Fine, Kit. 1970. "Propositional Quantifiers in Modal Logic." Theoria no. 36:336-346.
"In this paper Ï shall present some of the results I have obtained on modal theories which contain quantifiers for propositions. The
paper is in two parts: in the first part I consider theories whose non-quantificational part is S5; in the second part I consider theories whose
non-quantificational part is weaker than or not contained in S5. Unless otherwise stated, each theory has the same language L. This consists of a countable set
V of propositional variables p1, p2,, the operators v (or), ~ (not) and □ (necessarily), the universal quantifier (p), p a propositional variable, and brackets
( and ), The formulas of L are then defined in the usual way." (p. 336)
———. 1971. "The Logics Containing S4.3." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik no.
"In this paper I prove some general results on logics containing S 4.3. In section 2 I prove that they all have the finite model
property. Bull  has already proved thia result; but his proof is algebraic, whereas mine is semantic. In sections 3 and 4, I prove that they are all
finitely axiomatizable. It follows from these results that they are all decidable. Finally, in section 5, I show that the lattice of S 4.3 logics is isomorphic
to one on finite set of finite sequences of natural numbers. Needless to say, these results carry over to the extensions of the intermediate logic LC.
In a paper on logics containing K4, I shall present another semantic proof that S4.3 logics have the finite model property and thereby also
establish some results on compactness." (p. 371)
(1) R. A. Bull, "That All Normal Extensions of S4.3 Have the Finite Model Property", Zeitschrift für Mathematische Logik und
Grundlagen der Mathematik , 12, 1966, pp. 341-344.
———. 1971. "Counting, Choice and Undecidability." Manifold no. 11:17-22.
Abbreviations: Continuum Hypothesis = CH; Axiom of Choice = AC.
"In 1900 Hilbert stated 23 problems which he considered to be of crucial iaportance. The first of these was ’prove Cantor's Continuum
Hypothesis'. Gödel (1939) and Cohen (1963) have shown that the hypothesis can neither be proved nor disproved. Their proofs are expounded in:
P. J. Cohen, Set Theory and the Continuum Hypothesis , Benjamin 1966.
P. J. Cohen, "Independence results in set theory", in Studies ln Logic and the Foundations of Mathematics , North-Holland
1965, pp. 39-54.
K. Gödel, "The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the axiom of set theory", 1939,
4th printing, Princeton 1966·" (p. 71)
"The question now remains: what attitude should the working mathematician take towards CH? It is important to leave AC on one side at
this point because it possesses a degree of self-evidence that CH certainly lacks.
There are, I think, two main attitudes. On the one hand, one could say there is no sense in which CH is true or false and give up looking for
ways of settling the question. Instead, one would develop different set theories, with or without CH, somewhat in analogy to the alternative geometries of the
19th century. On the other hand, one could say that CH Is either true or false and look for new ways of determining which.
Two main ways suggest themselves. The first is to dispense with proof and to accept that hypothesis concerning transfinite cardinals which is
most fruitful in consequences. The second is to search for new self-evident axioms that might settle CH.
These new axioms might be formulated in terms of set-theoretic notions or in terms of a new notion or new notions altogether.
The adoption of non-deductive procedures or the discovery of non-set-theoretic notions would conflict with two common views about
mathematics, viz. that all mathematical knowledge is based upon proof and that all mathematical notions can be given a set-theoretic definition. Although it is
too early to talk of feasibility, it is worth noting that these common views are based upon an analysis of extant mathematics. There seems to be no reason, in
principle, why they should be true." (p. 82)
———. 1972. "In So Many Possible Worlds." Notre Dame Journal of Formal Logic no. 13:516-520.
"Ordinary modal logic deals with the notion of a proposition being true at least one possible world. This makes it natural to consider
the notion of a proposition being true in k possible worlds for any non-negative integer k. Such a notion would stand to Tarski's numerical
quantifiers as ordinary possibility stands to the existential quantifier.
In this paper (1) I present several logics for numerical possibility. First I give the syntax and semantics for a minimal such logic
(sections 1 and 2); then I prove its completeness (sections 3 and 4); and finally I show how to extend this result to other logics (section 5)." (p.
(1) The results of this paper are contained in my doctorate thesis, submitted to the University of Warwick in 1969. I am greatly indebted to
my supervisor, the late Arthur Prior. Without his help and encouragement this paper would never have been written.
———. 1972. "For So Many Individuals." Notre Dame Journal of Formal Logic no. 13:569-572.
"In , Tarski introduces the numerical quantifiers.
Because of their definability, the numerical quantifiers have rarely been considered on their own account. However, in this paper I consider
a predicate logic without identity which is enriched with numerical quantifiers as primitive. In section 1, I present the syntax and semantics for this logic;
and in sections 2 and 3, I establish its completeness." (p. 569)
 Tarski, A., Introduction to Logic, Oxford University Press (1946).
———. 1972. "Logics Containing S4 Without the Finite Model Property." In Conference in Mathematical Logic, London '70 ,
edited by Hodges, Wifrid, 98-102. Berlin: Springer Verlag.
"In , Harrop asked whether there were logics containing the intuitionistic logic IL which lack the finite model property [=fmp].
Jankov gave examples of such logics, but they were not finitely axiomatizable. By the Tarski-McKlnsey translation, Harrop's problem relates to the question of
whether there exist extensions of the modal logic S4 without fmp. Makinson  showed that there are extensions of the modal logic M without fmp, but he could
not extend his results to S4. In this paper, I shall exhibit logics containing both IL and S4 which lack fmp, but are finitely axiomatized and decidable."
"Finally, it should be noted that we can add axioms to the logics described above so as to obtain logics which are decidable, finitely
axiomatized, complete for their intended interpretation, and yet without fmp." (p. 101)
 Harrop, R., On the existence of finite models and decision procedures, Proceedings of the Cambridge Philosophical Society , vol.
54 (1958), 1-16.
 Makinson, D., A Normal Modal Calculus Between T and S4 Without the Finite Model Property, Journal of Symbolic Logic , voi. 34,
Number 1 (1969), 35-38.
———. 1972. "Some Necessary and Sufficient Conditions for Representative Decision on Two Alternatives." Econometrica no.
"A social decision rule is one that produces a social decision for each configuration of individuals' decisions. Such a rule is
representative if it produces a social decision that is the result of repeatedly applying the rule of simple majority decision to decisions obtained
by that rule. We give necessary and sufficient conditions for a social decision rule for two alternatives to be representative." (p.
"The central problem of this paper is to find an alternative characterization of the representative functions. May in  gave an
alternative characterization of the simple majority decision functions, and Murakami in [2 and 3] established that monotonicity and self-duality are necessary
conditions for being a representative or indirect majority decision function. (In fact, Murakami deals with what he calls democratic functions, i.e.,
representative functions which are non-dictatorial; but this latter condition may be added or left out at will.) However, he was not able to establish any
sufficient conditions. In this paper, I establish his conjecture that strong monotonicity with self-duality is a sufficient condition.
I use this result and a new property of not being "zigzag" to prove that monotonicity, self-duality, and not being zigzag are
necessary and sufficient conditions. (2)
Finally, I show that the monotonic, self-dual, and positive functions are exactly those definable in terms of the voting and jury
operators." (p. 1084)
(2) P. C. Fishburn independently solved this problem in his paper "The Theory of Representative Majority Decision,"
Econometrica , 39 (1971), pp. 273-284. However, he uses a completely different method of proof and a slightly different condition in place of
 MAY, K. O.: "A Set of Independent, Necessary and Sufficient Conditions for Simple Majority Decision," Econometrica ,
20 (1952), 680-684.
 MURAKAMI, Y.: "Formal Structure of Majority Decisions," Econometrica , 34 (1966), 709-718.
 MURAKAMI, Y.: Logic and Social Choice . London: Macmillan, and New York: Dover, 1968.
———. 1973. "Conditions for the Existence of Cycles under Majority and Non-Minority Rules." Econometrica no.
"This paper provides type I necessary and sufficient conditions for transitivity and quasi-transitivity under simple majority rule. (2)
For type II conditions, a master list of orderings is acceptable if the social rule generates a "rational" (e.g., transitive) social ordering
whenever the individuals select their preference orderings from the list. A list ordering may be selected any number of times, and, in particular, it may not
be selected at all. For type I conditions, on the other hand, each list ordering must be selected at least once, so that the list and the configuration of
individual orderings must exactly match in the kind of orderings they contain. Thus for type II conditions it is the absence of certain kinds of orderings that
blocks irrational social choice, whereas for type I conditions the presence of certain kinds of orderings may also contribute toward blockage.
Type II conditions have been determined for a wide variety of rules and under several definitions of rationality. Our main interest in this
paper is in type I conditions for simple majority rule with rationality defined in terms of transitivity or quasi-transitivity. However, our method of argument
will yield simple alternative proofs of some standard results on type I conditions and it will also yield the type I and type II conditions for transitivity
under non-minority rule.
Section 1 lays down some relevant definitions. Section 2 proves the "min-midmax" theorem, which is the basis for all that follows.
Sections 3 and 4, respectively, establish the conditions for transitivity and quasi-transitivity under majority rule.
Finally, Section 5 proves the min-mid-max theorem for the non-minority rule and establishes the condition for transitivity under that
rule." (p. 889)
(2) The terminology of type I and II conditions is Pattanaik's . Type II conditions were first proposed by Inada  and type I conditions
by Pattanaik .
 INADA, K.: "On the Simple Majority Decision Rule," Econometrica , 36 (1969), 490-506.
 PATTANAIK, P. K.: "SufficientConditions for the Existence of a Choice Set under Majority Voting," Econometrica , 38
 PATTANAIK, P. K.: Voting and Collective Choice . Cambridge: Cambridge University Press, 1971.
———. 1973. "Surveys on Deontic Logic, Mathematical Logic and the Philosophy of Mathematics." In UNESCO Survey of the Social
———. 1974. "An Ascending Chain of S4 Logics." Theoria no. 40:110-116.
"This paper shows that there exists a continuum of logics containing the modal logic S4. (1) §1 contains preliminary definitions and
results; §2 introduces the key notion of a frame formula; §3 establishes the main result and some consequences; and §4 establishes some further results."
(1) Jankov  has independently, and previously, proved the analogous result for intuitionistic sentential logic. His method of proof is
algebraic, whereas mine is semantic.
 V. A. Jankov, On the Extension of the Intuitionist Propositional Calculus to the Classical Calculus, and the Minimal Calculus to the
Intuitionist Calculus, Journal of Symbolic Logic 38, 1973, pp. 331-332.
———. 1974. "Models for Entailment." Journal of Philosophical Logic no. 3:347-372.
Reprinted in: Alan Ross Anderson, Nuel D. Belnap, Jr., with contributions by J. Michael Dunn ... [et al.], Entailment: The Logic
of Relevance and Necessity, Princeton: Princeton University Press, 1992 vol. II, pp. 208-231.
