Is the Humean defeated by induction? A reply to Smart. (What is a Law of Nature? Cambridge University Press, Cambridge, 1983; Dialogue 30

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1 This is a copy of the accepted manuscript for publication. Is the Humean defeated by induction? A reply to Smart Abstract: This paper is a reply to Benjamin Smart s (Philos Stud 162 (2): , 2013) recent objections to David Armstrong s solution to the problem of induction (What is a Law of Nature? Cambridge University Press, Cambridge, 1983; Dialogue 30 (4): , 1991). To solve the problem of induction, Armstrong contends that laws of nature are the best explanation of our observed regularities, where laws of nature are dyadic relations of necessitation holding between first-order universals. Smart raises three objections against Armstrong s pattern of inference. First, regularities can explain our observed regularities; that is, universally quantified conditionals are required for explanations. Second, if Humean s pattern of inference is irrational, then Armstrong s pattern of inference is also irrational. Third, universal regularities are the best explanation of our observed regularities. I defend Armstrong s solution of induction, arguing against these three claims. Key words: Armstrong, Explanation, Humeanism, Induction, Laws of Nature. 1 Introduction The problem of induction is the problem of justifying inductive inferences, such as the following: (1) All observed ravens are black. All ravens are black. David Armstrong (1983: cap. 4, 6; 1991) argues that the theory of nomic necessity 1

2 makes it possible to solve the problem of induction. To make this feasible, Armstrong follows Gilbert Harman s (1965) interpretation of induction. Induction is described as a special case of inference to the best explanation. (1) All observed ravens are black. (2) The best explanation for (1) is that it is a law of nature that all ravens are black. All ravens are black. According to Armstrong s proposal, laws of nature, symbolized by N(R,B), are firstorder universals, that is, laws of nature are dyadic relations of necessitation, N, holding between first-order universals (R and B). This natural necessitation between universals is the best explanation for our observed regularities. Recently, Armstrong s proposal has been challenged by some regularists. In general, the regularity theorist s strategy has two steps. First, the problem of induction is interpreted in a slightly different way from Armstrong s interpretation of the problem of induction. Second, an alleged missing link or extra step is inserted in Armstrong s solution to block Armstrong s justification of induction. For example, Beebee (2011) considers that what it is really required to explain is why all observed R s have been B s, so far. The expression so far is Beebee s new ingredient of the problem of induction. Then, she advances a competing explanation for this explanandum, a time-limited necessary connection. She argues that Armstrong s solution to the problem of induction, based on timeless necessity, is no longer valid because there is an alleged illicit inductive premise in Armstrong s solution, i.e., an extra step. (See Castro (2014) for a reply). 2

3 Smart follows this same strategy. Smart cooks up two universals: OR means observed-raven and OB means observed-black. For him, what it is really required to explain is why all observed-ravens are observed-black. These universals are Smart s new ingredient in the problem of induction. Three objections are raised from these cooked universals. First, Armstrong s solution to the problem of induction requires universally quantified conditionals to have explanatory value. Second, Armstrong s own pattern of inference fails by his own standards. Third, Armstrong s explanations are not the best explanations. This objection tries to argue that universal regularities are the best explanation of our observed regularities. Basically, Smart s ultimate claim is to show that Armstrong s inductive inferences are no better justified that the Humean s. The remainder of this paper is organised as follows. First, the solution of Armstrong to the problem of induction is presented. Second, the discussion follows. Smart s three objections to Armstrong s solution to the problem of induction are answered individually. 1 2 Armstrong s solution to the problem of induction Armstrong s solution to the problem of induction has two steps: the bottom-up step and the top-down step. The bottom-up step is a step of inference to the best explanation (IBE) that goes from the observational facts to laws. It is a law of nature that all Rs are Bs ; N(R,B) is the best explanation that all observed Rs are Bs. N(R,B) is the truthmaker of the law-statement that best explains our observations. The top-down step is an analytical or conceptual step. It goes from laws to observational facts. The law of nature that all Rs are Bs, N(R,B), entails and explains that all Rs are Bs, x(rx Bx), the law-statement, where all Rs are Bs is logically equivalent to all observed Rs are 1 I follow Smart s letters R and B instead of the traditional letters F and G. 3

4 Bs and all unobserved Rs are Bs. Here is a reconstruction of Armstrong s chain of explanation, where O refers to the predicate O_:_is observed. Bottom-up: Top-down: x((ox Rx) Bx) IBE N(R,B) N(R,B) x(rx Bx) where x(rx Bx) [ x((ox Rx) Bx) x(( Ox Rx) Bx)]. 3 First objection Let R, OR, B and OB be universals. R means raven, OR means observed-raven, B means black and OB means observed-black. Three relations may then be established: (1) N(OR,OB) x(orx OBx). (2) N(R,B) x(rx Bx). (3) N(R,B) x(orx OBx). According to Smart, propositions (1) and (2) follow Armstrongian logic, that is, a necessitation relation between two universals entails the universally quantified conditional between those two universals. However, proposition (3) does not follow Armstrongian logic, that is, a necessitation relation between two universals cannot entail the universally quantified conditional between two completely distinct universals (Smart 2013: 323). OR and OB are not subsets of the properties R and B, respectively. 2 2 Supposedly, for the Humean (or at least, for the nominalist Humean), OR and OB are subsets of R and B. 4

5 Armstrong s solution to the problem of induction fails because x(orx OBx) does not follow from N(R,B). That is, from it is a law of nature that all ravens are black, it cannot be deduced that all observed-ravens are observed-black. According to Smart, the necessitarian may try to solve this issue by suggesting that N(R,B) explains x(orx OBx). Smart justifies this explanation by claiming that N(R,B) does necessitate (in Armstrong s contingent necessitation sense) x(rx Bx). Thus, actually, there is a missing link between N(R,B) and x(orx OBx): it is missing x(rx Bx). At the end of the day, the necessitarian is committed to the following chain of explanation: 1. N(R,B) 2. x(rx Bx) 3. x(orx OBx) To explain our observations, Smart confronts the necessitarian with the dilemma: [The necessitarian] needs either to accept OR and OB as natural properties, and claim the natural necessitation relation stands between these universals at our world, or that regularities can bridge the gap between the proposed explanand and the explanandum. (Smart 2013: 324) 3 It is widely assumed that for an Armstrongian (and surely for anyone), OR and OB are inacceptable as natural properties. Thus, the necessitarian should follow the second horn of the dilemma. If this is the case, actually, x(rx Bx) explains (ORx OBx). Smart 3 There is a typo in the original paper. I wrote OB instead of BR. 5

6 (2013: 324) concludes that without accepting that universally quantified statements have explanatory value, there can be no explanatory relation between N(R,B) and the colour of observed ravens. 4 Reply to the first objection My reply has three points. First, I discuss the relations of explanation between N(R,B) and x(orx OBx) and between x(rx Bx) and x(orx OBx). I argue that neither N(R,B) nor x(rx Bx) explains x(orx OBx). Second, I discuss the entailment between N(R,B) and x(orx OBx). Third, I argue against the existence of the putative universals observed-raven and observed-black. 4 Thus, our explanandum is all observed ravens are black, x((ox Rx) Bx), instead of all observed-ravens are observed-black, x(orx OBx). In this section, to make the discussion more intelligible, I need to introduce a character, a particular sort of idealized necessitarian called absurd necessitarian. The absurd necessitarian accepts Smart s challenge. He wants to explain why all observedravens are observed-black, x(orx OBx). At the end of the discussion, the absurd necessitarian will be discarded, as all observed-ravens are observed-black, x(orx OBx), is an absurd explanandum. 4.1 either (R,B) nor x(rx Bx) explains x(orx OBx) Smart claims that N(R,B) indirectly explains x(orx OBx), via x(rx Bx). That is: (1) N(R,B) entails x(rx Bx); (2) x(rx Bx) explains x(orx OBx); thus, N(R,B) (indirectly) explains x(orx OBx). This reasoning seems to be incorrect for 4 In this paper, natural properties and universals are interchangeable terms. 6

7 several reasons. If it is argued that N(R,B) explains x(orx OBx), then, on the contrary to Smart, that relation of explanation cannot be a relation of explanation simpliciter. 5 The absurd necessitarian searches for the best explanation of our observed regularities. The Armstrong s down-up step is a step of inference to the best explanation. The down-up step is not a chain of explanation simpliciter. To justify induction, inference to the best explanation cannot be discarded. Otherwise, we are left with the classical enumerative induction, before Harman: all observed-ravens are observed-black; thus, all ravens are black. An obvious reply is to claim that N(R,B) is the best explanation of x(orx OBx). However, this claim is highly controversial. For the absurd necessitarian N(R,B) would be, at most, a potential explanation of x(orx OBx). However, N(R,B) would not be the best explanation of x(orx OBx). Clearly, N(OR,OB), instead of N(R,B), would be the best explanation of x(orx OBx). The best inference to laws must be about the types of universals that were observed. If it is alleged that we observed ORs constantly conjoined with OBs, then the absurd necessitarian would infer a relation between ORs and OBs. According to this reasoning, there would no longer be a missing link, x(rx Bx), between N(OR,OB) and x(orx OBx). N(OR,OB) would not entail x(rx Bx). The relation of explanation between N(OR,OB) and x(orx OBx) would be a direct relation. There is an additional issue regarding the claim that N(R,B) is the best explanation of x(orx OBx). Let us take a closer look at Smart s chain of explanation: (1) N(R,B) entails x(rx Bx); (2) x(rx Bx) explains x(orx OBx); 5 Smart s argumentation for the chain of explanation is silent about the best explanation. For instance, N(R,B) explains x(orx OBx) Smart (2011: 323) 7

8 thus, N(R,B) (indirectly) best explains x(orx OBx). Arguably, it is not clear that the second step of this chain of explanation is true. The problem here lies on the flawed Smart s support for his chain of explanation. That is, if N(R,B) is the best explanation of x(orx OBx) and N(R,B) entails x(rx Bx), it does not straightforwardly follow that x(rx Bx) explains x(orx OBx). This is what Lipton (2004: 63) calls the deductive consequence condition on inference. Observational data may be relevant to a hypothesis, but that hypothesis may not explain the observational data. Here is an example from physics. Newton s dynamical theory explains body dynamics. Newton s dynamical theory was an inference to the best explanation of the dynamics of observed bodies on Earth. It turns out that Newton s dynamical theory entails a lower level theory concerning laws of planetary orbit. These laws are a deductive consequence of Newton s dynamical theory. However, it is implausible to claim that the laws of planetary orbit also explain the dynamics of the observed bodies on Earth. It seems to me that, for the absurd necessitarian N(R,B) is not even a potential explanation of x(orx OBx). In light of Smart s grammar (with four distinct universals, R, B, OR and OB), it is implausible to claim that N(R,B) may explain x(orx OBx). Smart s objection against the entailment between N(R,B) and x(orx OBx) can be used against the alleged explanation relation between N(R,B) and x(orx OBx). It is implausible to claim that N(R,B) may explain x(orx OBx), as Smart (2013: 323) states, because this is similar to asserting that all massive objects curve spacetime may be explained by it is contingently necessary that electrons have charge 1. In other words, for the absurd necessitarian, N(R,B) would not be part of the pool of his potential explanations. Surprisingly, Smart does not commit to the idea that N(R,B) explains x(rx Bx): [t]hat all ravens are black is just an implication of his explanation 8

9 [N(R,B)] (Smart 2013: 322 my italic); a perfectly good explanation for this [our observations] might not serve as an explanation for why ravens in general are black (Smart 2013: 323); N(R,B) does not explain why whenever we observe a raven it will be observed to be black (Smart 2013: 324). Smart (2013: 322) speculates that the necessitarian might have a better chance of explaining this phenomenon [all ravens are black] than the Humean (although this is not obviously true). This position is very unusual in the literature. Armstrong (1983: 41; 1988: 225; 1993: 422) and several authors (Mumford 2004: 88; Beebee 2011: 507) consider that N(R,B) explains x(rx Bx). That is, N(R,B) entails and explains the law-statement, x(rx Bx). I believe that providing explanations for law-statements is a basic desideratum for any metaphysical theory of laws of nature. Let us assume that N(R,B) explains x(rx Bx). Then Smart s chain of explanation can be rewritten in the following terms: (1) N(R,B) explains x(rx Bx); (2) x(rx Bx) explains x(orx OBx); thus, by transitivity, N(R,B) explains x(orx OBx). If transitivity is a plausible structural principle of explanation and N(R,B) does not explain x(orx OBx) (as I tried to argue above), then N(R,B) does not explain x(rx Bx) or x(rx Bx) does not explain x(orx OBx). On Armstrong s top-down step, N(R,B) entails and explains x(rx Bx). Then, the blame is on the alleged relation of explanation between x(rx Bx) and x(orx OBx). That is, x(rx Bx) does not explain x(orx OBx). Moreover, according to Smart, R and B, and OR and OB are four distinct universals; thus, x(rx Bx) can hardly explain why x(orx OBx). As Smart (2013: 323) puts it, once again, this is to claim that all massive objects curve spacetime are explained by [all] electrons have charge 1. Finally, from the point of view of necessitarianism, if N(R,B) explained 9

10 x(orx OBx), that relation of explanation would not be justified by the claim that N(R,B) does necessitate (in Armstrong s contingent necessitation sense) x(rx Bx). 6 When we are trying to justify our explanations, we are not trying to prove that our explanation is true. Granting that our explanation is true, we try to justify that the explanation provides understanding of the observed phenomena (Lipton 2004: 21). The best guide for inference is total science. Based on observation, for example, science would say that it is rational to infer that there is a law of nature that all Rs are Bs. Laws of nature provide understanding for our observations. In this scenario, it would not make sense to invoke a missing link, x(rx Bx), between our observations, x(orx OBx), and our inference, N(R,B). N(R,B) would directly explain x(orx OBx). Science is enough to justify our explanatory inferences. 4.2 (R,B) x(orx OBx) Recall that Smart s starting problem was how to infer x(orx OBx) from N(R,B). However, even if Smart s chain of explanation is correct, for reasons beyond my knowledge, actually, this chain per se does not help the absurd necessitarian to infer x(orx OBx) from N(R,B). Smart s chain of explanation only rewrites Armstrong s bottom-up step, and it is silent about what happens in Armstrong s top-down step. As I see it, Smart s starting problem is a red herring. Nothing in Smart s dilemma depends on Smart s starting problem. It may be suggested that there is indeed one more missing link in Armstrong s top-down step. It is missing x(rx Bx) between N(R,B) and x(orx OBx): N(R,B) entails x(rx Bx); x(rx Bx) entails x(orx OBx); thus, by transitivity, N(R,B) 6 This relation of contingent necessitation between N(R,B) and x(rx Bx) implies an infinite regress. See Bird (2005). 10

11 entails x(orx OBx). However, arguably, x(rx Bx) does not entail x(orx OBx) because it is not evident how a universally quantified conditional between two universals (R and B) can entail a universally quantified conditional between two completely distinct universals (OR and OB). If x(rx Bx) does not entail x(orx OBx), in light of Smart s proposal, it continues to be a mystery how x(orx OBx) may follow from N(R,B). x(orx OBx) follows from N(R,B) if some modifications in Smart s universals ontology are made. The predicate O_:_is observed must correspond to a universal. 7 Let s say observeness is the property that all observational things have in common. If the predicate O_: _is observed is a universal, then the universals OR, observed-raven, and OB, observed-black, can be defined as conjunctive properties. Observed-raven is the conjunction of the universals observedness (O) and ravenhood (R); that is, OR=O&R. Observed-black is the conjunction of the universals observedness (O) and blackness (B); that is, OB=O&B. For example, if the particular a has the property O and a has the distinct property R, then a has the property O&R. However, on the contrary to Smart, this is not to say that O, R and OR are three wholly distinct properties. Nor is OR a property above the property O and the property R. The same goes for the properties O, B and OB. If the previous conjunctive properties are accepted, then x(orx OBx) follows from N(R,B). (1) N(R,B) x(rx Bx) Premise (2) x(rx Bx) x((ox Rx) (Ox Bx)) Premise 7 It might be argued that observeness is not a monadic universal but rather a dyadic universal a relational universal, i.e., something as such _observes_. It seems to me that this is not an issue for my proposal. 11

