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Inductive Logic 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 384 INDUCTIVE LOGIC [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 ..
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