A Probabilistic Theory of Coherence

Transcription

1 A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want C to be a quanttatve, probablstc generalzaton of the (deductve) logcal coherence of E. So, n partcular, we requre C to satsfy the followng ntutve desderatum. () C(E) s Maxmal > 0 0 < 0 Mnmal (postve, constant) (negatve, constant) f the E f E s postvely dependent (see below for def.) f E s ndependent (see below for def.) f E s negatvely dependent (see below for def.) f all subsets of are logcally equvalent (and E are unsatsfable E s satsfable) Desderatum () captures the qualtatve features that a probablstc generalzaton of logcal coherence should satsfy t requres C to respect the extreme deductve cases, and to be properly senstve to probablstc dependence (a general noton of probablstc dependence wll be defned precsely, and n a slghtly non-standard way, below). I propose a probablstc measure of coherence C based on a slght modfcaton of Kemeny and Oppenhem s (95) measure of factual support F. The formulaton of C s somewhat ntrcate. We begn wth some prelmnary defntons. Frst, we defne the two-place functon X,Y). X,Y) may be nterpreted as the degree to whch one proposton Y supports another proposton X (relatve to a fntely addtve, regular, Kolmogorov (956) probablty functon Pr). Pr( Y X ) Pr( Y X ) Pr( Y X ) + Pr( Y X ) X, Y ) 0 f X s contngent and Y s not a f X and Y are necessary truths f X s a necessary truth and Y s contngent f Y s a necessary falsehood necessary falsehood For smplcty, I am assumng that the probablty functon Pr s regular or strctly coherent n the sense of Shmony 95. That s, I assume that Pr assgns probablty only to necessary truths, and probablty 0 only to necessary falsehoods (and, therefore, that Pr(X Y) s extreme only f X and Y are logcally dependent). We could weaken ths assumpton, but t would complcate our defntons (as would a generalzaton to nfnte sets of propostons, and perhaps to countably addtve Pr). In the context of Bayesan epstemology, assgnng extreme probablty to contngent propostons s controversal (Jeffrey 99). Our F s more complete than Kemeny and Oppenhem s (95), whch s defned only for contngent X and Y. We need ths addtonal structure n F to ensure that C satsfes (). Even our X,Y) remans undefned n the case where X s a necessary falsehood but Y s not. We don t need to worry about ths case here (however, note that ths omsson does make C a partal functon). But, I thnk we can at least say that f X s contradctory, then X,Y) 0 (.e., that nothng supports a contradcton).

2 Next, we use F to defne the probablstc ndependence and postve/negatve dependence of a set of propostons E. Let P be the power set (sans null set) of the set E\{E } (unless E s a sngleton, n whch case P E). And, for each x P, let X be the conjuncton of the elements of x. Then, we defne: E s postvely dependent ndependent negatvely dependent ff for all E ff for all E ff for all E E and for all x P, E, X ) > 0 E and for all x P, E, X ) 0 E and for all x P, E, X ) < 0 Our defnton of ndependence s (nearly) logcally equvalent (except for certan extreme cases n whch E and/or X are non-contngent ) to the standard defnton of the ndependence of sets E seen n probablty texts (Kolmogorov 95: I.5). Interestngly, the concepts of postve and negatve dependence of a set E are not typcally defned (at all) n standard probablty textbooks (at least, I have not seen such defntons). Lke the standard defnton of ndependence, my defntons of postve/negatve dependence requre correlaton (or ant-correlaton) of all subsets of propostons n E (.e., parwse, 3-wse, 4-wse, etc.). Ths means that sets wth mxed correlatons or ant-correlatons (e.g., parwse but not 3-wse correlaton or ant-correlaton, etc.) wll not count as dependent sets on my defnton. Now, we re ready to ntroduce the components of our quanttatve measure of coherence C. Let S {{E,X) x P } E E}. In general, S wll have n ( n ) elements, where n > s the number of elements of E (f n, then S wll have just one element: E )). For nstance, f n 3, then S wll be {E,E ), E,E 3 ), E,E &E 3 ), E ), E,E 3 ), E &E 3,E &E )}, whch has 3 ( 3 ) elements. Fnally, we defne C as follows. 3 C(E) df mean(s) That s, C s smply the mean value of S. It s easy to verfy that C satsfes (). Ths s because F s a proper probablstc generalzaton of deductve support (Kemeny & Oppenhem 95, Ftelson 00: 3..3). In partcular, we have: The standard (Kolmogorov 95: I.5) defnton of ndependence says that E s ndependent of tself n cases where E s non-contngent. Ths s because n such cases Pr(E & E) Pr(E) Pr(E) or 0, f Pr(E) or 0, respectvely. I thnk ths s unntutve. Intutvely, all consstent propostons are (maxmally) postvely correlated wth (dependent on) themselves, and all contradctons are (maxmally) negatvely correlated wth (dependent on) themselves. Ths s what my defntons of ndependence and dependence ental, owng to my defnton of F on whch E,E) for all consstent E, and E,E) for all self-contradctory E. 3 Here, I take C to be the straght average of S. One could generalze our defnton of C so as to assgn dfferent weghts to dfferent types of (n)dependence (e.g., -wse (n)dependence vs 3-wse (n)dependence mght be weghted dfferently n the average, or negatve dependence among certan subsets of E mght be weghed more heavly than postve dependence, etc.). One can thnk of our defnton of C as assumng that all of the F- components of C (n S) have the same weght.