"This paper gives a modelling for Ackermann’s systems Π' and Π" , Anderson’s and Belnap’s system
E and R , and several of their subsystems. The distinctive feature of this modelling is a point-shift in the evaluation of negation and
entailment: the negation of a formula holds at a point if the formula itself fails to hold at a complementary point; and an entailment holds at a point if
whenever its antecedent holds at a point its consequent holds at an appropriately associated point. These rules enable negations of valid formulas to hold at a
point and valid formulas themselves to fail to hold at a point. They also provide a grip on certain axioms involving negation or nested entailment." (p.
347, notes omitted)
The first two sections present the deductive-semantic framework; §51.1 specifies the models, and §51.2 the logics. The following two sections
establish completeness; §51.3 for a minimal logic B, and §51.4 for Π', Π", E and the several subsystems. §51.5 outlines various alternative
versions of the modeling. The last two sections contain applications of the modeling: §51.6 to the admissibility of modus ponens; and §51.7 to the finite model
property and decidability. Many of the systems considered are shown to have these properties; see §63 for a further survey on decidability, and §65 for
fundamental undecidabilily results." (pp. 208-209 of the revised reprint)
———. 1974. "An Incomplete Logic Containing S4." Theoria no. 40:23-29.
"This paper uses the standard terminology of modal logic. It should suffice to say that: all logics contain the minimal logic K
and are closed under necessitation, substitution and modus ponens; frames consist of a relation defined on a non-empty set of points; models consist of a frame
with a valuation; and truth-at-a-point is defined and notated in an obvious way; with the formula □ A true at a point iff A is true at all
accessible points. The formula A is true in (satisfied by) a model if it is true in all (some) points of the model; A is strongly verified in
a model if all substitution-instances of A are true in the model; and A is valid in a frame if A is true in all models based upon
the frame, A set of formulas is true, strongly verified, or valid if all of its members are. Unless otherwise stated, all logics contain S4 and all
models and frames possess reflexive and transitive relations.
A logic is complete if any formula valid in all frames that validate the logic is in the logic. This paper exhibits a logic L containing S4
that is not complete." (p. 23)
———. 1974. "Logics Containing K4. Part I." Journal of Symbolic Logic no. 39:31-42.
"There are two main lacunae in recent work on modal logic: a lack of general results and a lack of negative results. This or that logic
is shown to have such and such a desirable property, but very little is known about the scope or bounds of the property. Thus there are numerous particular
results on completeness, decidability, finite model property, compactness, etc., but very few general or negative results.
In these papers I hope to help fill these lacunae. This first part contains a very general completeness result. Let In
> be the axiom that says there are at most n incomparable points related to a given point. Then the result is that any logic containing K4
and In > is complete.
The first three sections provide background material for the rest of the papers. The fourth section shows that certain models contain no
infinite ascending chains, and the fifth section shows how certain elements can be dropped from the canonical model. The sixth section brings the previous
results together to establish completeness, and the seventh and last section establishes compactness, though of a weak kind. All of the results apply to the
corresponding intermediate logics." (p. 31)
Fine, Kit, and Fine, Ben J. 1974a. "Social Choice and Individual Ranking I." Review of Economic Studies no.
"This paper investigates social positional rules. The rules are social in that they produce a social output for any configuration of
individual preference orderings. They are positional in that the output produced depends only upon the positions occupied by each alternative in the individual
preference orderings. (3)
Social rules may be distinguished by the form of their output, be it a quasi-ordering, choice structure or complete ordering. For each form
of output, we shall determine the class of social rules that satisfy certain desirable conditions. Part one deals with quasi ordering rules; part two will deal
with the other types of rules.
Indeed, this part shows that certain desirable conditions are uniquely satisfied by the so-called positional rule. One alternative is as good
as another by this rule if any individual's ranking of these cond alternative can be matched by as high a ranking of the first alternative by some possibly
different individual. The individuals'rankings should be as good for the one alternative as for the other." (p. 303)
(*) Some of the results of this paper are contained in B. Fine's B.Phil. thesis, Oxford1971. We should like to thank the editor and a referee
for many helpful suggestions.
(3) There have been several recent papers on positional rules. See , ,  and . However, most of the results of these papers
overlap with the material of Part II (which is forthcoming in this journal) rather than Part I. Further details will be given there, but let us note that Smith
 also has a variable number of individuals and a composition condition (his separability).
 Arrow, K. J. Social Choice and Individual Values (New York: Wiley, 1951; 2nd ed. 1963).
 Fisburn, P. C. "A Comparative Analysis of Group Decision Methods", Behavioural Science , 16 (1971).
 Fishburn, P. C. The Theory of Social Choice (Princeton UniversityPress, 1973).
 Gale, D. The Theory of Linear Economic Models (New York: McGraw-Hill, 1960).
 Gardenfors, P. "Positionalist Voting Functions", forthcoming in Theory and Decision . [September 1973, Volume 4, Issue
1, pp 1-24]
 Hansson, B. "On Group Preferences", Econometrica , 37 (1969).
 Sen, A. K. Collective Choice and Social Welfare (Holden-Day, 1970).
 Smith, J. H. "Aggregation of Preferences with Variable Electorate", forthcoming in Econometrica . [Vol. 41, No. 6
(Nov., 1973), pp. 1027-1041 ]
———. 1974b. "Social Choice and Individual Ranking II." Review of Economic Studies no. 41:459-475.
"In Part I of this paper it was shown that certain appealing conditions forced any social quasi-ordering rule to include the positional
rule, which is itself the intersection of all finite ranking (f.r.) rules. These conditions are slightly strengthened in the first three sections of this part,
but this allows us to characterize in Section 3 the rules that also satisfy the additional properties as the intersection of some set of f.r. rules. In case a
continuity property, which can be interpreted as a non-veto condition applied to groups, does not hold, the set of f.r. rules must be extended to include
transfinite weightings. Section 1 finds sufficient conditions for a quasi-ordering rule to be positional. This is used in Section 2 to prove the results
contained in Section 3 for the special case of a social ordering rule, when a single f.r. rule emerges. This special case is then generalized in Section 3.
In Section 4, for the first time in the paper, we analyse conditions that recognize social decision depending upon the number of
alternatives. Previously, only the number of individuals has been effectively allowed to vary. Again, simple and natural properties have powerful consequences,
and it is thereby shown that the Borda rule is a compelling choice for making social decision, given a veil of ignorance, that is no knowledge of the special
features of the individuals and alternatives concerned. In case only a quasi-ordering rule is required, social decision is based on the intersection of a set
of f.r. rules symmetrical about the Borda rule.
In Section 5 we turn to choice structureules. First a positional choice structure is defined. It is the strongest such rule containing all
the f.r. rules, since an alternative in a set belongs to the choice from that set iff for some f.r. it is best in the set. This last condition is shown to be
equivalent to demanding that the HC of that element does not belong to the convex hull of the HC of the other
alternatives in the set. Then an outline is made for a conditions analysis of the rule: it is found to be the weakest rule satisfying certain conditions, in
the sense that any other rule satisfying those conditions must be more decisive. In this, the method, results and analysis correspond to Part I's consideration
of the positional quasi-ordering rule.
Section 6 is devoted to an examination of some questions concerned with the independence of conditions and Section 7 contains concluding
remarks. The above only sketches the major results of thispaper. In addition,the analysis of normal social quasi ordering rules in Section 2 and Section 3 has
obvious relevanceto the theory of production and utility under risk in the presence of indivisibility.
Finally, it should be noted that throughou this part, individual preferences are assumed to be antisymmetrical.Whilst the complications posed
by individual indifference were overcome in Part I (Section6), a more general analysis becomes analytically cumbersome and presents more problems here.
Nevertheless many of the results, especially analysis by conditions, do apply more generally, though possibly ith slight modifications." (pp. 459-460)
(*) The first part of the paper [1974a] was written up by K. Fine and the second by B. Fine. Both authors have contributed to all sections of
the paper, though the first has contributed more to the material on the positional rule and the second to the material on normal social rules. Some of the
resultsfor ordering rules in this paper have been independently established by Smith [Smith, J. H. "Aggregation of Preferences with Variable
Electorate", Econometrica . Vol. 41, No. 6 (Nov., 1973), pp. 1027-1041].
Fine, Kit. 1975. "Vagueness, Truth and Logic." Synthese no. 30:265-300.
Reprinted in: Rosanna Keefe & Peter Smith, Vagueness: A Reader, Cambridge: MIT Press, 1996, pp. 119-150.
"My investigation of this topic began with the question "What is the correct logic of vagueness?" This led to the further
question "What are the correct truth-conditions for a vague language?" And this led, in its turn, to a more general consideration of meaning and
The contents of the paper are as follows. The first half contains the basic material. Section 1 expounds and criticizes one approach to the
problem of specifying truth-conditions for a vague language. The approach is based upon an extension of the standard truth-tables and falls foul of something I
call penumbral connection. Section 2 introduces an alternative framework, within which penumbral connection can be accommodated. The key idea is to consider
not only the truth-values that sentences actually receive but also the truth-values that they might receive under different ways of making them more precise.
Section 3 describes and defends the favoured account within this framework.
According to this account, as roughly stated, a vague sentence is true if and only if it is true for all ways of making it completely
precise. The second half of the paper then deals with consequences, complications and comparisons of the preceding half. Section 4 considers the consequences
that the rival approaches have for logic. The favoured account leads to a classical logic for vague sentences; and objections to this unpopular position are
met. Section 5 studies the phenomenon of higher-order vagueness: first, in its bearing upon the truth-conditions for a language that contains a
definitely-operator or a hierarchy of truth-predicates; and second, in its relation to some puzzles concerning priority and eliminability.
Some of the topics tie in with technical material. I have tried to keep this at a minimum.
But the reader must excuse me if the technical undercurrent produces an occasional unintelligible ripple upon the surface. Many of the more
technical passages can be omitted without serious loss in continuity." (p. 265)
———. 1975. "Normal Forms in Modal Logic." Notre Dame Journal of Formal Logic no. 16:229-237.
"There are two main methods of completeness proof in modal logic.
One may use maximally consistent theories or their algebraic counterparts, on the one hand, or semantic tableaux and their variants, on the
other hand. The former method is elegant but not constructive, the latter method is constructive but not elegant.
Normal forms have been comparatively neglected in the study of modal sentential logic. Their champions include Carnap , von Wright ,
Anderson [l] and Cresswell . However, normal forms can provide elegant and constructive proofs of many standard results. They can also provide proofs of
results that are not readily proved by standard means.
Section 1 presents preliminaries. Sections 2 and 3 establish a reduction to normal form and a consequent construction of models. Section 4
contains a general completeness result. Finally, section 5 provides normal formings for the logics T and K4." (p. 229)
 Anderson, A. R., "Improved decision procedures for Lewis's calculus S4 and Van Wright's calculus M," The Journal of
Symbolic Logic, vol. 34 (1969), pp. 253-255.
 Bull, R. A., "On the extension of S4 with CLMpMLp," Notre Dame Journal of Formal Logic, vol. VIII (1967), pp.
 Carnap, R., "Modalities and quantification," The Journal of Symbolic Logic, vol. 11 (1946), pp. 33-64.
 Cresswell, M. J., "A conjunctive normal form for S3.5," The Journal of Symbolic Logic, vol. 34 (1969), pp.
 Wright, G. H. von, An Essay in Modal Logic, Amsterdam (1951).
———. 1975. "Review of David Lewis ' Counterfactuals '." Mind no. 84:451-458.
Reprinted in: Modality and Tense. Philosophical Papers, as chapter 10, pp. 357-365.
"This is an excellent book. It combines shrewd philosophical sense with fine technical expertise; the statement of views is concise and
forthright; and the level of argument is high." (p. 451)
"Lewis suggests that merely possible worlds are like the actual world, ‘differing not in kind but only in what goes on at them’. Indeed,
for him there is no absolute difference between the actual world and the others: the difference is relative to a particular possible world as point of
reference. A similar view has been held about the present time, but it is hard to accept for possible worlds. On the logical construction view, the actual
world is distinguished by the property that all of its propositions are true. Here ‘true’ is an absolute term. It is not defined as truth in the actual world
but, on the contrary, truth-in-a-world is defined as set-theoretic membership." (p. 455).
———. 1975. "Some Connections between Elementary and Modal Logic." In Proceedings of the Third Scandinavian Logic Symposium
, edited by Kanger, Stig, 15-31. Amsterdam: North-Holland.
"A common way of proving completeness in modal logic is to look at the canonical frame. This paper shows that the method is applicable
to any complete logic whose axioms express a XA-elementary condition or to any logic complete for a A-elementary class of frames. We also prove two mild
converses to this result. (1) The first is that any finitely axiomatized logic has axioms expressing an elementary condition if it is complete for a certain
class of natural subframes of the canonical frame. The second result is obtained from the first by dropping ‘finitely axiomatized’, and weakening ‘elementary’
Classical logic is used in the formulation and proof of these results.
The proofs are not hard, but they do show that there may be a fruitful and non-superficial contact between modal and elementary logic.
Hopefully, more work along these lines can be carried out.
§ 1 outlines some basic notions and results of modal logic. For simplicity, this is taken to be mono-modal. However, the results can be
readily extended to multi-modal logics and, in particular, to tense logic.
§ 2 proves the first of the above results and a related result as well; § 3 proves the second of the above results; and finally, § 4
constructs counterexamples to some plausible looking converse results." (pp. 15-16)
(1) After writing this paper, I discovered that A.H. Lachlan had already proved the first of these ‘mild converses’ in . His proof uses
Craig’s interpolation theorem, whereas mine uses the algebraic characterization of elementary classes. R.I. Goldblatt  independently hit upon this latter
proof at about the same time as I did.
He also has a counter-example to the converse of this result. It is similar to the one in § 4.
I should like to thank Steve Thomason for the references above and for some helpful comments on the paper.
 R.I. Goldblatt, First-order definability in modal logic, [ The Journal of Symbolic Logic , Vol. 40, No. 1 (Mar. 1975), pp.
 A.H. Lachlan, A note on Thomason’s refined structures for tense logic, Theoria , [Vol. 40, No. 2 (Aug. 1974), pp. 117–120]
———. 1976. "Review of The Nature of Necessity ' (A. Plantinga)." The Philosophical Review no.
Reprinted in: Modality and Tense. Philosophical Papers, as chapter 11, pp. 366-370.
"This book discusses several topics in the theory of modality: the de re/de dicto distinction, possible worlds, essences,
names, possible objects, and existence. In the final two chapters, the preceding material is applied to the problem of evil and the ontological argument. In
its philosophical (though not theological) parts, the book is close to Kripke’s Naming and Necessity .
There are similar accounts of the a priori/necessary distinction, proper names, transworld identity, and the identity theory." (p.
———. 1976. "Completeness for the Semi-Lattice Semantics. Abstract." Journal of Symbolic Logic no. 41:560.
———. 1976. "Completeness for the S5 analogue of Ei. Abstract." Journal of Symbolic Logic no. 41:559-560.
———. 1977. "Properties, Propositions and Sets." Journal of Philosophical Logic no. 6:135-191.
"This paper presents a theory of cxtensional and intensional entities. The entities in question belong to a hierarchy that begins with
individuals, sets, properties and propositions. The hierarchy extends to higher orders, both extensional and intensional. Thus it contains sets of
propositions, properties of sets, properties of such properties, and, in general, it contains relations-in-intension and relations-in-extension over types of
entities already in the hierarchy.
The theory does not say what a proposition or property is. Rather, a possible worlds account of these entities is taken for granted. Thus a
proposition is regarded as a set of possible worlds, a property as a set of world-individual pairs, and similarly for the other intensional entities.
What the theory does is to characterize and investigate various properties of the entities in terms of possible worlds. These properties
include existence, being purely general or qualitative, being logical, having an individual constituent, and being essentially modal. Thus the theory is
ontological rather than linguistic. Its main concern is with the ontological status of the various entities and not with their relation to language." (p.
