Ruth Barcan Marcus

Reuben Post Halleck Professor of Philosophy
Yale University, New Haven, USA

2. What example(s) from your work illustrates the role formal methods can play in philosophy?

I had a pre-formal interest in the modal concepts of necessity and possibility which can be traced back to Aristotle.  This led me to study, with the encouragement of my undergraduate tutor JC. McKinsey, the work of C.I. Lewis who had developed modal systems of propositional logic. As a graduate student at Yale this interest was pursued with the support of F.B. Fitch.  At that time, and through the early forties, a prevailing view about interest in modal logic was that of W.V. Quine who claimed that its source was in a misguided confusion between use and mention. It was also claimed that any attempt to develop a system of quantified modal logic would lead to deep problems of interpretation, including incoherence.

I set about extending some of the Lewis systems to first and then second order (JSL, 11, 1946 and JSL, 12, 1947) under Ruth C. Barcan included is an attribute forming abstraction operator and the "Barcan formula" taken as an axiom:

◊($x)Fx Þ ($x)◊Fx

On the given formalizations the converse is provable. The “Barcan Formula” and its converse remains a subject for debate.

The system Quantified Modal Logic (QML) had some outcomes which answered to some non­formal speculation and debate about modalities such as.

a. A formal proof of the necessity of identity.

b. Despite Lewis’ claim that strict implication was supposed to 'capture the relation of deducibility or logical consequence  the strict analogue of the deduction theorem in unmodalized predicate logic was not provable. A weaker more plausible deduction theorem is provable in some of the Lewis systems. JSL 1946. (See also JSL 18, 1953 for a more extended account.)

c. A substitution theorem which proscribes the substitution of non-necessary equivalences in the scope of the necessity operator.

B. Interpretation of QML

a. I defend Smullyan’s claim that Quine’s examples of substitution failures in modal contexts are mistaken, Russell’s formal theory of descriptions is adopted as well as a linguistic theory of direct reference for proper names, (Proper names are not taken to be disguised descriptions.) JSL, 13, 1948.

b. Modalites and Intensional Languages, Synthese, 1961. Several issues are addressed using formal methods.

i. A direct reference view for genuine proper names is defended. I called them tags since they are unmediated. On such a direct reference view of proper names in conjunction with the theory of descriptions, the much advertised; "failures" of substitution for identity in modal contexts are dispelled within interpreted QML.

ii. The necessity of identity is defended. (The "=" in identity formulae is flanked by individual variables or constants. Only proper names are individual constants.)

iii. A substitutional alternative to objectual quantification is proposed for consideration in some contexts. It is seen as of interest but not urged. See also "Quantification and Ontology" Nous, VI, no. 3, 1972 where Quine’s claim that the substitu1ional account leads to a contradiction is formally shown to be fallacious.

iv. A non-formal account of essentialism challenges the claim that essentialism is "invidious". Formal accounts within QML are given in Nous l, 1967,  some versions of which have been adopted, often without attribution.

v. A sample model theoretic semantics with fixed domain is given in which the Barcan formula is provable. Metaphysically understood, domains of interpretation contain only actual objects. Possibility concerns properties actual objects might have. My arguments against possibilia are linguistic and empirical and discussed in The Proceedings of The American Philosophical Association, 1975, Grazer Philosophicshe Studien, vol. 25, 1985/86, Revue lnternationle de Philosophie, 1997.

vi. The paper here considered was delivered in 1962 and was followed by comments from Quine and a discussion with Quine, Kripke , Follesdaal et al which were published in Synthese, 1962. There Quine says that "the distinction between names and descriptions is a red herring." That distinction (the red herring) was later adopted by Kripke (1971) and others. The historical chain account of the name transmission was first proposed by Peter Geach in "The Perils of Pauline", Revue of Metaphysics, 1969.

C. Moral dilemmas and consistency
An analogue of the formal model theoretic definition of consistency is used to dispel the received claim that when a moral code can mandate incompatible actions in a particular case, it must be inconsistent. But, what is formally required for consistency is that there is a possible world which is dilemma free, e.g. where keeping promises does not conflict with saving lives as in Plato' s example. There are ethical conclusions to be drawn; we ought to arrange our lives and institutions to minimize occasions of conflict. Journal of Philosophy, LXXVIT, no 3, 1980 and in Homer Mason editor, Moral Dilemmas and Moral Theory, OUP, 1996. This account runs counter to many formalizations of deontic logic where conflicts of obligations in a particular case entails inconsistency.

D. Set theory in a Modal Framework
The syntax and semantics of QML can be enriched to accommodate a theory of collections described by inventory and sets described as satisfying some attribute not given by inventory e.g.

^x(x is a planet)


^x(x=N Ú x=V Ú x=M Ú x=P Ú x=Ma Ú x=U Ú x=S Ú x=E Ú x=J)

where the upper case letters are names of the actual planets.

Any two attributes given by inventory, if they describe the same collection then they are necessarily equivalent. No axiom of extensionality is required. This is just an extension of the necessity of identity for the singular case. That does not hold for attributes which may be satisfied by the 8ame things where the equivalence is not necessary but analogous to a material equivalence.

This paper was written in two versions; there was a problem of making the vocabulary and symbolism perspicuous. Acta Philosoophica Fennica, XVI, 1963,  American Philosophical Quarterly, 1974.

The list above is partial account of the way that formal methods have been essential to my philosophical work. In recent years I have been writing on epistemological issues such as belief and rationality.  Here I have rejected most received accounts of propositional attitudes and attempts to systematize epistemic logic. The entire project is in a formative stage so it has been omitted from the discussion

Read the remaining part of Ruth Barcan Marcus' interview in the book Formal Philosophy

ISBN-10    87-991013-1-9    hardcopy
ISBN-10    87-991013-0-0    paperback
Published by Automatic Press ● VIP, 2005

Order now from Amazon!    US    UK    CA     DE