3 C(E) s > 0 0 < 0 f the E f E s postvely dependent f E s ndependent f E s negatvely dependent f all subsets of are logcally equvalent (and E are unsatsfable E s satsfable). C vs Shogenj s Measure of Coherence Interestngly, Shogenj s rato measure of coherence (Shogenj 999, Akba 000, Shogenj 00) does not satsfy (). 4 In the case where the E are logcally equvalent (hence Pr(E ) p, for all ), Shogenj s measure reduces to Pr(E &L & E n ) Pr(E ) L Pr(E n ) p p n p n whch s not a constant (nor s t maxmal on the [0, ) scale of Shogenj s measure), and stll depends on the uncondtonal probabltes of the E. Ths s unntutve, as ths should be a case n whch the degree of coherence s maxmal, and does not depend on the prors of the E. Here, Shogenj s measure of coherence nherts an undesrable feature of the rato measure of degree of support or confrmaton (Pollard 999, Ftelson 00: 3..3). It s also nterestng to note that Shogenj s measure s based only on the n-wse (n)dependence of the set E. It s well known that a set E can be j-wse ndependent, but not - wse ndependent, for any j (ndeed, we can have dsagreement for any combnatons of - and j-wse ndependence as well see (Pfeffer 994: 4.) for several concrete examples). Snce Shogenj s measure s based only on n-wse ndependence (dependence), n cases where a set s n-wse ndependent (dependent), but not j-wse (for some j n) ndependent (dependent), Shogenj s measure does not take nto account the mxed nature of the coherence (ncoherence) of E (and ts subsets), and t judges E as havng the same degree of coherence (ncoherence) as a fully ndependent (or fully dependent) set. Ths seems ncorrect to me. I thnk t s mportant for a measure of coherence to be senstve to the (n)dependences mplct n all subsets of E. 3. Akba s Crtcsms of Shogenj s Coherence Measure Akba s (999) crtcsms of Shogenj s coherence measure do not apply to our C. For nstance, Akba complans that f E entals E, then Shogenj s measure says that the degree of coherence of the set {E,E } s /Pr(E ), whch he fnds unntutve, snce t only depends on the 4 The fact that Shogenj s measure s always postve s a merely conventonal (and therefore nsgnfcant) volaton of (). Ths can be fxed smply by takng the logarthm of Shogenj s measure. The problems wth Shogenj s measure dscussed below are not merely conventonal.

4 uncondtonal probablty of E. I agree wth Akba s ntuton here; and, so does C. In such a case, we have (f E and E are contngent 5 ): C({ E, E }) mean( S) E, E ) + E, E ) E, E ) + + Pr( E E ) So, C supports Akba s ntuton that the degree of coherence n ths case should depend on the precse relatonshp between E and E, and not merely on Pr(E ) alone (smlar thngs happen when C s appled to Akba s other examples nvolvng more than two events). Akba also dscusses the problem of sngleton sets of propostons. He ponts out that Shogenj s measure of coherence judges the self-coherence of all propostons to be the same. Ths s unntutve, snce (for nstance) necessary truths should be vewed as more self-coherent than necessary falsehoods. Our measure C captures ths ntuton, snce: C({E}) E,E) f E s a necessary truth f E s a necessary falsehood Nonetheless, t s stll true that C judges the degree of coherence to be the same (+) for all satsfable sngleton sets propostons. However, snce there seem to be no clear countervalng ntutons about what a probablstc theory of degree of coherence should say n the contngent, sngleton case (Akba 000, Klen & Warfeld 994, Shogenj 999), I do not vew ths as a shortcomng of my proposed probablstc measure of coherence. Here, I am n agreement wth Shogenj (999) n understandng coherence as a relaton between propostons. Intutvely, all propostons cohere wth themselves (maxmally), except for necessary falsehoods. 6 References San José State Unversty San José, CA branden@ftelson.org Akba, K Shogenj s probablstc measure of coherence s ncoherent. Analyss 60: Ftelson, B. 00. Studes n Bayesan Confrmaton Theory. Ph. D. thess, Unversty of Wsconsn Madson (Phlosophy). The thess can be downloaded n ts entrety from the followng URL Jeffrey, R. 99. Probablty and the Art of Judgment. Cambrdge: Cambrdge Unversty Press. Kemeny, J. and P. Oppenhem. 95. Degrees of factual support. Phlosophy of Scence 9: Klen, P. and T. Warfeld What prce coherence? Analyss 54: 9 3. Kolmogorov, A Foundatons of the Theory of Probablty (second englsh edton). New York: AMS Chelsea Publshng. 5 If E and E are contngent and E entals E, then the set E {E, E } s postvely dependent, and we have E, E ) Pr(E E ) / Pr(E E ), snce Pr(E E ) 0. 6 Thanks to Luc Bovens, Alan Hájek, Stephan Hartmann, Jm Hawthorne, and Tomoj Shogenj for useful dscussons about coherence and probablty. And, thanks to an anonymous referee for ther comments and suggestons on prevous drafts of ths paper.

5 Pfeffer, P Probablty for Applcatons. New York: Sprnger-Verlag. Pollard, S Mlne s measure of confrmaton. Analyss 59: Shmony, A Coherence and the axoms of confrmaton. Journal of Symbolc Logc 0: 8. Shogenj, T Is coherence truth-conducve? Analyss 59: Shogenj, T. 00. On the probablstc measure of coherence. Reply to Shogenj s probablstc measure of coherence s ncoherent by K. Akba. Analyss 6: