INDUCTIVE LOGIC
INDUCTIVE LOGIC
The idea of inductive logic as providing a gene- called ‘‘confirmation,’’ which is a quantitative gen- ral, quantitative way of evaluating arguments is a eralization of deductive entailment (See Carnap, relatively modern one. Aristotle’s conception of Rudolf; Confirmation Theory). ‘induction’ (epagogZ )—which he contrasted with Carnap (1950) gives some insight into the mod- ‘reasoning’ (sullogismoB)—involved moving ern project of inductive logic and its relation to only from particulars to universals (Kneale and classical deductive logic: Kneale 1962, 36). This rather narrow way of think- ing about inductive reasoning seems to have held Deductive logic may be regarded as the theory of the relation of logical consequence, and inductive logic as the sway through the Middle Ages and into the seven- theory of another concept [‘‘c’’] which is likewise objec- teenth century, when Francis Bacon (1620) devel- tive and logical, viz., ...degree of confirmation. (43) oped an elaborate account of such reasoning. During the eighteenth and nineteenth centuries, More precisely, the following three fundamental the scope of thinking about induction began to tenets have been accepted by the vast majority of broaden considerably with the description of more proponents as desiderata of modern inductive logic: sophisticated inductive techniques (e.g., those of 1. Inductive logic should provide a quantitative Mill [1843]), and with precise mathematical ac- generalization of (classical) deductive logic. counts of the notion of probability. Intuitive and That is, the relations of deductive entailment quasi-mathematical notions of probability had long and deductive refutation should be captured been used to codify various aspects of uncertain as limiting (extreme) cases with cases of ‘‘par- reasoning in the contexts of games of chance and tial entailment’’ and ‘‘partial refutation’’ statistical inference (see Stigler 1986 and Dale 1999), lying somewhere on a continuum (or range) but a more abstract and formal approach to proba- between these extremes. bility theory would be necessary to formulate the 2. Inductive logic should use probability (in its general modern inductive-logical theories of nonde- modern sense) as its central conceptual build- monstrative inference. In particular, the pioneering ing block. work in probability theory by Bayes (1764), Laplace 3. Inductive logic (i.e., the nondeductive relations (1812), Boole (1854), and many others in the eigh- between propositions that are characterized teenth and nineteenth centuries laid the ground- by inductive logic) should be objective and work for a much more general framework for logical. inductive reasoning. (Philosophical thinking about the possibility of inductive knowledge was most (Skyrms 2000, chap. 2, provides a contemporary famously articulated by David Hume 1739–1740 overview.) In other words, the aim of inductive and 1758) (See Problem of Induction). logic is to characterize a quantitative relation (of The contemporary idea of inductive logic (as a inductive strength or confirmation), c, which satis- general, logical theory of argument evaluation) did fies desiderata 1–3 above. The first two of these not begin to appear in a mature form until the late desiderata are relatively clear (or will quickly be- nineteenth and early twentieth centuries. Some of come clear below). The third is less clear. What the most eloquent articulations of the basic ideas does it mean for the quantitative relation c to be behind inductive logic in this modern sense appear objective and logical? Carnap (1950) explains his in John Maynard Keynes’s Treatise on Probability. understanding as follows: Keynes (1921, 8) describes a ‘‘logical relation be- tween two sets of propositions in cases where it is That c is an objective concept means this: if a certain c not possible to argue demonstratively from one to value holds for a certain hypothesis with respect to a certain evidence, then this value is entirely independent another.’’ Nearly thirty years later, Rudolf Carnap of what any person may happen to think about these (1950) published his encyclopedic work Logical sentences, just as the relation of logical consequence Foundations of Probability, in which he very clearly is independent in this respect. [43] ... The principal explicates the idea of an inductive-logical relation common characteristic of the statements in both fields
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[deductive and inductive logic] is their independence of denote unconditional and conditional probability the contingency of facts [of nature]. This characteristic functions, respectively. Informally (and roughly), justifies the application of the common term ‘logic’ to ‘‘P( p)’’ can be read ‘‘the probability that proposi- both fields. [200] tion p is true,’’ and ‘‘P( p q)’’ can be read ‘‘the j This entry will examine a few of the prevailing probability that proposition p is true, given that modern theories of inductive logic and discuss proposition q is true.’’ The nature of probability how they fare with respect to these three central functions and their relation to the project of induc- desiderata. The meaning and significance of these tive logic will be a central theme in what follows. desiderata will be clarified and the received view about inductive logic critically evaluated. A Naive Version of Basic Inductive Logic and the Received View Some Basic Terminology and Machinery for Inductive Logic According to classical deductive propositional logic, the argument from {P1, ..., Pn}toC is It is often said (e.g., in many contemporary intro- valid iff (‘‘if and only if ’’) the material conditional ductory logic texts) that there are two kinds of P16; ...; 6Pn C is (logically) necessarily argument: deductive and inductive, where the pre- ðtrue. Naively, oneÞ ! might try to define ‘‘inductively mises of deductive arguments are intended to guar- strong’’ as follows: The argument from {P1, ..., antee the truth of their conclusions, while inductive Pn}toC is inductively strong iff the material condi- arguments involve some risk of their conclusions tional P16 ...6Pn C is (logically?) probably being false even if all of their premises are true (see, true. Moreð formally,Þ ! one can express this naive e.g., Hurley 2003). It seems better to say that there inductive logic (NIL) proposal as follows: is just