———. 1977. "Prior on the Construction of Possible Worlds and Instants." In Worlds, Times and Selves , 116-168. London:
Postscript to ' Worlds, Times and Selves ', by Arthur Norman Prior, reprinted in: Modality and Tense. Philosophical
Papers, as chapter 4.
"Fundamental to Prior’s conception of modality were two theses:
The ordinary modal idioms (necessarily, possibly) are primitive (1)
Only actual objects exist (2)
The first thesis might be called Modalism or Priority, in view of its nature and founder. The second thesis is sometimes called Actualism,
and the two theses together I call Modal Actualism." (p. 116)
"My aim in this chapter is to carry out this programme of reconstruction, at least in outline. I have often followed the lead of Prior,
much of whose later work (3) arose from this programme. However, I cannot be sure that he would have approved of all of the steps I take." (p. 118)
(1) Many references might be given. See e.g. ‘Modal Logic and the Logic of Applicability’ Theoria , 34 (1968), reprinted as Chapter
(2) See Papers on Time and Tense , p. 143
(3) See the chapter of this book, Ch. XI of Papers on Time and Tense, and V of Past, Present and Future.
———. 1978. "Model Theory for Modal Logic Part I: The ' de re / de dicto' Distinction." Journal of Philosophical
Logic no. 7:125-156.
"It is an oddity of recent work on modality that the philosopher's main concern has been with quantificational logic whereas the
logician's has been with sentential logic. There have, perhaps, been several reasons for this divergence of interest. One is that the area of sentential modal
logic is already rich in logical problems; and another is that the semantics for quantified modal logic has been in an unsettled state. But whatever the
reasons have been in the past, the time would now seem ripe for a more fruitful interaction between these two approaches to the study of modality.
My aim in these papers has been to bring the methods of model theory closer to certain common philosophical concerns in modal logic. Indeed,
most of the results answer questions that arise from some definite philosophical position. In this respect, my approach differs from that of Bowen  and
others, who attempt to extend the results of classical model theory to modal logic. Although this approach has its attractions, it also suffers from two
drawbacks. The first is that most of its results are devoid of philosophical interest; and the second is that many standard results of classical model theory,
such as the Interpolation Lemma, do not apply to some standard modal logics, such as quantified S5 (see my paper ).
The philosophical position that underlies the results of the first two parts of this paper may be called de re scepticism. It is the
doctrine that quantification into modal contexts does not, as it stands, make sense. Call a sentence de dicto if, in it, the necessity operator never
governs a formula that contains a free variable. Then for the de re sceptic, only de dicto sentences, or their equivalents, are
legitimate." (p. 125)
 Bowen, K. A., 1975, ‘Normal Modal Model Theory’, Journal of Philosophical Logic 4, 2, 97–131.
 Fine, K., ‘Failures of the Interpolation Lemma in Modal Logic’, Journal of Symbolic Logic , (44), 1979, pp. 201-206.
———. 1978. "Model Theory for Modal Logic Part II: The Elimination of ' de re' Modality." Journal of Philosophical
Logic no. 7:277-306.
"In the first part of this paper, two philosophical positions were introduced: de re scepticism; and anti-Haecceitism.
According to the first, quantification into modal contexts does not, as it stands, make sense; and according to the second, the identity or non-identity of
individuals in distinct possible worlds is a matter of convention. It was shown that the two positions are equivalent in the sense that whatever first-order
modal sentence is legitimate for the one is also legitimate for the other.
A soft and hard version of each of these positions may be distinguished. According to the soft de re sceptic, it is possible to make
sense of de re modal discourse; and according to the soft anti-Haecceitist, it is possible to define coherent identity conditions for individuals
across possible worlds. Both of the soft positions, then, are compromising ones in that they allow that ordinary modal discourse may be reconstructed. The hard
versions of the positions, on the other hand, deny that any such reconstruction is possible.
The soft de re sceptic may reconstruct ordinary modal discourse in various ways. One way is to reinterpret either quantification or
modality (or both) so that each de re sentence is equivalent to one that is de dicto . Although this method has been prominent in the
literature, I shall deal with it only incidentally here. I hope to deal with it more fully elsewhere. Another way is to add axioms to the standard modal logic
so that two conditions are satisfied. The first (eliminability) is that every de re sentence should have a de dicto equivalent relative to
the resulting system. The second (conservativeness) is that no dicto sentence should be provable in the resulting system that is not already a theorem
of standard modal logic." (p. 277)
———. 1979. "Failures of the Interpolation Lemma in Quantified Modal Logic." Journal of Symbolic Logic no. 44:201-206.
"Beth’s Definability Theorem, and consequently the Interpolation Lemma, fail for the version of quantified S5 that is presented in
Kripke’s . These failures persist when the constant domain axiom-scheme ∀x □ φ ≡□ ∀x φ is added to S5 or, indeed, to any weaker extension of
quantificational K .
§1 reviews some standard material on quantificational modal logic. This is in contrast to quantified intermediate logics for, as Gabbay 
has shown, the Interpolation Lemma holds for the logic CD with constant domains and for several of its extensions. §§2—4 establish the negative results for the
systems based upon S5. §5 establishes a more general negative result and, finally, §6 considers some positive results and open problems. A basic knowledge of
classical and modal quantificational logic is presupposed." (p. 201)
 D. Gabbay, Craig's interpolation lemma for modal logics, Conference in Mathematical Logic , London, 1970, Lecture Notes in
Mathematics , no. 255, Springer-Verlag, Berlin and New York, 1972, pp. 111-127.
