Ibn Sina’s Anticipation of Burdian and Barcan Formulas
Zia Movahed
1. Contemporary attributions of some theses in QML
In 1947 Carnap wrote: [p. 196][1]
Any system of modal logic without quantification is of interest only as a basis for a wider system including quantification. If such a wider system were found to be impossible, logicians would probably abandon modal logic entirely.
Surprisingly Carnap’s anticipation turned out to be right, but exactly in the opposite directiona! What captured the attention of modal logicians most, was propositional modal logic, and quantified modal logic (QML) has remained underdeveloped. Of course there are many systems of, and many approaches to QML.[2] But apart from provability interpretation of it, other treatments of the subject have been of interest to mostly philosophical logicians, and QMLs have become a battle ground for heated controversies over philosophy problems such as essentialism, transworld identity, existence, possible worlds and the very notion of necessity.
But among many versions of QML, the one I am interested in here is a system in which all the following formulas are derivable:
BF: " xðFx ® ð"xFx
CBF: "xFx ® "xðFx
BUF: a"xFx ® "xa Fx
BF and CBF stand for the Barcan formula and the Coverse Barcan formula. These names drive from Ruth C. Barcan (later Mrs. J. A. Marcus), who called attention to the formulas in her paper of 1946.[3] Also BUF stands for the Buridan formula, attributed to Jean Buridan, French philosopher of the 14^{th} century, by Alvin Plantinga [p. 58].[4] BF and CBF have important semantical properties. To see how, I begin with the simplest model validating all the three formulas.
As to the language of QML, it contains individual constants and variables, and n place predicates (n ³ 0). Well formed formulas of QML are inductively constructed out of Ø ®, ", and . Furthermore, a and $ are defined in the standard way. The simplest model of QML is the sequence M = < W, R, D, I >, where W is a nonempty set of possible worlds, R a binary equivalence relation over the members of W, D a nonempty set of individuals and I an interpretation function which assigns to every term (variable or constant) a member of D : I(t)eD, and to each nplace predicate letter A a set of ntuples of D : I(A_{t1}, … , A_{tn}) is < 0_{1}, … , 0_{n} > where 0_{ie}D. The truthvalue of each wff at a world we W is given by the following rules:
I(A) : I(A_{t1 }, … , A_{tn}) is true in w iff < I(t_{1}), … , I(t_{n}) > e I(A) in w;
I(ØA) : I(ØA) is true in w iff I(A) is false in w,
I(A ® B): I(A ® B) is true in w iff if I(A) is true w;
then I(B) is true in w;
I("xAx): I("xAx) is true in w iff for every I´ which is just like
I except possibly that I(x) ¹ I´ (x) I´ (Ax) is true in w;
I(A): I(A) is true in w iff for every w´ such that wRw´, I(A) is true in w´.
It is easy to verify that BF, CBF and BUF are all valid in M. Other logically and philosophically interesting aspects of this simple semantics are as follow:
1. BF imposes the condition that whenever wRw´, then D(w´) Í D(w) and CBF the condition that whenever wRw´, then D(w) Í D(w´). So if we have both BF and CBF, then wRw´, implies D(w) = D(w´). Therefore to have both formulas, the domains of all worlds should have the same objects. M has this property, so it is called a constant model of QML.
2. According to BF and CBF, to move from one world to another nothing comes into existence and nothing passes out of existence.
3. Since R in an equivalence binary relation, all S5 modal axioms are valid in M, in fact in the frame < W, R,> of M.
Finally to obtain the simplest QML respecting our semantics, it suffices to add the laws and rules of classical quantification theory to the propositional modal logic S5 [pp. 245247].[5] This is a kind of systems that philosophical logician like R. Barcan, E.N. Zalta, B. Linsky[6] and T. Williamson[7] argue that it best represents our intuitive understanding of metaphysical nature of alethic modality.
2. Ibn Sina’s anticipations
According to the received texts of the ancient Greek and the carly Islamic period Ibn Sina (Avicenna, 9801037) was the first logician who realized that the meanings of modal quantified propositions are sensitive to the order of their quantifiers and modal operators and extensively discussed what are now called Buridan and Barcan formulas. So in Kitab al‘Ibara, the third volume of his major work alShifa (in Latin known as Sufficientia) Ibn Sina writes: [pp. 115116][8]
If we say that [*] every human being possibly is a writer, …, no one would doubt it, …, but if we say that [**] possibly every human being is a writer, and possibility comes before quantifier, then that would be really doubtful. (My translation from Arabic.)
Here, Ibn Sina, after some illuminating remarks, concludes that [**] implies [*] but not vice versa. So he accepts BUF and rejects its converse CBUF as a modal principle.
As to BF and CBF he writes: [pp. 115116 again][9]
But to say that some people possibly are not writers is modally the same as saying that possibly some people are not writers.
In modern symbolism this is formulized as:
$xaØW(x) « a$xØW(x)
and by taking negations of both sides, we have
"xðW(x) « "xW(x)
which is the conjunction of BF and CBF in one biconditional.
Finally, and this is what I regard as Ibn Sina’s most important innovation in the subject, he discovered that there are two kinds of modality and coined the names “mode of predication” (Jahate Haml) and “mode of quantifier” (Jahate sour) for what, more than one hundred years later, medieval logicians called respectively de re and de dicto modality. This is a distinction which has become the subject of many discussions in modern philosophy of logic and language.
In conclusion I think, to do justice to the pioneer work of Ibn Sina, it would be fair to rename Buridan and Barcan formulas as Ibn Sina Buridan and Ibn Sina Barcan formulas.
Acknowledgment
I would like to thank Professor W.D. Hart and Professor Timothy Wiliamson for encouraging me to bring these historical facts to light.
Notes
[1]. R. Carnap, Meaning and necessity, University of Chicago Press, 1997.
[2]. J.W. Garson, Quantification in modal logic, in: Ilandbook of Philosophical Logic eds. D. Gabby and F. Guenthner, Dordrecht, Reidel, 2 (1989), 249307.
[3]. R.C. Barcan, A functional calculus of first order based on strict implication, J. Symbolic Logic, 11 (1946), 116.
[4]. A. Plantinga, The nature of necessity, Oxford, (1974).
[5]. G.E. Hughes and M. J. Cresswell, A new introduction to modal logic, Routledge (1996).
[6]. E.N. Zalta, and B. Linsky, In defense of the simplest quantified modal logic, in philosophical perspectives (8): Philosophy of logic and language, ed. J. Tomberline, Ridgeview, 1994.
[7]. T. Williamson, Bare possiblia, Erkenntnis, 48 (1998), 257273.
[8]. Ibn Sína, alIbara (Interpretation), ed. M. ElKodeiri, DaralKatib al‘Arabí, Cairo, 1970, Part 3 of alShifa.
[9]. W.C. Kneal and M. Kneal, The development of logic, Oxford, (1962).
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