one kind of argument: An argument is a set c C; P1; ...; Pn is high iff of propositions, one of which is the conclusion, the ð f gÞ P P16; ...; 6Pn C is high: rest are premises. There are many ways of evaluat- ðð Þ ! Þ ing arguments. Deductive logic offers strict, quali- There are problems with this first, naive attempt tative standards of evaluation: the conclusion to use probability to generalize deductive validity either follows from the premises or it does not, quantitatively. As Skyrms (2000, 19–22) points out, whereas inductive logic provides a finer-grained there are (intuitively) cases in which the material (and thereby more liberal) quantitative range of conditional P16; ...; 6Pn C is probable but evaluation standards for arguments. One can also ð Þ ! the argument from {P1, ..., Pn}toC is not a define comparative and/or qualitative notions of strong one. Skyrms (21) gives the following exam- inductive support or confirmation. Carnap (1950, ple: }8) and Hempel (1945) both provide penetrating discussions of the contrast between quantitative (P) There is a man in Cleveland who is 1,999 years and and comparative/qualitative notions. For simplici- 11 months old and in good health. (C) No man will live to be 2,000 years old. ty, the focus here will be on quantitative approa- ches to inductive logic, but most of the main issues Skyrms argues that P(P C) is high, simply and arguments discussed below can be recast in because P(C) is high and! not because there is comparative or qualitative terms. any evidential relation between P and C. Indeed, Let {P1, ..., Pn} be a finite set of propositions intuitively, the argument from (P)to(C) is not constituting the premises of an (arbitrary) argument, strong, since (P) seems to disconfirm or counter- and let C be its conclusion. Deductive logic aims support (C). Thus, P P16 ...6Pn C being ðð Þ ! Þ to explicate the concept of validity (i.e., deductive- high is not sufficient for c C; P1; ...; Pn being ð f gÞ logical goodness) of arguments. Inductive logic high. Note also that P P16 ...6Pn C cannot ðð Þ ! Þ aims to explicate a quantitative generalization of serve as c C; P1; ...; Pn ; since it violates desider- ð f gÞ this deductive concept. This generalization is often atum 1. If P1; ...; Pn refutes C, then Pr P16 f g ðð called the ‘‘inductive strength’’ of an argument ...6Pn C Pr Ø P16 ...6Pn)); which is (Carnap 1950 uses the word ‘‘confirmation’’ here). not minimal,Þ! sinceÞ¼ theð ð conjunction of the premises Following Carnap, the notation c(C,{P1, ..., Pn}) of an argument need not have probability one. will denote the degree to which {P1, ..., Pn} jointly Skyrms suggests that the mistake that NIL makes inductively support (or ‘‘confirm’’) C. is one of conflating the probability of the material As desideratum 2 indicates, the concept of conditional Pr P16 ...6Pn C with the con- ðð Þ ! Þ probability is central to the modern project of in- ditional probability of C, given P16 ...6 Pn; that ductive logic. The notation P( ) and P( ) will is, P C P16 ...6Pn :. According to Skyrms, it is j ð j Þ
385 INDUCTIVE LOGIC the latter that should be used as a definition of Following Kolmogorov, define conditional pro- c C; P1; ...; Pn : The reason for this preference bability P( ) in terms of unconditional probabili- ð f gÞ j is that P P16 ...6Pn C fails to capture the ty P( ), as follows: evidentialðð relation betweenÞ ! theÞ premises and conclu- Pr X Y Pr X6Y =Pr Y ; sion, since P P16 ...6Pn C can be high sole- ð j Þ¼ ð Þ ð Þ ðð Þ ! Þ provided that Pr Y 0: ly in virtue of the unconditional probability of (C ) ð Þ 6¼ being high or solely in virtue of the unconditional A probability model M = < B; PM > consists of probability of P16 ... 6Pn being low. As Skyrms a Boolean algebra B of propositions (or sentences (20) stresses, c C; P1; ...; Pn should measure the in some language), together with a particular prob- ð f gÞ ‘‘evidential relation between the premises and the ability function PM( ) over the elements of B. conclusion.’’ This leads Skyrms (and many others) These axioms (and the definition of conditional to defend the following account, which might probability) say what the mathematical properties be called the received view (RV) about inductive of probability models are, but they do not say logic: anything about the interpretation or application of such models. The latter issue is philosophi- c C; P1; ...; Pn Pr C P16 ...6Pn : ð f gÞ¼ ð j Þ cally more central and controversial than the for- The idea that c C; P1; ...; Pn should be mer (but see Popper 1992, appendix *iv, Roeper ð f gÞ identified with the conditional probability of C, and Leblanc 1999, and Ha´jek 2003 for dissenting given P16 ...6Pn; has been nearly universally views on the formal theory of conditional proba- accepted by inductive logicians since the inception bility). There are various ways in which one can of the contemporary discipline. Recent pedagogical interpret or understand probabilities (see Probabil- advocates of the RV include Copi and Cohen ity for a thorough discussion). The two interpreta- (2001), Hurley (2003), and Layman (2002); and tions that are most commonly encountered in the historical champions of various versions of the context of applications to inductive logic are the so- RV include Keynes (1921), Carnap (1950), Kyburg called ‘‘epistemic’’ and ‘‘logical’’ interpretations of (1970), and Skyrms (2000), among many others. probability. There are nevertheless some compelling reasons to doubt the correctness of the RV. These reasons, Epistemic Interpretations of Probability which are analogous to Skyrms’s reasons for reject- In epistemic interpretations of probability, ing the NIL, will be discussed below. But before P (H) is (roughly) the degree of belief that an one can adequately assess the merits of the NIL, M epistemically rational agent assigns to H, according RV, and other proposals concerning inductive to a probability model M of the agent’s epistemic logic, one needs to say more about probability state. A rational agent’s background knowledge models and their relation to inductive logic (see K is assumed (in orthodox theories of epistemic Probability). probability) to be ‘‘included’’ in any epistemic probability model M, and therefore K is assumed Probability: Its Interpretation and Role in to have an unconditional probability of 1 in M. Traditional Inductive Logic PM(H E) is the degree of belief an epistemically rationalj agent assigns to H upon learning that E The Mathematical Theory of Probability is true (or on the supposition that E is true; see For present purposes, assume that a probability Joyce 1999, chap. 6, for discussion), according to a function P( ) is a finitely additive measure func- probability model M of the agent’s epistemic state. tion over a Boolean algebra of propositions (or According to standard theories of epistemic proba- sentences in some formal language). That is, as- bility, agents learn by conditionalizing on evidence. sume that P( ) is a function from a Boolean algebra So, roughly speaking, the probabilistic structure of B of propositions (or sentences) to the unit interval a rational agent’s epistemic state evolves (in time t) [0,1] satisfying the following three axioms (this through a series of probability models {Mt}, where is Kolmogorov’s (1950) axiomatization), for all evidence learned at time t has probability 1 in all propositions X and Y in B: subsequent models Mt0 ; t0 > t: Keynes (1921) seemsf tog be employing an episte- i. P X 0: mic interpretation of probability in his inductive ii. Ifð XÞis a (logically) necessary truth, then logic when he says: P X 1: ð Þ¼ iii. If X and Y are mutually exclusive, then Let our premises consist of any set of propositions h, and P XVY Pr X Pr Y). our conclusion consist of any set of propositions a, then, ð Þ¼ ð Þþ ð
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if a knowledge of h justifies a rational degree of belief in logical interpretations of probability require a a of degree x, we say that there is a probability-relation brief discussion. of degree x between a and h P a h x . 4 ½ ð j Þ¼ ð Þ It is not obvious that the RV can satisfy desidera- Logical Interpretations of Probability tum 3—that c be logical and objective—if the prob- Philosophers who accepted the RV and were ability function P that is used to explicate c in the concerned about the inductive-logical ramifications RV is given an epistemic interpretation of this kind. (mainly, regarding the satisfaction of desideratum 3) After all, whether ‘‘a knowledge of h justifies a of interpreting probabilities epistemically began rational degree of belief in a of degree x’’ seems to to formulate logical interpretations of probability. depend on what one’s background knowledge K is. In such interpretations, conditional probabilities And while this is arguably an objective fact, it also P(X Y) are themselves understood as quantitative seems to be a contingent fact and not something generalizationsj of a logical entailment (or deduci- that can be determined a priori (on the basis of a bility) relation between propositions Y and X. The and h alone). As Keynes (1921) explains, his proba- motivation for this should be clear—it seems like bility function P(a h) is not subjective, since ‘‘once the most direct way to guarantee that an RV-type the facts are givenj which determine our knowledge theory of inductive logic will satisfy desideratum 3. [background and h], what is probable or improbable If P( ) is itself logical, then c( , ), which is de- [viz., a] in these circumstances has been fixed objec- fined byj the RV as P( ), should also be logical, tively, and is independent of our opinion’’ (4). But and the satisfaction of desideratumj 3 (as well as the he later suggests that the function is contingent on other two) seems automatic. Below it will become what the agent’s background knowledge K is, in the clear that RV logical probability is not the only sense that P(a h) can vary ‘‘depending upon the way (and not necessarilyþ the best way) to satisfy the knowledge to whichj it is related.’’ three desiderata for providing an adequate account Carnap (1950, }45B) is keenly aware of this prob- of the logical relation of inductive support. In prep- lem. He suggests that Keynes should have charac- aration, the notion of logical probability must be terized P(a h) as the degree of belief in a that is examined in some detail. justified byj knowledge of h—and nothing else (the Typically, logical interpretations of probability reader may want to ponder what it might mean attempt to define Pr(q p), where p and q are sen- for an agent to ‘‘know h and nothing else’’). As tences in some formalj first-order language L,in Keynes’s remarks suggest (and as Maher 1996 terms of the syntactical features of p and q (in L). explains), the problem is even deeper than this, The most famous logical interpretations of proba- since even a complete specification of an agent’s bility are those of Carnap. It is interesting to note background knowledge K may not be sufficient to that Carnap’s (1950 and 1952) systems are almost pick out a unique (rational) epistemic probability identical to those described 20–30 years earlier by model M for an agent. (Keynes’s reaction to this W. E. Johnson (1921 and 1932) (Paris 1994; was to conclude that sometimes quantitative judg- Kyburg 1970, Ch. 5). His later work (Carnap ments of inductive strength or degree of condition- 1971 and 1980) became increasingly complicated, al probability are not possible and that in these involving two-dimensional continua, and was less cases one must settle for qualitative or comparative tightly coupled with the syntax of L (Maher 2000 judgments.) The problem here is that ‘‘P(X K )’’ and 2001; Skyrms 1996 discusses some recent (‘‘the probability of X, given background knowl-j applications of Carnapian techniques to Bayesian edge K ’’) will not (in general) be determined unless statistical models involving continuous random an epistemic probability model M is specified, variables; Glaister 2001 and Festa 1993 provide which (a fortiori) gives PrM(X), for each X in M. broad surveys of Carnapian theories of logical And, without a determination of these fundamen- probability and inductive logic). tal or a priori probabilities PM (X), a general (quan- Begin with a standard first-order logical language titative) theory of inductive logic based on L containing a finite number of monadic predicates epistemic probabilities seems all but hopeless. F, G, H, ... and a finite or denumerable number This raises the problem of specifying an appropri- of individual constants a, b, c, .... Define an un- ate a priori probability model M. Keynes (1921, conditional probability function P( ) over the sen- chap. 4) and Carnap (see below) both look to the tences of L. Finally, following the standard principle of indifference at this point, as a guide to Kolmogorovian approach, construct a conditional choosing a priori probability models. Before dis- probability function P( ) over pairs of sentences cussing the role of the principle of indifference, of L, using the ratioj definition of conditional
387 INDUCTIVE LOGIC probability given above. To fix ideas, consider a Pr Fa6Ga6ØGb ð Þ very simple toy language L with only two monadic Pr Fa6Ga6Fb6ØGb V Fa6Ga6ØFb6ØGb ¼ ðð Þ ð ÞÞ predicates, F and G and only two individual con- Pr Fa6Ga6Fb6ØGb Pr Fa6Ga6ØFb6ØGb : ¼ ð Þþ ð Þ stants a and b. In this language, there are only Now, it is only a brief step to the definition of the sixteen possible states of the world that can be conditional probability function P( ) over pairs described. These sixteen maximally specific descrip- of sentences in L. Using the standard,j Kolmogor- tions are called the state descriptions of L, and they ovian ratio definition of conditional probability, are as follows: for all pairs of sentences X, Y in L: Fa6Ga6Fb6Gb ØFa6Ga6Fb6Gb P X Y P X6Y =Pr Y ; provided that P Y 0: ð j Þ¼ ð Þ ð Þ ð Þ 6¼ Fa6Ga6Fb6ØGb ØFa6Ga6Fb6ØGb Thus, once the unconditional probability func- Fa6Ga6ØFb6Gb ØFa6Ga6ØFb6Gb tion P( ) is specified for the state descriptions of a Fa6Ga6ØFb6ØGb ØFa6Ga6ØFb6ØGb language L, all probabilities both conditional and Fa6ØGa6Fb6Gb ØFa6ØGa6Fb6Gb unconditional are thereby determined over L. And, Fa6ØGa6Fb6ØGb ØFa6ØGa6Fb6ØGb this gives one a logical probability model M over Fa6ØGa6ØFb6Gb ØFa6ØGa6ØFb6Gb the language L. The unconditional, logical pro- Fa6ØGa6ØFb6ØGb ØFa6ØGa6ØFb6ØGb bability functions so defined are typically called Two state descriptions S and S are said to be measure functions. Carnap (1950) discusses two 1 2 ‘‘natural’’ measure functions. permutations of each other if S1 can be obtained { from S by some permutation of the individual The first Carnapian measure function is m , 2 which assumes that each of the state descriptions constants. For instance, Fa6ØGa6ØFb6Gb can is equiprobable a priori: If there are N state descrip- be obtained from ØFa6Ga6Fb6ØGb by permut- tions in L, then m{ assigns 1 to each state descrip- ing a and b. Thus, Fa6ØGa6ØFb6Gb and ØFa6 N tion. While this may seem like a very natural Ga6Fb6ØGb are permutations of each other (in L). A structure description in L is a disjunction of measure function, since it applies something like the principle of indifference to the state descriptions state descriptions, each of which is a permutation { of the others. In the toy language L, there are the of L (see below for discussion), m has the conse- following ten structure descriptions: quence that the resulting probabilities cannot reflect learning from experience. Consider the following Fa6Ga6Fb6Gb Fa6ØGa6ØFb6Gb ð Þ simple example. Assume that one adopts a logical V ØFa6Ga6Fb6ØGb probability function P( ) based on m{ as one’s own a ð Þ Fa6Ga6Fb6ØGb (Fa6ØGa6ØFb6ØGb priori degree of belief (or credence) function. Then, ð Þ Þ V Fa6ØGa6Fb6Gb V ØFa6ØGa6Fb6ØGb one learns (by conditionalizing) that an object a is F, ð Þ ð Þ that is, Fa. Intuitively, one’s conditional degree of Fa6Ga6ØFb6Gb ØFa6Ga6ØFb6Gb ð Þ credence P(Fb Fa) that a distinct object b also is F, V ØFa6Ga6Fb6Gb j ð Þ given that a is F, should not always be the same as Fa6Ga6ØFb6ØGb ØFa6Ga6ØFb6ØGb one’s a priori degree of credence that b is F. That is, ð Þ ð Þ V ØFa6ØGa6Fb6Gb V ØFa6ØGa6ØFb6Gb the fact that one has observed another F object ð Þ ð Þ should (at least in some cases) make it more proba- Fa ØGa Fb ØGb ØFa ØGa ØFb ØGb 6 6 6 6 6 6 ble (a posteriori) that b will also be F (i.e., more Now assign nonnegative real numbers to the state probable than Fb was a priori). More generally, if descriptions, so that these sixteen numbers sum to 1. oneobservesthatalargenumberofobjectshavebeen Any such assignment will constitute an uncondi- F, this should raise the probability that the next tional probability function P( ) over the state object one observes will also be F. Unfortunately, descriptions of L. To extend P( ) to the entire lan- no a priori probability function based on m{ is con- guage L, stipulate that the probability of a disjunc- sistent with learning from experience in either sense. tion of mutually exclusive sentences is the sum of the To see this, consider the simple case Pr(Fb Fa): j probabilities of its disjuncts. Since every sentence P Fb Fa m{ Fb6Fa =m{ Fa in L is equivalent to some disjunction of state ð j Þ¼1 ð Þ ð Þ m{ Fb Pr Fb : descriptions, and every pair of state descriptions ¼ 2 ¼ ð Þ¼ ð Þ is mutually exclusive, this gives a complete un- So, if one assumes an a priori probability function conditional probability function P( ) over L. For based on m{, the fact that one object has property F instance, since Fa6Ga6ØGb is equivalent to cannot affect the probability that any other object the disjunction Fa6Ga6Fb6ØGb V Fa6Ga6 will also have property F. Indeed, it can be shown ð Þ ð ØFb6ØGb ; one will have: (Kyburg 1970, 58–59) that no matter how many Þ
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objects are assumed to be F,thiswillbeirrelevant from which the definitions of P( ) in terms of m* (according to probability functions based on m{)to and m{ fall out as special cases. Thisj continuum of the hypothesis that a distinct object will also be F. conditional probability functions depends on a pa- The fact that (on the probability functions rameter l, which is supposed to reflect the ‘‘speed’’ generated by the measure m{) no object’s having with which learning from experience is possible. In certain properties can be informative about other this continuum, l 0 corresponds to the ‘‘straight objects also having those same properties has been rule’’ of induction,¼ which says that the probability viewed as a serious shortcoming of m{ (Carnap that the next object observed will be F, conditional 1955). As a result, Carnap formulated an alterna- upon a sequence of past observations, is simply the tive measure function m*, which is defined as frequency with which F objects have been observed follows. First, assign equal probabilities to each in the past sequence; l yields a conditional structure description (in the toy language above, n probability function much¼þ like1 that given above by 1 { 10). Then, each state description belonging to a assuming the underlying logical measure m (i.e., given structure description is assigned an equal l implies that there is no learning from ¼þ1 portion of the probability assigned to that struc- experience). And setting l k (where k is the num- ture description). For instance, in the toy lan- ber of independent families¼ of predicates in Car- guage, the state description Fa6Ga6ØFb6Gb nap’s more elaborate 1952 linguistic framework) 1 1 gets assigned an a priori probability of 20 (2 of yields a conditional probability function equivalent 1 10), but the state description Fa6Ga6Fb6Gb to that generated by the measure function m*. 1 1 1 receives an a priori probability of 10 (1 of 10). But even this l– continuum has problems. First, To further illustrate the differences between m{ and none of the Carnapian systems allow universal gen- m*, here are some numerical values in the toy eralizations to have nonzero probability. This language L: problem was addressed by Hintikka (1966) and Hintikka and Niiniluoto (1980), who provided var- ious alterations of the Carnapian framework that Measure function m{ Measure function m* allow for nonzero probabilities of universal gener- alizations. Moreover, Carnap’s early systems did 1 mà Fa6Ga m{ Fa6Ga6ØFb6Gb ð not allow for the probabilistic modeling of analog- 16 1 ð Þ¼ 6Fb6Gb Þ¼10 ical effects. That is, in his 1950–1952 systems, the fact that two objects share several properties in m{ Fa6Ga6ØFb6Gb mà Fa6Ga6ØFb common is always irrelevant to whether they ðð Þ 1 ð 1 share any other properties in common. Carnap’s V ØFa6Ga6Fb6Gb 6Gb ðð ÞÞÞ ¼ 8 Þ¼20 more recent (and most complex) theories of logical probability (1971, 1980) include two additional { 1 1 adjustable parameters ( and ), designed to pro- m Fa mà Fa g ð Þ¼2 ð Þ¼2 vide the theory with enough flexibility to overcome these (and other) limitations. Unfortunately, no { 1 3 1 Pr Fa Fb mà Fa Prà Fa Fb > Carnapian logical theory of probability to date ð j Þ¼2 ¼ ð Þ ð j Þ¼5 2 { has successfully dealt with the problem of analogi- Pr Fa mà Fa Prà Fa ¼ ð Þ ¼ ð Þ¼ ð Þ cal effects (Maher 2000 and 2001). Moreover, as Putnam (1963) explains, there are further (and Unlike m{, m* can model learning from experience, some say deeper) problems with Carnapian (or, since in the simple language more generally, syntactical) approaches to logical probability, if they are to be applied to inductive 3 1 P Fa Fb > Pr Fa inference generally. The consensus now seems to be ð j Þ¼5 2 ¼ ð Þ that the Carnapian project of characterizing an if the probability function P is defined in terms of adequate logical theory of probability is (by his the logical measure function m*. Although m* does own standards and lights) not very promising have some advantages over m{, even m* can give (Putnam 1963; Festa 1993; Maher 2001). counterintuitive results in more complex languages This discussion has glossed over technical details (Carnap 1952). in the development of (Carnapian) logical interpre- Carnap (1952) presents a more complicated frame- tations or theories of probability since 1950. To work, which describes a more general class (or recapitulate, what is important for present purposes ‘‘continuum’’) of conditional probability functions, is that Carnap (along with the other advocates
389 INDUCTIVE LOGIC of logical probability) was an RV theorist about that these relations are logically alike. These relations are inductive logic. He identified the concept c( , )of obviously alike if the evidence has a symmetrical structure inductive strength (or inductive support) with the with respect to their possible events. The statement of equi- concept of conditional probability P( ). And he probability asserts nothing more than the symmetry. (22) thought (partly because of the problemsj he saw with epistemic interpretations) that in order for Carnap seems to be saying that the principle of an RV account to satisfy desideratum 3, it needed indifference is to be applied only to possible events to presuppose a logical interpretation (or theory) that exhibit certain a priori symmetries with respect of probability. This led him, initially, to develop to some rational agent’s background evidence. But various logical measures (e.g., the a priori logical this appears no more logical than Keynes’s episte- probability functions m{ and m*), and then to de- mic approach to probability. It seems that the resulting probabilities P( ) will not be logical in fine conditional logical probability Pr( ) in terms j of these underlying a priori logical measures,j using the sense Carnap desired (at least no more so than the standard ratio definition. This approach ran Keynes’s epistemic probabilities were), unless Car- into various problems when it came to the applica- nap can motivate—on logical grounds—the choice tion of P( ) to inductive logic. These difficulties of an a priori probability model. To that end, mainly hadj to do with the ability of Carnap’s Carnap’s application of the principle of indiffer- P( ) to undergird learning from experience and/ ence is not very useful. Recall that the goal of or certainj kinds of analogical reasoning (for other Carnap’s project (of inductive logic) was to expli- philosophical objections to Carnap’s logical prob- cate the confirmation relation, which is itself sup- ability project, see Putnam 1963). In response to posed to reflect the evidential relation between these difficulties, Carnap began to fiddle directly premises and conclusions (Carnap 1950 uses the with the definition of P( ). In 1952, he moved to a locutions ‘‘degree of confirmation’’ and ‘‘weight parameterized definition ofj P( ), which contained of evidence’’ synonymously). How is one to under- j stand what it means for evidence not to ‘‘favor any an ‘‘index of inductive caution’’ (l) that was sup- posed to regulate the speed with which learning of the possible events’’ in a way that does not from experience is reflected by P( ). Later, Car- require one to already understand how to measure j the degree to which the evidence confirms each of nap (1971, 1980) added g and to the definition of P( ), as noted above, in an attempt to further the possible events? Here, Carnap’s discussion of generalizej the theory and allow for sensitivity to the principle of indifference presupposes that de- certain kinds of analogical effects. Ultimately, no gree of confirmation is to be identified with degree such theory was ever viewed by Carnap (or others) of conditional probability. In that reading, ‘‘not as fully adequate for the purposes of grounding an favoring’’ just means ‘‘conferring equal probability RV conception of inductive logic. on,’’ and Carnap’s unpacking of the principle of At this point, it is important to ask, In what sense indifference reduces directly to a mathematical are Carnap’s theories of logical probability (especial- truth (which, for Carnap, is good enough to render ly his later ones) logical ? His early theories (based on the principle logical ). If one had independent the measure functions m{ and m*) applied something grounds for thinking that conditional probabilities like the principle of indifference to the state and/or were the right way to measure confirmation (or structure descriptions of the formal language L in weight of evidence), then Carnap would have a rath- order to determine the logical probabilities P( ). er clever (albeit not terribly informative) way to In this sense, these early theories assume that certainj (logically) ground his choice of a priori probability sentences of L are equiprobable a priori. Why is such models. Unfortunately, as will be seen below, there an assumption logical ? Or, more to the point, how are independent reasons to doubt Carnap’s presup- is logic supposed to tell one which statements are position here that degree of confirmation should be equiprobable a priori ? Carnap (1955) explains that identified with degree of conditional probability. Without that assumption, Carnap’s principle of in- the statement of equiprobability to which the principle of difference is no longer logical (by his own lights), indifference leads is, like all other statements of inductive and the problem of the contingency (nonlogicality) probability, not a factual but a logical statement. If the of the ultimate inductive-logical probability assign- knowledge of the observer does not favor any of the possi- ments returns with a vengeance. There are indepen- ble events, then with respect to this knowledge as evi- dent and deep problems with any attempt to dence they are equiprobable. The statement assigning consistently apply the principle of indifference to equal probabilities in this case does not assert anything about the facts, but merely thelogicalrelationsbetween contexts in which hypotheses and/or evidence in- the given evidence and each of the hypotheses; namely, volve continuous magnitudes (van Fraassen 1989).
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Carnap’s later theories of P( ) introduce even as an account of inductive strength or inductive further contingencies, in the formj of adjustable support? This will lead to a fourth material desid- parameters, the ‘‘proper values’’ of which do not eratum for measures of inductive support, and seem to be determinable a priori (Carnap 1952, ultimately to a concrete alternative to the RV. 1971, 1980). In particular, consider Carnap’s (1952) l-continuum. The parameter l is supposed Rethinking the Received View to indicate how sensitive P( ) is to learning from j experience. A higher value of l indicates slower How to Ensure the Transparent Satisfaction of learning, and a lower l indicates faster learning. Desideratum 3 As Carnap (1952) concedes, no one value of l is The existing attempts to use the notion of prob- best a priori. Presumably, different values of l are ability to explicate the concept of inductive support appropriate for different contexts in which confir- (or inductive strength) c have foundered on the mational judgments are made (see Festa 1993 for a question of their contingency (which threatened contextual Carnapian approach to confirmation). violation of desideratum 3). It may be that these It seems that the same must be said for the addi- contingencies can be eliminated (in general) only by tional parameters g and (Carnap 1971, 1980). The making the notion of inductive support explicitly moral here seems to be that it is only relative to a relational. To follow such a plan, in the case of the particular assignment of values to l, g, and that RV one should rather say: probabilistic (and/or confirmational) judgments The inductive strength of the argument from are objectively and noncontingently determined in {P1, ..., Pn} to C relative to a probability model Carnap’s later systems. This is analogous to the M
391 INDUCTIVE LOGIC sentences in a formal language. The deductive logi- say that there is a strong evidential relation be- cian is not in the business of telling people which tween P and C? According to proponents of the languages they should use, since this (presumably RV, one should say just that. This seems wrong, pragmatic) question is ‘‘external’’ to deductive because intuitively PM C P PM C : That is, ð j Þ¼ ð Þ logic. However, once a language has been specified, PM(C P) is high solely because PM(C) is high, the deductive relations among sentences in that and notj because of any evidential relation between language are determined objectively and noncon- P and C. This is the same kind of criticism that tingently, and it is up to the deductive logician to Skyrms (2000) made against the NIL proposal. explicate these relations. In the approach to induc- And it is just as compelling here. The problem tive logic just described, the same sort of thing can here is that P is irrelevant to C. Plausibly, it seems be said for the inductive logician. It is not the that if P is going to be counted as providing evi- business of the inductive logician to tell people dence in favor of C, then P should raise the proba- which probability models they should use (presum- bility of C (Popper 1954 and 1992; Salmon 1975). ably, that is an epistemic or pragmatic question), This leads to the following fourth material desider- but once a probability model is specified, the in- atum for c: ductive logical relations in that model (viz., c) are . c(C,{P , ..., P }) should be sensitive to the determined objectively and noncontingently. In the 1 n probabilistic relevance of P ... P to C. present approach, the duty of the inductive logician 1 6 6 n is (simply) to explicate the c-function—not to de- In particular, desideratum 4 implies that if P1 cide which probability models should be used in raises the probability of C1, but P2 lowers the which contexts. probability of C , then c C1; P1 > c C2; P2 : This 2 ð Þ ð Þ One last analogy might be useful here. When the rules out P C P16 ...6Pn as a candidate for ð j Þ theory of special relativity came along, some people c C; P1; ...; Pn ; and it is therefore inconsistent were afraid that it might introduce an element of withð f the RV. ManygÞ nonequivalent probabilistic- subjectivity into physics, since the velocities of relevance measures of support (or confirmation) objects were now determined only relative to a satisfying desideratum 4 have been proposed and frame of reference. There was no physical ether defended in the philosophical literature (Fitelson with respect to which objects received their abso- 1999 and 2001). lute velocities. However, the velocities and other One can combine desiderata 1–4 into the follow- values were determined objectively and noncontin- ing single probabilistic inductive logic. This unified gently once the frame of reference was specified, desideratum gives constraints on a three-place prob- which is the reason Einstein originally intended to abilistic confirmation function c C; P1; ...; Pn ; ð f g call his theory the theory of invariants. Similarly, it M ; which is the degree to which {P1, ..., Pn} seems that there may be no logical ether with re- inductivelyÞ supports C, relative to a specified prob- spect to which pairs of propositions (or sentences) ability model M
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. the measure c is defined relative to a proba- confirmation as increase in firmness, where the bility model, and so its values are determined former is properly explicated using just conditional objectively and noncontingently by the values probability (a` la the RV) and does not require in the specified model; and relevance of the premises to the conclusion, while . sensitivity to P-relevance is built into the de- the latter presupposes that the premises are proba- sideratum (probabilistic inductive logic). bilistically relevant to the conclusion. Strangely, Carnap does not even mention Kemeny and Interestingly, almost all relevance measures pro- Oppenheim’s measure (of which he was aware) as posed in the confirmation theory literature fail to a proper measure of confirmation as increase in satisfy probabilistic inductive logic (Fitelson 2001, firmness. Instead, he suggests for that purpose a 3.2.3). One historical measure that does satisfy } relevance measure that does not satisfy desidera- probabilistic inductive logic was independently tum 1 and so is not even a proper generalization of defended by Kemeny and Oppenheim (1952) as deductive entailment. This puzzling but crucial se- the correct measure of confirmation (in opposition quence of events in the history of inductive logic to Carnap’s RV c-measures) within a Carnapian may explain why relevance-based approaches (like framework for logical probability: that of Kemeny and Oppenheim) have never c C; P1; ...; Pn ; M enjoyed as many proponents as the RV. ð f g Þ¼ PM P16 ...6Pn C PM P16 ...6Pn ØC BRANDEN FITELSON ð j ÞÀ ð j Þ : PM P16 ...6Pn C PM P16 ...6Pn ØC ð j Þþ ð j Þ References Indeed, of all the historically proposed (probabi- listic) measures of degree of confirmation (and Bacon, F. (1620), The Novum Organon. Oxford: The Uni- versity Press. there have been dozens), the above measure is the Bayes, T. (1764), ‘‘An Essay Towards Solving a Problem in only one (up to ordinal equivalence) that satisfies the Doctrine of Chances,’’ Philosophical Transactions of all four of the material desiderata. The four simple the Royal Society of London 53. desiderata are thus sufficient to (nearly uniquely) Boole, G. (1854), An Investigation of the Laws of Thought, determine the desired explicandum c, or the degree on Which Are Founded the Mathematical Theories of Logic and Probabilities. London: Walton & Maberly. of inductive strength of an argument. There are Carnap, R. (1950), Logical Foundations of Probability. Chi- other measures in the literature, such as the log- cago: University of Chicago Press. likelihood ratio, that differ conventionally from, ——— (1952), The Continuum of Inductive Methods. Chi- but are ordinally equivalent to, the above measure cago: University of Chicago Press. (for various other virtues of measures in this fami- ——— (1955), Statistical and Inductive Probability and In- ductive Logic and Science (leaflet). Brooklyn, NY: Galois ly, see Fitelson 2001, Good 1985, Heckerman 1988, Institute of Mathematics and Art. Kemeny and Oppenheim 1952, and Schum 1994). ——— (1962), Logical Foundations of Probability, 2nd ed. Chicago: University of Chicago Press. ——— (1971), ‘‘A Basic System of Inductive Logic, I,’’ in Historical Epilogue on the Relevance of R. Carnap and R. Jeffrey (eds.), Studies in Inductive Relevance Logic and Probability, vol. 1. Berkeley and Los Angeles: University of California Press, 33–165. In the second edition of Logical Foundations of ——— (1980), ‘‘A Basic System of Inductive Logic, II,’’ in Probability, Carnap (1962) acknowledges that R. Jeffrey (ed.), Studies in Inductive Logic and Probabili- probabilistic relevance is an intuitively compelling ty, vol. 2. Berkeley and Los Angeles: University of Cali- fornia Press, 7–155. desideratum for measures of inductive support. Copi, I., and C. Cohen (2001), Introduction To Logic, 11th This acknowledgement was in response to the tren- ed. New York: Prentice Hall. chant criticisms of Popper (1954), who was one of Dale, A. (1999), A History of Inverse Probability: From the first to urge relevance as a desideratum in Thomas Bayes To Karl Pearson, 2nd ed. New York: this context (see Michalos 1971 for a thorough Springer-Verlag. Festa, R. (1993), Optimum Inductive Methods. Dordrecht, discussion of this important debate between Pop- Netherlands: Kluwer Academic Publishers. per and Carnap). But instead of embracing rele- Fitelson, B. (1999), ‘‘The Plurality of Bayesian Measures of vance measures like Kemeny and Oppenheim’s Confirmation and the Problem of Measure Sensitivity,’’ (1952) (and rewriting much of the first edition of Philosophy of Science 66: S362–S378. Logical Foundations of Probability), Carnap (1962) ——— (2001), Studies in Bayesian Confirmation Theory. PhD. dissertation, University of Wisconsin–Madison simply postulates an ambiguity in the term ‘‘confir- (Philosophy). mation.’’ He now argues that there are two kinds Glaister, S. (2001), ‘‘Inductive Logic,’’ in D. Jacquette (ed.), of confirmation: confirmation as firmness and A Companion to Philosophical Logic. London: Blackwell.
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Good, I. J. (1985), ‘‘Weight of Evidence: A Brief Survey,’’ in Laplace, P. S. M. d. (1812), The´orie Analytique des Prob- J. Bernardo, M. DeGroot, D. Lindley, and A. Smith abilite´s.. Paris: Ve. Courcier. (eds.), Bayesian Statistics, 2. Amsterdam: North-Holland, Layman, C. S. (2002), The Power of Logic, 2nd ed. New 249–269. York: McGraw-Hill. Ha´jek, A. (2003), ‘‘What Conditional Probabilities Could Maher, P. (1996), ‘‘Subjective and Objective Confirma- Not Be,’’ Synthese 137: 273–323. tion,’’ Philosophy of Science 63: 149–174. Heckerman, D. (1988), ‘‘An Axiomatic Framework for Be- ——— (2000), ‘‘Probabilities for Two Properties,’’ Erkennt- lief Updates,’’ in L. Kanal and J. Lemmer (eds.), Uncer- nis 52: 63–91. tainty in Artificial Intelligence 2. New York: Elsevier ——— (2001), ‘‘Probabilities for Multiple Properties: The Science Publishers, 11–22. Models of Hesse and Carnap and Kemeny,’’ Erkenntnis Hempel, C. (1945), ‘‘Studies in the Logic of Confirmation,’’ 55: 183–216. parts I and II, Mind 54: 1–26 and 97–121. Michalos, A. (1971), The Popper–Carnap Controversy. The Hintikka, J. (1966), ‘‘A Two-Dimensional Continuum of Hague: Martinus Nijhoff. Inductive Methods,’’ in J. Hintikka and P. Suppes Mill, J. (1843), A System of Logic, Ratiocinative and Induc- (eds.), Aspects of Inductive Logic. Amsterdam: North- tive, Being a Connected View of the Principles of Evidence Holland. and the Methods of Scientific Investigation. London: Hintikka, J., and I. Niiniluoto (1980), ‘‘An Axiomatic Parker. Foundation for the Logic of Inductive Generalization,’’ Paris, J. (1994), The Uncertain Reasoner’s Companion: A in R. Jeffrey, Studies in Inductive Logic and Probability, Mathematical Perspective. Cambridge: Cambridge Uni- vol. 2. Berkeley and Los Angeles: University of Califor- versity Press, chap. 12. nia Press. Popper, K. (1954), ‘‘Degree of Confirmation,’’ British Jour- Hume, D. (1739– 1740), A Treatise of Human Nature: Being nal for the Philosophy of Science 5: 143–149. an Attempt to Introduce the Experimental Method of ——— (1992), The Logic of Scientific Discovery. London: Reasoning into Moral Subjects, vols. 1–3. London: John Routledge. Noon (1739), Thomas Longman (1740). Putnam, H. (1963), ‘‘‘Degree of Confirmation’ and Induc- ——— (1758), An Enquiry Concerning Human Understand- tive Logic,’’ in P. A. Schilpp (ed.), The Philosophy of ing in Essays and Treatises on Several Subjects. London: Rudolf Carnap. La Salle, IL: Open Court Publishing, A. Millar. 761–784. Hurley, P. (2003), A Concise Introduction to Logic, 8th ed. Roeper, P., and H. Leblanc (1999), Probability Theory and Melbourne, Australia, and Belmont, CA: Wadsworth/ Probability Logic. Toronto: University of Toronto Press. Thomson Learning. Salmon, W. C. (1975), ‘‘Confirmation and Relevance,’’ in Johnson, W. E. (1921), Logic. Cambridge: Cambridge Uni- G. Maxwell and R. M. Anderson Jr. (eds.), Induction, versity Press. Probability, and Confirmation: Minnesota Studies in the ——— (1932), ‘‘Probability: The Deductive and Inductive Philosophy of Science, vol. 6. Minneapolis: University of Problems,’’ Mind 49: 409– 423. Minnesota Press, 3–36. Joyce, J. (1999), The Foundations of Causal Decision Theory. Schum, D. (1994), The Evidential Foundations of Probabilis- Cambridge: Cambridge University Press. tic Reasoning. New York: John Wiley & Sons. Kemeny, J., and P. Oppenheim (1952), ‘‘Degrees of Factual Skyrms, B. (1996), ‘‘Carnapian Inductive Logic and Bayes- Support,’’ Philosophy of Science 19: 307–324. ian Statistics,’’ in Statistics, Probability and Game Theo- Keynes, J. (1921), A Treatise on Probability. London: Mac- ry: Papers in Honor of David Blackwell. IMS Lecture millan. Notes, Monograph Series 30: 321–336. Kneale, W., and M. Kneale (1962), The Development of ——— (2000), Choice and Chance. Melbourne, Australia, Logic. Oxford: Clarendon Press. and Belmont, CA: Wadsworth/Thomson Learning. Kolmogorov, A. (1950), Foundations of the Theory of Prob- Stigler, S. (1986), The History of Statistics. Cambridge, ability. New York: Chelsea. MA: Harvard University Press. Kyburg, H. E. (1970), Probability and Inductive Logic. Lon- van Fraassen, B. (1989), Laws and Symmetry. Oxford: don: Macmillan. Oxford University Press, chap. 12.
INNATE/ACQUIRED DISTINCTION
Arguments about innateness center on two distinct introduced in Plato’s Meno and it took center stage but overlapping theoretical issues. One concerns in seventeenth- and eighteenth-century debates be- the explanation of the origin of ideas in the tween rationalists and empiricists. More recently, it human mind. This ancient question was famously has seen a sophisticated revival in arguments about
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