 S. Kripke, Semantical considerations on modal logic, Acta Philosophica Fennica , vol. 16 (1963), pp. 83-94.
———. 1979. "Analytic Implication." In Papers on Language and Logic , edited by Dancy, Jonathan, 64-70. Keele: Keele
Reprinted in: Notre Dame Journal of Formal Logic , 27, 1986, pp. 169-179.
"Parry presented a system of analytic implication in  and , Dunn  gave an algebraic completeness proof for an extension of this
system and Urquhart  later gave a semantic completeness proof for Dunn's system with necessity. This paper establishes completeness for Parry's original
system, (*) thereby answering a question of Gödel , and then, on the basis of the completeness result, derives decidability; it also deals with
quantificational versions and other modifications of his system.
Section 1 contains some informal remarks on the notion of analytic implication.
They are not strictly relevant to the later analysis, although they may help to place it in perspective. Section 2 presents the semantics and
Section 3 exhibits a system of analytic implication. Section 4 helps to demonstrate that the system is equivalent to Parry's, and Section 5 establishes
completeness. Finally, Section 6 outlines the theory for some related systems." (p. 64)
(*) I mean the full system of  with adjunction, A14 and A15.
 Anderson A. R. and N. D. Belnap, Jr., "A simple treatment of truth-functions," The Journal of Symbolic Logic, vol. 25
(1959), pp. 301-302.
 Dunn, J. M., "A modification of Parry's analytic implication," Notre Dame Journal of Formal Logic, vol. 13, no. 2
(1972), pp. 195-205.
 Epstein, D., "The semantic foundations of logic," to appear.
 Hughs, G. E. and M. J. Cresswell, An Introduction to Modal Logic, Methuen, London, 1968.
 Kielkopf, C. F., Formal Sentential Entailment, University Press of America, Washington, D.C., 1977.
 Parry, W. T., "Ein Axiomsystem fur eine neue Art von Implication (analytische Implication)," Ergebrisse eines
Mathematischen Colloquiums, vol. 4 (1933), pp. 5-6.
 Parry, W. T., "The logic of C. I. Lewis," pp. 115-154 in The Philosophy of C. I. Lewis, ed., P. A. Schilpp, Cambridge
University Press, 1968.
 Parry, W. T., "Comparison of entailment theories," The Journal of Symbolic Logic, vol. 37 (1972), pp. 441 f.
 Post, E. L., The Two-Valued Iterative Systems of Mathematical Logic, Princeton, University Press, Princeton, New Jersey,
 Urquhart, A., "A semantical theory of analytical implication," Journal of Philosophical Logic, vol. 2 (1973), pp.
———. 1980. "First-Order Modal Theories. [II. Propositions]." Studia Logica no. 39:159-202.
Abstract. "This paper is part of a general programme of developing and investigating particular first-order modal theories. In the
paper, a modal theory of propositions is constructed under the assumption that there are genuinely singular propositions, ie. ones that contain individuals as
constituents. Various results on decidability, axiomatizability and definability are established."
"In some recent work (, , , ), I have attempted to carry out a dual programme of developing a general model-theoretic
account of first-order modal theories, on the one hand, and of studying particular theories of this sort, on the other. The two parts of the programme are
meant to interact, with the second providing both motivation and application for the first. The present paper belongs to the second part of the programme and
deals with the question of giving a correct essentialist account of propositions.
My approach is distinctive in two main ways, one linguistic and the other metaphysical. On the linguistic side, I have let the variables for
propositions be both nominal and objectual. That is to say, the variables occupy the same position as names and are interpreted in terms of a range of objects,
which, in the present case, turn out to be propositions. This approach stands in contrast to the earlier work of Prior , Bull , Fine , Kaplan 
and Gabbay , , in which the variables are sentential (they occupy the same position as sentences) and are interpreted either substitutionally or in
terms of a range of intensional values." (p. 159)
 R. A. Bull, On modal logic with propositional quantifiers, Journal of Symbolic Logic 34 (1969), pp. 257–263.
 K. Fine, Propositional quantifiers in modal logic, Theoria 36 (1970), pp. 336–346.
 —, Model theory for modal logic, part I, Journal of Philosophical Logic 7 (1978), pp. 125–156.
 —, Model theory for modal logic, part II — The elimination of de re modality, Journal of Philosophical Logic 7 (1978),
 —, First-order modal theories, part I — Modal set theory, to appear in Nôus . 
 —, Model theory for modal logic, part III — Existence and predication, to appear in Journal of Philosophical Logic . 
 D. M. Gabbay, Modal logic with propositional quantifiers, Zeitschrift für Mathematische Logic und Grundlagen der Mathematik 18
(1972), pp. 245–249
 —, Investigations in Modal and Tense Logics , D. Reidel, Holland, 1976.
 D. Kaplan, S5 with quantifiable propositional variables, Journal of Symbolic Logic 35 (1970), pp. 355.
 A.N. Prior, Egocentric logic, Nôus , vol. II, no. 3 (1968), pp. 191–207.
———. 1981. "First-Order Modal Theories. I: Sets." Noûs no. 15:177-205.
"The aim of this paper is to formalize various metaphysical theories within a first-order modal language. The first part deals with
modal set theory. The later parts will deal with propositions, possible worlds, and facts.
Such an undertaking is relevant both to logic and to metaphysics.
Its relevance to logic lies mainly in its bearing on the model theory for first-order modal languages. I have begun to develop such a theory
in . The consideration of particular theories can then provide both an application of and motivation for general results in this field. There is already a
fruitful interaction between the proof of general results and the consideration of particular first-order theories within classical model theory; and the hope
is that there should be as beneficial an interaction within modal logic.
The relevance of the undertaking to metaphysics consists mainly in the general advantages that accrue from formalizing an intuitive theory.
First of all, one thereby obtains a clearer view of its primitive notions and truths. This is no small thing in a subject, such as metaphysics, that is so
conspicuously lacking in proper foundations.
But once a formalization is given, one can establish results about the theory as a whole and thereby obtain that overall view of a subject
that philosophers often strive for but rarely obtain." (p. 177)
"The plan of this part of the paper is as follows. §1 contains an informal discussion and justification of our axioms for modal set
theory. §2 then presents the formal theories. §3 develops a proof- and a model-theory for class abstracts in modal set theory and establishes a useful result
on transferring abstracts from classical set theory into a modal context. In §4, it is shown that the formal theories are equivalent in that any two of them
share the same theorems in their common language. The proof of equivalence contains general result on when the possible worlds semantics for a given modal
theory can be represented within that theory itself. The next section discusses the adequacy of our formalizations and shows that, in a certain sense, they
capture all of the essential truths about sets as such. The last section is concerned with the identity of sets and places the problem within a general account
of the identity of objects." (p. 178)
 Fine, K., 'Model Theory for Modal Logic I, II, III', The Journal of Philosophical Logic , (1978) 125-56, (1978) 277-306, and to
appear. [1981, 293-307]
———. 1981. "Model Theory for Modal Logic. Part III: Existence and Predication." Journal of Philosophical Logic no.
"This paper is concerned with the technical implications of a certain view connecting existence to predication. This is the view that in
no possible world is there a genuine relation among the nonexistents of that world or between the nonexistents and the existents. (1) The meaning of the term
‘genuine’ here may be variously explained. On an extreme interpretation, all relations are ‘genuine’, so that none of them are to relate non-existents.
On a milder interpretation, the genuine relations are those that are simple or primitive in some absolute sense. But even without appeal to
an absolute concept of simplicity, we can require that all relations should be analyzable in terms of some suitable set of relations, relating only existents
In order to make our results applicable to the thesis, we shall suppose that the primitive non-logical predicates of our language correspond
to the genuine relations, whatever they might be taken to be. Thus, the linguistic formulation of the thesis becomes that the primitive predicates of the
language should only be true, in each world, of the existents of that world.
Of course, the thesis might have been given a linguistic formulation, without any reference to relations, in the first place.
The thesis is an instance of what has been called Actualism. This is the ontological doctrine that ascribes a special status to actual or
existent objects. Another form of the doctrine, so-called World Actualism, says that the behaviour of nonexistents is supervenient upon the behaviour of the
existents, that two possible worlds which agree in the latter respect cannot differ in the former respect. The present thesis, by contrast, might be called
Predicate Actualism . It should be clear that Predicate Actualism implies World Actualism, at least if the predicates used to describe the world are
to express ‘genuine’ relations; for then there are no relationships involving nonexistents by which two worlds might be distinguished. On the other hand, World
Actualism does not, as it stands, imply Predicate Actualism." (p. 293)
(*) This paper is the third and final part of a series (see the references below). It was completed and submitted to the Journal of
Philosophical Logic in 1977, at about the same time as the other parts. But because of some mishap in the mail, its publication was delayed. The present
part is independent from the other parts in its results, but draws upon the terminology of Section 2 of Part I.
I should like to thank the editor, R. Thomason, for many valuable remarks on the earlier version of the paper.
(1) I have briefly discussed this thesis elsewhere. The reader may like to consult Section 7 of [ 11], pp. 151 and 156-160 of , p. 564 of
, and Section 8 of [7b].
There has been a fair amount of recent literature on the topic. I cannot give a complete survey, but the reader may like to consult Chapters
IV-V of , p. 86 of , Chapters VII-VIII of , and pp. 333-336 of [ 11].
[l] Fine, K., Postscript to Worlds, Times and Selves (with A. N. Prior), Duckworth, England (1977).
 Fine, K., ‘Propositions, Sets and Properties’, Journal of Philosophical Logic 6 (1977), 135-191.
 Fine, K., ‘Review of ‘, Philosophical Review 86 (1977), 562-66.
 Fine, K., ‘Model Theory for Modal Logic - Part I’, Journal of Philosophical Logic 7 (1978), 125-156.
[S] Fine, K., ‘Model Theory for Modal Logic - Part II’, Journal of Philosophical Logic 7 (1978), 277-306.
 Fine, K., The Interpolation Lemma Fails for Quantified S5’, Journal of Symbolic Logic 44 (1979), 201-206.
[7a, b, c] Fine, K., ‘FirstaIder Modal Theories - I Sets, II Propositions, III Facts’, in Nous (1981), Studia Logica XXXIX,
2/3 (1980) 159-202, and Synthese (1981), respectively.
 Kripke, S., Semantical Considerations on Modal Logic’, Acta Philosophica Fennica 16,83-94. Reprinted in L. Linsky (ed.),
Reference and Modality , Oxford University Press (1971).
 Plantinga, A., The Nature of Necessity , Clarendon Press,Oxford (1974).
 Prior, A. N., Time and Modality , Clarendon Press, Oxford (1957).
 Stalnaker, R., ‘Complex Predicates’, Monist 60 (1977), 327-339.