.1 Definitions
To block the Barcan consequence, it is necessary to relativize universes of dis-course to possible worlds. In fact, we could have developed the semantics for Q1by linking UDs to worlds, with the proviso that each world have the same UD asall the others in a given interpretation. Hughes and Cresswell add to the defini-tion of a Q1 -interpretation a function q from worlds to subsets of D. Then I =〈W,R,D,q,v〉. Suppose, for example, that in a given interpretation, D={1,2}and W={w1, w2}. Then we might have q(w1)={1} and q(w2)={1,2}. Toindicate the world wi with which a domain D is associated, we shall write Dwi .So in our example, Dw1={1} and Dw2={1,2}.
A constraint on QPL frames is that the UDs be “nested”, in the sense thatevery object in the domain at a world is in the UD of every accessible world,though the converse need not hold. The idea is that the UD can “grow” as onemoves to accessible worlds, but it cannot “shrink”.
Nested Universes of DiscourseIf Rwiwj , then Dwi ⊆ Dwj
We shall see that this requirement makes the converse Barcan consequence holdin the QPL systems.
The allowance of world-relative UDs affects the interpretation of the con-stants of the modal language.13 First, because it allows that some objects arein the UD of some worlds but not others, a constant which designates an objectin the domain of one world may fail to designate an object in the domain ofanother. That is, in some cases, v(a, w) is undefined.
As a result, the semantical rule Designation for Constants must be madedisjunctive. Rather than stating that a constant assigns a member of the UDto an object at a world, the rule must state that it either assigns a member ofthe UD or it assigns nothing at all.
Limited Designation for Constantsv(a, w) ∈ Dw or v(a, w) is undefined
12Other excluded systems are those that result from adding symmetry to the semanticalsystems K, D or T.
13Hughes and Cresswell do not include constants in their syntax.
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Next, the rule Non-Empty Designation must be modified to take Limited Des-ignation into account. At most what can be required is that there be a worldwhich has a domain containing an object which the constant designates.
Rigid Designation as well has to be re-worked to fit the semantics. Instead ofsaying that every constant picks out the same object at every world, we will saythat it picks out the same object at every world at which that object is in theUD.
To sum up the rules, each constant is assigned at least one object in the UD ofat least one world. Moreover, the same assignment is made to that constantsat all worlds at which that object is in the UD.
In the example above, suppose that ‘a’ is assigned 1 at w1. Then it mustbe assigned 1 at w2 as well. Now suppose that ‘b’ is assigned 2 at w2. Thenbecause it is a rigid designator, it would be assigned 2 at w1 as well if 2 werein the UD of wi, but it is not. So v(b, w1) is undefined.
The semantics for QPL does not make any changes in the basic way variable-assignments are made. They are all based on D as before. But recall oursimplifying convention that vd(x,w)=d(x). Since d(x) may not be in Dw, weshall require that vd(x,w) be undefined in that case, just as in the case wherea constant does not designate a member of Dw. So we shall say that if d(x) ∈w, then vd(x,w)=d(x). Otherwise, vd(x,w) is undefined.
This means that the notion of a variant has to be modified. For the semanticsfor Q1, we could say that vd[ui/x](x, w)=ui. But since a variant is itself avariable-assignment, in some cases, what d[ui/x] assigns to ‘x’ will not be inDw. So we shall need another limited notion.
Limited Variant AssignmentIf ui ∈ Dw, then vd[ui/x](x, w1)=ui; otherwise, vd[ui/x](x, w1) is undefined
Given these modifications in the semantics for terms, changes will have tobe made in the truth-definitions for formulas. We begin with atomic formulas.Suppose that ‘a’ does not designate anything in the UD at a world w and wewish to know whether the sentence ‘Fa’ is true at that world. Since v(a, w) isundefined, it is natural to follow Hughes and Cresswell and stipulate that v(Fa,w) is undefined.14
Identity formulas are special cases of atomic formulas, and they will behandled the same way. If at least one of its terms is non-designating at aworld, the formula will have no truth-value. The truth-functional connectivesare interpreted similarly: if the truth-value one of the component formulas is
14In their later work, A New Introduction to Modal Logic, Hughes and Cresswell use adifferent approach, but in any event, they do not treat constants.
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undefined, then the truth-value of the formula of which it is a component is alsoundefined. The modal sentences require that their component sentences have atruth-value at all accessible worlds in order to have a truth-value themselves.
Finally, we can give truth-definitions for quantified formulas. Here we mustdistinguish between those variants that assign a member of the UD at the worldto a variable and those that do not. A universally quantified formula (∀x)α(x)will get the value T at a world w given a variable assignment d if α(x) is trueon all the x-variants of d that designate a member of the UD at w. It gets thevalue F at w if α(x) is false on some x-variant that designates a member of w’sUD. Otherwise, the value is undefined. Similar remarks hold for the existentialquantifier.
Before turning to examples, we shall state these truth-definitions formally.
vd(α&β, w)=T is defined iff both vd(α, w) and vd(β, w) are defined
If vd(α&β, w) is defined, thenvd(α&β, w)=T if vd(α, w)=T and vd(β, w)=Tvd(α&β, w)=F if vd(α, w)=F or vd(β, w)=F
Disjunctive Formulas
vd(α ∨ β, w) is defined iff both vd(α, w) and vd(β, w) are defined
If vd(α ∨ β, w)=T is defined, thenvd(α ∨ β, w)=T if vd(α, w)=T or vd(β, w)=Tvd(α ∨ β, w)=F if vd(α, w)=F and vd(β, w)=F
Conditional Formulas
vd(α ⊃ β, w) is defined iff both vd(α, w) and vd(β, w) are defined
If vd(α ⊃ β, w)=T is defined, thenvd(α ⊃ β, w)=T if vd(α, w)=F or vd(β, w)=Tvd(α ⊃ β, w)=F if vd(α, w)=T and vd(β, w)=F
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Biconditional Formulas
vd(α ≡ β, w) is defined iff both vd(α, w) and vd(β, w) are defined
If vd(α ≡ β, w) is defined, thenvd(α ≡ β, w)=T if vd(α, w)=vd(β, w)vd(α ≡ β, w)=F if vd(α, w) 6= vd(β, w)
Necessity Formulas
vd(�α, w) is defined iff for all wi such that Rwwi, vd(α, wi) is defined
If vd(�α, w) is defined, thenvd(�α, w)=T if for all wi such that Rwwi, vd(α, wi)=Tvd(�α, w)=F if for some wi such that Rwwi, vd(α, wi)=F
Possibility Formulas
vd(♦α, w) is defined iff for all wi such that Rwwi, vd(α, wi) is defined
If vd(♦α, w) is defined, thenvd(♦α, w)=T if for some wi such that Rwwi, vd(α, wi)=Tvd(♦α, w)=F if for all wi such that Rwwi, vd(α, wi)=F
Universal Formulas
vd((∀x)α(x), w)=T iff for all x-variants d[ui/x] of d such that ui ∈ Dw,vd[ui/x](α(x), w) = T
vd((∀x)α(x), w)=F if for some x-variant d[ui/x] of d such that ui ∈ Dw,vd[ui/x](α(x), w) = F; otherwise, vd((∀)xα(x), w) is undefined
Existential Formulas
vd((∃x)α(x), w)=T iff for some x-variant d[ui/x] of d such that ui ∈ Dw,vd[ui/x](α(x), w) = T
vd((∃x)α(x), w)=F if for all x-variants d[ui/x] of d such that ui ∈ Dw,vd[ui/x](α(x), w) = F; otherwise, vd((∃x)α(x), w) is undefined
As with Q1, we will say that a sentence is true if it is true on all variableassignments. This is not a biconditional, though, since a sentence may fail tobe true on all variants if its truth-value is undefined on some variant. So we willsay that a sentence is false just in case it is false on all variants. Otherwise, thesentence lacks a truth-value.
The fact that sentences can lack truth-values on the QPL semantics af-fects the definition of semantic entailment. Recall that a set of sentences Γsemantically entails a sentence α in a frame just in case at any world in anyinterpretation where all the members of Γ are true, α is true. We shall modifythis definition by adding a condition: that α have a truth-value.
Semantical Entailment in a Frame Fr (QPL)
{γ1, . . . , γn} �Fr α iff for any I based on Fr and any w in W in Fr,if v(γ1, w)=T and · · · , and v(γn, w)=T and v(α, w) is defined, then
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v(α, w)=T.
Finally, a sentence is valid just in case on every interpretation and at any worldin that interpretation, the sentence is true.
Validity in a Frame Fr (QPL)
�Fr α iff for any I based on Fr and any w in W in Fr,if v(α, w) is defined, then v(α, w)=T.
As with Q1, we will here give alternative truth-definitions for quantifiedformulas. Note how closely these parallel the alternative truth-definitions forQ1 : the only modification is that the UD it mentions is world-bound.
Alternative Truth-Definition for ∀ (QPL)vd((∀x)α(x), w)=T iff (∀ui)(ui ∈ Dw ⊃ vd[ui/x](α(x), w)=T).
Alternative Truth-Definition for ∃ (QPL)vd((∃x)α(x), w)=T iff (∃ui)(ui ∈ Dw& vd[ui/x](α(x), w)=T).
4.1.2 Quantifiers and Modalities in QPL
We will now provide a counter-example to an instance of the Barcan conse-quence. An interpretation will be given in which ‘(∀x)�Fx’ is true but ‘�(∀x)Fx’is false at a single world. On a K -interpretation I, let D={1,2} and W={w,w1}. Suppose further that Rww1 and no other accessibility relation holds. Wecan satisfy the Nested UD constraint by allowing that Dw={1} and Dw1={1,2}.Finally, we give the extension of ‘F’ as 1 at both w and w1: v(F, w)={〈1〉} andv(F, w1)={〈1〉}.
First we will show that on this interpretation, vd((∀x)�Fx, w)=T for any d.To evaluate the sentence, we need consider only those x-variants which assignto ‘x’ a member of the UD at w. Since there is only one such member, 1, allx-variants are the same, assigning 1 to ‘x’. So consider any x-variant d[1/x] ofd. We can say that vd[1/x](Fx, w1)=T, since 1 is in the extension of ‘F’ at w1.Now w1 is the only world accessible to w, so vd[1/x](�Fx, w)=T. And since thisholds for any x-variant of d that assigns a member of Dw to ‘x’, vd((∀x)�Fx,w)=T.
To determine vd(�(∀x)Fx,w), we first look for the value of ‘(∀x)Fx’ at w1.To determine this value, we must not only consider the x-variant d[1/x], whichsatisfies ‘Fx’, but also the x-variant d[2/x], since 2 is in the UD at w1. Thisvariant does not satisfy ‘Fx’ because 2 is not in the extension of ‘F’ at w1. So,vd((∀x)Fx, w1)=F. Then vd(�(∀x)Fx, w)=F, which was to be proved.
It worth noting here that vd[2/x](Fx, w) is undefined. The reason for thisis that vd[2/x](x, w) /∈ Dw: the member of the UD assigned to ‘x’ is not inthe UD at w. So by the truth-definition for atomic sentences, ‘Fx’ can receiveneither the value T nor the value F at w, and so it has no value there given thevariable-assignment d[2/x].
On the positive side, it can be shown that an instance of the converse Barcanconsequence holds. So assume that vd(�(∀x)Fx, w)=T for an arbitrary d and
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w. From the truth-definition for necessity sentences, it follows that vd((∀x)Fx,wi)=T for all accessible worlds wi. So, for all x-variants d[ui/x], such that ui∈ Dwi , vd[ui/x](Fx, wi)=T.
Since wi is an arbitrary world accessible to w, we can say that for all x-variants d[ui/x], such that ui ∈ Dwi , vd[ui/x](�Fx, w)=T. By the conditionthat UDs be nested, if ui ∈ Dw, then ui ∈ Dwi . So, for all x-variants d[ui/x],such that ui ∈ Dw, vd[ui/x](�Fx, w)=T. Hence, vd((∀x)�Fx, w)=T. This iswhat was to be proved. The nesting condition is what enabled the satisfactionof the clause in the truth-definition for the universal quantifier that requiresthat the assignment to ‘x’ be in the UD at w. That is what makes the proofwork.
We shall here modify the original formal proof of this instance of the converseBarcan consequence to accommodate the conditions in the truth-definitions.
Although the above proof establishes only that the converse Barcan consequenceholds for a single instance, it can be shown that all instances are semanticalentailments in QPL.
4.1.3 Identity in QPL
When we turn to identity, we get a very peculiar result. First, we still have itthat:
�QPL−x ai = aj ⊃ �ai = aj.
�QPL−x (∀x)(∀y)(x = y ⊃ �x = y).
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The reason these hold is because there is no way to make them false at a world,given Rigid Designation and the fact that variable assignments are indifferent topossible worlds. However, it is easy to construct an interpretation where, say thetruth-value of ‘a=b ⊃ �a=b’ is undefined. All that is required is that ‘a’ or ‘b’not designate at a world. However, if they both designate the same individualat a world, then, by the requirement of Nested UDs and Rigid Designation,they designate that individual at all accessible worlds, and so the truth-value of‘�a=b’ will be defined. And in fact it is true at the original world.
The same remarks hold for the schema �(∃x)x=a. There are interpreta-tions on which a given constant a does not designate at a world, in which case(∃x)x=a has no truth-value. But if such a world is accessible, then �(∃x)x=aalso lacks a truth-value, which makes it irrelevant to whether a substitution in-stance of the schema is valid. Where there is a truth-value, (∃x)x=a is alwaystrue, and by closure, �(∃x)x=a is always true as well, and so it is valid.
Similar considerations apply to the related schema (∃x)�x=a. The formulax=a will lack a truth-value at those worlds at which a does not designate any-thing, in which case (∃x)�x=a will lack a truth-value as well. On the otherhand, if x=a has a truth-value at a world, it is because a designates a mem-ber of the UD at that world. By the nesting requirement for UDs and RigidDesignation, there is a member of the UD of each accessible world which a des-ignates, namely, the same member designated at the world where a=x has atruth-value. Now there will be at least one variable-assignment d which assignsto x the same object that a designates. Since d assigns x the same object at allthe worlds where there is such an object in the UD, it assigns it the same objectas a designates at all the accessible worlds. So given d and the truth-definitionfor identity, x=a is true at all the accessible worlds, and �x=a is true at theoriginal world. Since there is at least one such assignment, (∃x)�x=a is true atthe original world.
Another version of the peculiarity of the definition of validity involves infer-ence. We could have the situation where a universally quantified sentence, forexample, ‘(∀x)Fx’, is true at a world, while a substitution instance ‘Fa’ lacks atruth-value at that world, because ‘a’ does not designate anything at that world.Must we say, then, that the universal instantiation (∀ Elimination) is invalid?The answer is “no”, because we have defined the entailment relation in termsonly of sentences whose truth-values are defined. The only way an entailmentfails to hold is if all the sentences in a set are true (at a world) and the targetsentence is false (at that world). Entailment of α by Γ is a negative notion, atbottom: all it states is that it is not possible for all the members of Γ to betrue, while α is false.
If one wishes for a more satisfying notion of entailment, one might turn tofree logic. The inference from ‘(∀x)Fx’ to ‘Fa’ is invalid in free logic, though onecan obtain a valid inference by adding a premise to the effect that the constant‘a’ designates an object. Without free logic, we could supply the needed premise,‘(∃x)x = a’. But this is valid in QPL systems and so it does not add anythingto the inference that was not already available.
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4.1.4 The Derivational Systems QPL
To form the derivational systems QPL, the Q1 quantifier rules are modified.In the Q1 systems, there are no restrictions on the introduction or elimina-tion of quantifiers except that they may not be used across scope lines. Thiswas justified in part because of the unitary UD in the Q1 semantics. In theworld-relative semantics for QPL systems, the requirement of nested domainspreserve some of the characteristics of a unitary UD. This has the effect thatQ1 derivational rules of ∀ Elimination and ∃ Introduction will continue to besound (though according to a modified definition of soundness). But the rulesof ∀ Introduction and ∃ Elimination will have to be restricted even further.
At first blush, it appears that the rules of ∀ Elimination and ∃ Introductionare not sound even in non-modal contexts. Suppose there is a world in aninterpretation where ‘a’ does not designate a member of its UD. And suppose‘(∀x)Fx’ is true at that world. Then ‘Fa’ lacks a truth-value. Thus, when ∀Elimination is used to get ‘Fa’, a move has been made from a true sentence (ata world) to a sentence that has no truth-value (at that world). This violatesany notion of soundness that holds that a sound inference rule may sanctiononly moves from true sentences to true sentences.
In the MSL and Q1 semantical systems, this is the case. But in thosesystems, there is no difference between saying that a rule moves from truesentences only to true sentences and saying that a rule moves from true sentencesonly to sentences that are not false. For every sentence in those systems is eithertrue or false. But in the QPL semantical systems, this is not the case: a sentencecan fail to be true without being false. The definition of semantical entailmentfor QPL exploits this distinction by requiring only that a set of true sentencessemantically entail sentences that are not false.
So if we understand soundness of derivational rules in the standard sensethat every use of the rule is a semantical entailment, the rule of ∀ Elimination issound in non-modal contexts. If ‘a’ does designate at the world, then if ‘(∀x)Fx’is true at that world, so is ‘Fa’. But if ‘a’ does not designate at that world, ‘Fa’has no truth-value.
Nothing changes in modal contexts. If ‘(∀x)�Fx’ is true at a world w, the‘�Fa’ is either true or undefined at the world. Suppose ‘a’ does not designatethere. Then by the nesting condition, ‘a’ does not designate at any accessibleworld either. And by the truth-definition for the � operator, ‘�Fa’ lacks atruth-value at w. If, on the other hand, ‘a’ does designate an object in theUD of w, then it designates that object in the UDs of all accessible worlds,again by the nesting condtion. The sentence ‘�Fa’ is also true at w, given that‘(∀x)�Fx’ is true at w. This is because ‘�Fx’ is satisfied by all x-variants of agiven variable-assignment d. Hence whatever value ‘a’ has, one of the x-variantswill give ‘x’ that same assignment.
Exercise. Justify the soundness in QPL of the inference from ‘�Fa’ to ‘(∃x)�Fx’.
With respect to ∀ Introduction, however, the Q1 rule is not sound, even bythe relaxed standards of QPL. Take, for example, a sentence ‘Fa’ that occurs
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on a line in a derivation and the move to ‘(∀x)Fx’. With the previous systems,we assumed that if a sentence occurs on a line in a derivation, it is taken to betrue on that line. But as we have seen, even ∀ Elimination allows us to writedown a sentence that is neither true nor false. So we cannot anymore take itthat a sentence written down on a line of a derivation is true.
This is a factor in the derivation of instances of the Barcan consequence. Webegin with ‘(∀x)�Fx’ and use ∀ Elimination to get ‘�Fa’. This sentence maybe true or it may lack a truth-value. It lacks a truth-value (at a world on aninterpretation) just in case ‘a’ does not designate at that world. Now supposethat Strict Reiteration is used, resulting in the occurrence of ‘Fa’ to the rightof a restricted scope line. Then we might wish to use ∀ Introduction.
1 (∀x)�Fx Assumption2 �Fa 1 ∀ E3 Fa 2 SR4 (∀x)Fx 3 ∀ I
But this move might lead to a falsehood. The fact that every object in theUD at a world w is in the extension of ‘F’ at all accessible worlds does not implythat every object in every accessible world is in the extension of ‘F’. It may bethat at another world, there are additional objects, some of which are not inthe extension of ‘F’. So we could go from what is taken to be a truth (at step1) to what is a falsehood.
The problem here clearly lies in the fact that ‘a’ is not suitable as an arbitraryconstant for generalization by virtue of step 3. When Strict Reiteration is used,‘a’ occurs in a new environment, so to speak, where it loses its arbitrariness. Itis only arbitrary when its occurrence is confined to part of the derivation whereit is introduced—in this case, to the left of the restricted scope line. So weshall set down as a new restriction on ∀ Introduction that it may not be usedon a constant that appears in a “live sentence” to the left of a restricted scopeline. A live sentence is one which does not lie in the scope of any undischargedassumption or unended modal scope line.
Universal Introduction (QPL)
α(a/x)
∴ (∀x)α(x)
Provided:(i) a does not occur in an undischarged assumption.(ii) a does not occur in (∀x)α(x).(iii) If α(a/x) occurs within a restricted scope line, a does not occur ina live sentence in the scope of a modal operator outside that line.
Similar remarks apply to ∃ Elimination. When we instantiate ‘(∃x)Fx’ to’Fa’, we want the occurrence of ‘a’ to be arbitrary. If ‘a’ occurs in a sentenceoutside a restricted scope line where ‘(∃x)Fx’ occurs, then it is already, as it were,
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“taken”, just as if it had occurred in an assumption. Because of Nested UDsand Rigid Designation, ‘a’ designates an individual in the UD in the accessibleworld. To illustrate, we shall give the long derivation of a Barcan consequencethat was earlier shortened using QN.
The key step here is 9, where ‘a’ was used to introduced as an “arbitrary”constant. It qualifies according to the Q1 rules because ‘a’ does not occur inany undischarged material assumptions. But step 10 introduces informationfrom outside the restricted scope line, involving the constant ‘a’. The fact thatan arbitrary individual that ‘a’ designates at a world is not in the extension of‘F’ at a world does not rule out that an arbitrary individual designated by ‘a’at an accessible world is in the extension of ‘F’ at that world.
So for ∃ Elimination in QPL, there must be a further restriction on the useof the constant that is supposed to be arbitrary. The restriction is the same aswith ∀ Introduction. The constant must not occur in any live sentence to theleft of a restricted scope line within which the rule is used.
Existential Elimination (QPL)
(∃x)α(x)
α(a/x)
β∴ β
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Provided:(i) a does not occur in an undischarged assumption.(ii) a does not occur in (∃x)α(x).(iii) a does not occur in β.(iv) If (∃x)α(x) occurs within a restricted scope line, a does not occur inin a live sentence in the scope of a modal operator.
The two rules of � = Introduction are sound in the semantics for QPLsystems. If a=b is true at a world w, then a=b is true at all accessible worlds,since by the nesting requirement a and b both designate at all accessible worlds,and by Rigid Designation, they designate the same thing at all of them. If thevalue of a=b is undefined at a world, i.e., if neither a nor b designate a memberof the UD at that world, this fact is irrelevant to the soundness of the rule.
If ∼a=b is true at a world, then a=b has a truth-value at that world, and thevalue is false. So a and b both designate, and they designate different objectsin the UD of that world. Given nesting, they designate in all accessible worlds,and by Rigid Designation, they designate different things at all such worlds. So∼a=b is true a all accessible worlds. Again, if ∼a=b has no truth-value at aworld, that fact is irrelevant to soundness.
So the derivational systems QPL-x (where x is weaker than KB) are like thederivational systems Q1-x, except for the restrictions on the rules of UniversalIntroduction and Existential Elimination.
Derivational Systems QPL-xModal operator rules as with modal system x
No PL rules may be used across restricted scope lines∃ Introduction, ∀ Elimination as for PL, with no restrictions in modal contexts
∀ Introduction, ∃ Elimination specially for QPL= Elimination may be used only on atomic formulas
� = Introduction
Derivational systems QPL-x are weaker than the corresponding Q1-x sys-tems.15 In particular, the Barcan consequence does not hold in the QPL-xderivational systems. They are contained in the Q1-x systems, however.16 rulesare Q1 rules. An inspection of the derivational proofs of the converse Barcanconsequence shows that every move that is made given the rules for Q1 canbe made with the QPL derivational rules. Finally, in all QPL systems at leastas strong as QPL-KB, the Barcan consequence holds. Inspection of the earlierderivation for Q1-KB shows that no QPL restrictions are violated.
4.1.5 Remarks on QPL Systems
The systems we have been developing in this section are more flexible than arethe Q1 systems. They allow for possible individuals that do not exist at a given
15Completeness is not being assumed here.16For the derivational system, it need only be noted that that all QPL
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world. But the price that must be paid for this luxury is rather steep. It is notthat truth-value gaps are in themselves implausible. Philosophers have arguedthat there are declarative sentences without truth-values. What is implausibleis re-definition of semantic entailment as that which does not lead from truth tofalsehood. This yields the very curious result that a true sentence can entail asentence that has no truth-value whatsoever. Generally, entailment is thoughtof as a guarantee of truth rather than just a prophylactic against falsehood.An alternative to this anemic notion of entailment is to change the underlyingpredicate logic so that all sentences have truth-values and entailment is definedin the normal way. We will examine a system of “free” Modal Predicate Logicin the next section.
Another alternative is to relativize the use of constants to worlds at whichthey designate. Such constants would then be “localized”. This has the ratherdrastic consequence that what functions syntactically as a term is not a term ata world where its designee does not exist. If we want modal logic to symbolizeEnglish, for example, ‘Pegasus’ would not function as a term, and we could notmeaningfully assert “Pegasus has wings” using the name ‘Pegasus’.17 Garsonpoints out another difficulty with this approach.
Furthermore, we have been assuming that terms are rigid, so termsmust have the same referent in all worlds. So the demand that termsbe local entails that any term must have an extension which exists inall the worlds. In fact, the only objects at which the domains mightvary are ones which are never named in any world. This undercutsthe whole point of introducing world-relative domains, namely toaccommodate terms that refer to things that may not exist in otherpossible worlds. (“Quantification in Modal Logic”, p. 259)
There are other approaches that attempt to give semantics for sentencescontaining terms that may not designate, but we will not canvass them now.See Garson’s “Quantification in Modal Logic” for more details.
5 Systems Q1R
A family of systems weaker than QPL systems, but contained in them, is gen-erated by dropping the Nested UDs restriction from the QPL semantics. Suchsystems will be called Q1R systems because they are like the Q1 systems onlywith worlds completely relativized to UDs.18 The QPL systems are only partialrelativizations.
Without the requirement that UDs be ordered by the accessibility rela-tion, the converse Barcan consequence is no longer a semantical entailment.A counter-example can be created by letting Rww1 exhaust the accessibility
17One might view it as a definite description, but Garson points out difficulties with thisapproach.
18Garson uses this name for a system to be discussed in the next section, which is based onfree logic.
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relation and letting Dw=1 and Dw1=2. Finally, suppose that 1 /∈ v(F,w) and2 ∈ v(F,w1).
Suppose that v(�(∀x)Fx,w=T. Then at w1, ’(∀x)Fx’ is true at w1. So 2satisfies ‘Fx’ at w1. Then since w1 is the only world accessible to w, for everyvariable-assignment that assigns a member of the UD of wi to ‘x’, ‘�Fx’ is trueat w. However, a variable-assignment that assigns 1 to ‘x’ at w does not satisfy‘�Fx’ at w, and in that case, ‘(∀x)�Fx’ is false at w.
The derivational rules for Q1R systems would have to be weakened to reflectthe weakening of the semantics. Consider the derivation of an instance of theconverse Barcan consequence.
1 �(∀x)Fx Assumption2 (∀x)Fx 1 SR3 Fa 2 ∀ E4 �Fa 2-3 � I5 (∀x)�Fx 4 ∀ I
As was noted in the discussion of Q1 systems, step 5 is the only step whichcombines modality and quantification. As our semantical reasoning showed, itis step 5 that must be blocked in Q1R derivational systems. Even if ‘�Fa’ holdsfor ‘a’ which is arbitrary in the context of an accessible world, ‘a’ might fail todesignate an object at the home world. So we will add a fourth restriction tothe use of ∀ Introduction, that α(a/x) may not be the result of the use a rulethat carries it across a restricted scope line.
Universal Introduction (Q1R)
α(a/x)
∴ (∀x)α(x)
Provided:(i) a does not occur in an undischarged assumption.(ii) a does not occur in (∀x)α(x).(iii) If α(a/x) occurs within a restricted scope line, a does not occur ina live sentence in the scope of a modal operator outside that line.(iv) a does not cross a restricted scope line in the derivation of α(a/x).
Note that this rule does not prevent entirely the use of ∀ Introduction in thescope of a modal operator. For example, if one begins with ‘(∀x)�Fx’ andderives ‘�Fa’ using ∀ Elimination, one could then use ∀ Introduction to infer‘(∀y)�Fy’.
The rule of ∃ Elimination will have to be modified similarly, as it is subjectto the same counter-example as is ∀ Introduction. This can be seen from lookingat the ♦/∃ form of the converse Barcan consequence. Suppose only 1 is in theUD of w and only 2 is in the UD of the only accessible world w1, and that 1does not satisfy ‘Fx’ at w, while 2 satisfies ‘Fx’ at w1. In that case, And thereis a variable assignment to ‘x’ which makes ‘♦Fx’ true at w, since 2 satisfies‘Fx’ at w1. Also, ‘(∃x)Fx’ is true at w1, in which case ‘♦(∃x)Fx’ is true at w.
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However, there is no variable assignment to ‘x’ which assigns a member of theUD at w and makes ‘♦Fx’ true at w, so ‘(∃x)♦Fx’ is false at w.
Let us see how the derivation of this instance of the converse Barcan formulaworks with the Q1 rules.
Here, the problematic step is 2. Suppose it is given that something at a world ispossibly F. Suppose further that ‘a’ designates an object at that world. Adapt-ing the counter-example, we can say that ‘a’ does not designate an object atan accessible world, which makes step 4 illegitimate. To prevent this from oc-curring, we will add yet another restriction to the list for the QPL derivationalsystem to disallow the ending of the existential scope line after line 5. Although‘♦(∃x)Fx’ does not contain ‘a’, it is the result of a kind of illegitimate assump-tion about ‘a’, that it designates at the accessible world. (This restriction willalso block the inference from ‘(∃x)�Fx’ to ‘�(∃x)Fx’).
Existential Elimination (Q1R)
(∃x)α(x)
α(a/x)
β∴ β
Provided:(i) a does not occur in an undischarged assumption.(ii) a does not occur in (∃x)α(x).(iii) a does not occur in β.(iv) If (∃x)α(x) occurs within a restricted scope line, a does not occur inin a live sentence in the scope of a modal operator.(v) a does not cross any restricted scope lines used in the derivation of β
The Q1R systems differ from the QPL systems in that systems stronger thatB can be superimposed on the Q1R semantics without the system collapsinginto a Q1 system. Even if the accessibility relation is symmetrical, there needbe no unitary domain, since there is no nesting requirement. Thus the Q1Rsystems are more flexible than the QPL systems. However, because it mustallow truth-value gaps, the Q1R semantics still requires the weak definition ofsemantic entailment as the impossibility of true sentences having false sentencesas their consequences.
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6 Free Modal Predicate Logic
6.1 Free Predicate Logic
In the semantics for Predicate Logic, every constant designates an object ina unitary universe of discourse. This semantical rule was carried over intothe semantics for the Q1 systems but was abandoned in the QPL and Q1Rsystems, where the UD is relativized to worlds. In non-modal Predicate Logic,there are no possible worlds that can serve as locations where a constant doesnot designate. So some other device is needed to get the same effect as we hadin the QPL and Q1R semantics.
Lablanc and Thomason have proposed a semantics for non-modal PredicateLogic which distinguishes two UDs, one “inner” and one “outer”, which shareno members in common.19 The inner UD contains the objects which are takento exist, while the outer UD contains those objects that are taken not to exist.We will assume here that the inner UD is not empty, though the outer maybe.20 If D is the inner UD in an interpretation, we will call D′ the outer UD.
The extensions of predicates and the designations of constants are taken fromthe union D ∪ D′ of the two UDs. That is, they are taken from the combinationof all the objects, both existent and non-existent. This has the consequence thata sentence ‘Fa’ can be true at a world on an interpretation when ‘a’ designatesan object in the outer UD and that object is in the extension of ‘F’, even thoughit does not exist at that world.
The assignments to the variables are taken from the comprehesive set D∪ D′. But the evaluation of open sentences is based only on the objects inthe inner UD. As a result, the sentence ‘Fa’ might be true, while ‘(∃x)Fx’ isfalse on an interpretation on which v(a)∈D′, i.e., ‘a’ designates an object in theouter domain. Since such an object is not in the inner UD, although there isa variable assignment to ‘x’ such that v(a)=d(x), that assignment is irrelevantto the evaluation of a sentence containing ‘x’ free. In the limiting case wherenothing in the UD is in the extension of ‘F’, ‘(∃x)Fx’ is false. In ordinaryEnglish, we might assert “Pegasus is a flying horse” without being willing toassert, “There are flying horses”.
We can, however, get a limited version of existential generalization by intro-ducing a new “existence” predicate E 1 into the syntax of PL.21 The extensionof this predicate, on an interpretation, is the set of (one-tuples of) the membersof the inner UD: v(E )=D1. Now we can say that if ‘Fa’ and Ea are true on aninterpretation, then ‘a’ designates a member of D. So if v(a)=ui, then thereis some variable-assignment d[ui/x] which satisfies ‘Fx’, since v(a)=d[ui/x](x).Since this is the case, ‘(∃x)Fx’ is true on variable assignment d on that inter-pretation. If I assert, “Carrie Webb is a dominant female professional golfer”,
19Hughes Lablanc and Richmond H. Thomason, “Completeness Theorems for SomePresupposition-Free Logics”, Fundamenta Math. 62, 125-164. See also Ermanno Bencivenga,“Free Logics”, in Gabbay and Guenther (eds), Handbook of Philosophical Logic, Vol. III, pp.373-426.
20Bencivenga notes that the inner UD may be empty on the Lablanc-Thomason semantics.21We shall in general drop the place-index ‘1’.
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and “Carrie Webb exists”, then I am entitled to assert, “There is a dominantfemale professional golfer”.
Existence Predicate
‘E 1’ is a predicate letter of Free Modal Logic
The existence predicate is needed to generate a sound version of anotherquantifier rule, ∀ Elimination. To say that everything is F, given the semantics,is only to say that every existing thing is F, so ‘Fa’ may not be true if ‘a’ doesnot designate an existing thing.
No changes need to be made to the rules of ∀ Introduction and ∃ Elimination.For the former rule, if we can get a result that holds of any arbitrary individualthat ‘a’ might designate in the combined UDs, then that result holds for anythingthat it might designate in the inner UD. Similar remarks apply for the existentialrule. If we assume that a condition holds when an arbitrary constant is used todesignate an individual, that constant is assumed to stand for any individual inthe inner or outer UD.
Identity in non-modal Predicate Logic is treated semantically by the simplerule that an identity sentence is true on an interpretation just in case its twoterms designate the same individual. Because a constant can fail to designate anobject in the inner UD, we may have a formula ‘x=a’, where on an interpretationI, ‘a’ designates a member of the outer UD. In that case, for any variableassignment d assigning a member of the inner UD to ‘x’, d(x) is not the sameas v(a), so the identity formula is false. Then ‘(∃x)x=a’ is false. If ‘(∃x)x=a’is false at a world, then so is ‘Ea’, since ‘a’ fails to designate an object in theinner domain. Conversely, if ‘Ea’ is true, it does designate such an object at theworld where it is evaluated. In that case, there will be a variable-assignmentwhich assigns ‘x’ to the same member of the UD as v assigns ‘a’. Then ‘x=a’ istrue on that assignment, and so (∃x)x=a is true at that world.
In general, a formula Ea is true at a world just in case (∃x)x=a is true atthat world. The latter form is generally considered in Predicate Logic as statingthat a exists. The existence predicate is introduced specially because of the roleit plays in the derivational system for FPL and because of the role it might playin more sophisticated systems of free logic.
Below we give the essentials of the semantics for Free Predicate Logic. Notlisted are semantical rules that carry over directly from the semantics for Pred-icate Logic, such as the truth-definitions for truth-functional compounds. Itshould be noted that every term has a value and every formula has a truth-value, unlike in the QPL semantics.
vd((∀x)α(x))=T iff (∀ui) such that ui ∈ D, vd[ui/x](α(x))=T
vd((∃x)α(x))=T iff (∃ui) such that ui ∈ D and vd[ui/x](α(x))=T
v(E ) ∈ D1
The derivational system for Free Predicate Logic is obtained by adding anextra clause to two of the four quantifier rules, as discussed above. In each case,the constant a involved must occur in the sentence Ea prior to the execution ofthe rule.
Universal Elimination (FPL)
(∀x)α(x)
Ea
∴ α(a/x)
Existential Introduction (FPL)
α(a/x)
Ea
∴ (∃x)α(x)
Of particular note is that the derivation of ‘(∃x)x=a’ from ‘a=a’ is blockedunless ‘Ea’ occurs as an earlier step. This is because, from a semantical pointof view, even though ‘a=a’ can be true because ‘a’ designates a member ofthe outer UD, ‘x=a’ would then be false no matter what variable-assignment ismade to ‘x’, since such an assignment can only designate a member of the innerUD, which does not share any members with the outer UD.
6.2 Systems FQ1R
The melding of the semantics for Free Predicate Logic with semantics for modallogics will be done in a different way depending on the system in question. Forthe Q1 systems, the adaptation is very straightforward. The Predicate Logicside of the Q1 semantics is exactly like that of non-modal Predicate Logic, witha unitary UD. So all that need be added to a Q1 -interpretation is the set D′
of non-existent objects and a modification in the truth-definitions for quantifiedformulas.
We will focus here on the superimposition of the Q1R semantics onto thesemantics for Free Predicate Logic. The reason is that in a free version of Q1,
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non-existing objects are, so to speak, absolutely non-existing: they do not existat any possible world. The Q1R semantics gives us real motivation for adoptingfree logic, in that it countenences the non-existence of any object at any givenworld. In the process of describing the free Q1R (FQ1R) semantics, a numberof comparisons will be made with the semantics for Q1.
The key to adapting free logic to the Q1R semantics is the relativization ofinner and outer UDs to possible worlds. This can be done in a very naturalway. The inner UD can be identified with the UD at the world, so it would beDw for world w. The outer UD at a world is simply what remains in D (theset specified in the frame) when the objects in the inner UD are removed. Sowe can say that D′w = D - Dw. So in the simple case used above, we can haveD={1,2}, Dw={1}, and Dw1={2}. Then D′w={2} and D′w1={1}.22
All the other elements of the semantics for free logic are smoothly adapted tothe semantics for modal logic. All value-assignments made by v are relativized toworlds. And many of the restrictions laid down in the QPL and Q1R semanticscan be cast aside.
There is always a value for a constant because such values are taken fromthe set D specified by the frame: v(a,w) ∈ D, the rule we called Designation forConstants in the Q1 semantics. This value can be defined even if the designatedobject is in the outer domain because it is the essence of free logic that con-stants are not assumed to designate objects that exist. The other original Q1semantical rules Non-Empty Designation and Rigid Designation do not requirethe limitations made in the QPL and Q1R semantics, either.
In all the systems examined so far variable assignments assign to each vari-able x a member of D: d(x)∈D. For the Q1 systems, we made the simplifyingassumption that the value of a variable at a world given a variable-assignment djust is the member of D assigned to the variable by d: vd(x,w)=d(x). This as-sumption had to be dropped in the relative-UD systems, so that if d(x) assignsan object not in the UD of a world, vd(x,w) is undefined.
With free logic, the simplifying assumption is restored, and valuations basedon variable-assigments are also always defined, as in Q1. Sometimes the assign-ment will be made by d to an object in the inner UD of a world, sometimes toa member of the outer UD, but in either case there is a value for the variableat a world given d. The restoration of this simplifying assumption also meansthat the semantical rule Variant Assignment is once again unlimited.
Predicates are assigned values as with all the systems. The extension of ann-place predicate at a world is defined on D, so it may be defined on objects thatare only in the outer UD. Since the value assignments made by v to terms arealso exactly as in Q1, we can say that the truth-definitions for atomic formlasand atomic sentences are just as they are for Q1, and there are no more truth-value gaps.
As with the case of the identity constant, a special semantical rule was needed22In practice, D′w need not be defined. It is only added to the semantics for non-modal
Predicate Logic to overcome the limitation on the designation of constants given a unitaryUD. With world-relative UDs, there is no such limitation, as constants may not designate ata world.
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in free Predicate Logic for the predicate constant ‘E ’. This is easily adapted tomodal semantics by requiring that the value of ‘E ’ at world w is just the set ofone-tuples whose members are in the inner UD at w: v(E,w)∈ Dw1.
Identity sentences are also treated in the same way as with Q1, and theirtruth-values are always defined. Formulas which whose main connectives aremodal or non-modal sentential operators are also treated exactly as in Q1, andagain they always have a truth-value.
The one way in which the semantics for FQ1R differs from that of Q1 is inthe treatment of quantified sentences. Here, reflecting the relativization of UDsto possible worlds, the truth-definitions are much more like those in QPL andQ1R. We want to say that a universally quantified formula is true at a worldjust in case the formula without the quantifier is satisfied by all objects in theinner UD of that world, which in the modal semantics is just the UD associatedwith the world. Similarly, an existentially quantified formula is to be true givend just in case the formula without the quantifier is satisfied by some objectin the inner UD of the world. There will never be a case where a universallyquantified formula lacks a truth-value.
Because of the lack of truth-value gaps, we can restore our original definitionsof semantical entailment and validity. This overcomes the weakness of the Q1Rsemantics, but it should be recalled that it does so by making the main inferencerules of non-free Predicate Logic unsound. It must be recognized that it doesso by allowing that sentences can be true of an object at worlds where it doesnot exist, as in the case of “Pegasus is a winged horse” asserted at the actualworld.
Summary of Systems FQ1R-xI=〈W, R, D, v〉
v defined for �, ♦, and non-modal operators as for KModal restrictions on R for system x
vd((∀x)α(x),w)=T iff (∀ui) such that ui ∈ Dw, vd[ui/x](α(x),w)=Tvd((∃x)α(x))=T iff (∃ui) such that ui ∈ Dw and vd[ui/x](α(x),w)=T
v(E,w) ∈ Dw1
These semantical rules block both the Barcan consequence and its converse.These results are established the counter-example given in the last section. Re-call that we let Rww1 exhaust the accessibility relation and let Dw=1 andDw1=2. Finally, we supposed that 1 /∈ v(F,w) and 2 ∈ v(F,w1).
For the converse Barcan consequence, we know that vd[2/x](Fx, w1)=T.Since 2 is the only member of the UD at w1, vd((∀x)Fx, w1)=T. And sincew1 is the only world accessible to w, vd(�(∀x)Fx,w)=T. On the other hand,vd[1/x](Fx,w1)=F. Therefore, vd[1/x]((�Fx, w)=F. Since 1 is in Dw, we haveit that vd((∀x)�Fx,w)=F.
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The a variation on this interpretation can be used to show that the Bar-can consequence fails. The difference is that we shall allow that Finally, wesupposed that 2 /∈ v(F,w) and 1 ∈ v(F,w1). We begin with the fact thatvd[1/x](Fx,w1)=T, since 〈1〉 is in the extension of ‘F’ at w1. So vd[1/x](�Fx,w)=T.And since 1 is the only object in the UD at w, vd((∀x)�Fx,w)=T. But we haveit that vd[2/x](Fx,w1)=F, so vd((∀x)Fx, w1)=F. Hence vd(�(∀x)Fx,w)=F.
The same interpretation can be used to show the invalidity of ‘(∃x)�x=a’,if we let vd(a, w1)=2. Then vd[1/x](x=a,w1)=F. Hence, vd[1/x](�x=a,w)=F.Since d[1/x] is the only x-variant of d that is relevant at w, vd((∃x)�x=a,w)=F.
Exercise. Show that ‘�(∃x)x=a’ is invalid in any FQ1R system.
Because of the retention of Rigid Designation, we still have the validity ofsentences of the form ai=aj ⊃ �ai=aj . The reason for this is that the valueof any constant is taken from D and so the value of a constant at a world isnot affected by its having a more limited UD. This affects only the valuationof sentences with quantifiers. The sentence ‘a=a’ is true at all worlds for thisreason as well.
Exercise. Show the validity in all FQ1R systems of ‘(∀x)(∀y)(x=y ⊃ �x=y)’.
The derivational systems FQ1R-x are obtained in the following way. Beginwith the MSL rules of system x (e.g., system T ). Then add the Q1 derivationalrules for identity. Use the Free Predicate Logic versions of ∀ Elimination and∃ Introduction. Finally, use the non-modal PL rules for ∀ Introduction and ∃Elimination. The QPL restrictions need not be added to block the derivationof the Barcan consequence as in Q1.
1 (∀x)�Fx Assumption2 �Fa 1 ∀ E3 Fa 2 SR4 (∀x)Fx 3 ∀ I5 �(∀x)Fx 2-4 � I
The derivation fails at step 2, which is not permitted in free logic. Moreover, wecannot salvage the derivation by assuming that ‘a’ designates an existing thing.
1 (∀x)�Fx & Ea Assumption2 (∀x)�Fx 1 & E3 Ea 1 & E4 �Fa 2 3 Free ∀ E5 Fa 4 SR6 (∀x)Fx 5 ∀ I
Step 6 is illegitimate because ‘a’ occurs in the assumption made at step 1.The converse of the Barcan consequence, as derived in Q1, is is similarly
blocked.
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1 �(∀x)Fx Assumption2 (∀x)Fx 1 SR3 Fa 2 ∀ E4 �Fa 2-3 � I5 (∀x)�Fx 4 ∀ I
This time, it is step 3 that is not permitted in free logic. And if, again, we addan existence assumption, the generalization at step 5 is not permitted.
The derivational system can be summarized as follows.
Derivational Systems FQ1R-xModal operator rules as with modal system x
No PL rules may be used across restricted scope lines∀ I and ∃ E as for QPL, with no restrictions in modal contexts∀ E and ∃ I as for PL, with no restrictions in modal contexts
= Elimination may be used only on atomic formulas� = Introduction
As the FQ1R systems are rather weak, there are not many interesting deriva-tions to examine. The earlier aborted derivation is instructive for allowing usto derive what is not blocked by the restricted rule of ∀ Introduction.
1 (∀x)�Fx & Ea Assumption2 (∀x)�Fx 1 & E3 Ea 1 & E4 �Fa 2 3 Free ∀ E
This illustrates the use of Free ∀ Elimination in a modal context, which is oneof the few interactions between quantifiers and modal operators still permitted.
The derivation is sound. Suppose vd((∀x)�Fx & Ea,w)=T. Then vd(a, w)is in the UD of w and any ui in that UD satisfies ‘�Fx’. So at every accessibleworld, each of these ui satisfies ‘Fx’, i.e., 〈ui〉 is in the extension of ‘F’ at thatworld. Since ‘a’ is a rigid designator, what it designates at w, one of the uiobjects, is also designated by it at any accessible world. So that object is in theextension of ‘F’ at any accessible world, in which case ‘�Fa’ is true at w.
To illustrate how the Free ∀ Introduction rule functions in a modal context,we might begin a derivation in this way.
1 �Fa & Ea Assumption2 �Fa 1 & E3 Ea 1 & E
But note that Free ∀ Introduction cannot be used as the next step, because ‘a’occurs in an undischarged assumption. It is hard to see how ‘Ea’ could appearon a line in a derivation in any other way, though. It cannot be the result of ∀Elimination, since one would need ‘Ea’ on a line to be able to instantiate to ‘a’in the first place. And there are no theorems of the form ‘Ea’ in free logic.
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7 Conclusion
The most promising family of systems of Modal Predicate Logic is the family ofFQ1R systems. They provide the most flexibility in dealing with non-existentobjects. Their only requirement is that at least one thing must exist in the UDof each of the worlds. The price of flexibility is weakness. Very few quanti-fier/modality interactions are required by the systems. But it should be notedthat strength can be added by placing restrictions on the accessibility relation.The system FQ1R-S5 is a very strong system modally, but that modal strengthdoes not force us into existential requirements.
Even greater flexibility might be sought in a system of Modal PredicateLogic. As we saw above, one might wish that constants not be rigid designators,for example. Such systems have been developed, but they are considerablymore complicated than the systems considered here.23 One might, perhaps fordifferent reasons, which to make identity contingent, so that what is identical isnot necessarily identical. This leads to yet other complications.24
In both cases, what is required is a change in the semantics. Constants andvariables must designate “intensional objects” or “individual concepts” ratherthan the standard objects required in the semantical systems we have beenexamining. A good case can be made that this is needed to produce the mostdesirable systems of modal logic.25 But an examination of these systems willnot be made here.
23See Garson discussion of Q3 in “Quantification in Modal Logic”.24See Hughes and Cresswell, A New Introduction to Modal Logic, pp. 334-335.25See especially Garson on this issue.
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