12 (3) OR=O&R Premise (4) OB=O&B Premise (5) x(rx Bx) x(orx OBx) From (2)-(4) N(R,B) x(orx OBx) From (1) and (5) The previous solution is not satisfactory. It seems to me that being observed is not a universal. We are not forced to admit in our ontology the property being observed. To be or not to be observed depends on some other properties that particulars must have. Armstrong (1978: 11) notes that a necessary condition for something s being a property ( ) that property must endow the particular with some specific causal power. Visual stimuli are caused by electromagnetic radiation emitted by bodies. Different bodies have different electromagnetic spectrums. Some bodies are directly observed by humans; others are indirectly observed, via appropriately technology. It seems that the causal power of electromagnetic radiation is a property endowed to objects of perception such as ravens. Furthermore, it would be bizarre that a particular raven would not instantiate the universal being observed before first-time observation; afterwards, it would instantiate the universal being observed. That is, a particular raven would begin to instantiate a universal without any modification in its own physical structure. A genuine universal must result in some causal difference to the particular that instantiates it. To sum up, in light of Smart s proposal, it is a mystery how to infer x(orx OBx) from N(R,B). Smart s starting problem is a red herring. The next section shows that Armstrong does not need to infer x(orx OBx) from N(R,B). 4.3 OR and OB 12

13 Let us return to Smart s universals ontology. If Smart s chain of explanation is incorrect, as I demonstrated above, given Smart s dilemma, then the absurd necessitarian must bite the bullet and accept OR and OB as natural properties to explain our observations. That is, N(OR,OB) explains x(orx OBx). This outcome is not a surprise. After all, if the absurd necessitarian and Armstrong depart from two different explanandum, then two different explanans should be inferred. On the one hand, the absurd necessitarian wants to explain why all observed-ravens are observed-black, x(orx OBx). 8 Then, the absurd necessitarian must infer that it is a law of nature that all observed-ravens are observed-black, N(OR,OB). On the other hand, Armstrong wants to explain why all observed ravens are black, x((ox Rx) Bx). Then, Armstrong must infer that it is a law of nature that all ravens are black, N(R,B). Needless to say, N(OR,OB) does not seem to be a law of nature at all. The lawstatement all ravens are black, x(rx Bx), simply does not follow from N(OR,OB). Something is going wrong here. The root of the problem concerns the nature of OR and OB. Smart s discussion begins from an incorrect explanandum. We are not trying to explain why all observed-ravens are observed-black, x(orx OBx). What we are really trying to explain is why all observed ravens are black, x((ox Rx) Bx). Our observed data concern ravenhood (R) and blackness (B). Our observed data do not concern ORhood and OBness. Predicates do not stand in a one-one relation to universals. OR and OB are manmade things. OR and OB are cooked-up predicates that do not correspond to universals. Being observed-raven and being observed-black are not genuine properties. They are artificial properties. Actually, it seems to me that observed-raven and observed- 8 Actually, Smart also notes that we want to explain why all ravens in our sample have been observed to be black (Smart 2013: 323), but this formulisation of explanandum does not reappear in the remainder of section 2 of the paper, where this point is discussed. 13

14 black are a camouflage for three distinct predicates: O_:_is observed, R_:_is raven and B_:_is black. According to our best total science, prima facie, the predicates R_:_is raven and B_:_is black refer to the universals ravenhood and blackness, respectively. There is no corresponding universal to the predicate O_:_is observed, as stated above. Thus, the statement all observed ravens are black can be formulised as x((ox Rx) Bx). This is the correct explanandum that needs to be explained. The absurd necessitarian must now be discarded. It is absurd to try to explain an incorrect explanandum. Here is how things run. The scientific enterprise begins by discovering universals and relations between those universals. We observe that a is R and a is B, b is R and b is B, c is R and c is B, etc. Science assumes that R and B are universals. Science hypothesizes that there is a relation between Rs and Bs that explains why all observed Rs are Bs. That is, it is a law of nature that all Rs are Bs explains why all observed Rs are Bs. This step is a step of inference to the best explanation. Then, predictions are entailed by the law. The law entails and explains that all Rs are Bs. All Rs are Bs is logically equivalent to all observed Rs are Bs and all unobserved Rs are Bs. If we try to express the sentence all unobserved ravens are black in Smart s grammar of universals, once again, it is evident that observed-raven and observedblack are a camouflage for three distinct predicates. The sentence all unobserved ravens are black is required for our inductive inferences. Crucially, we need to infer something about the unobservable to amplify our reasoning, and prediction enters onto the stage. Following Smart s grammar, we may paraphrase all unobserved ravens are black by all unobserved ravens are unobserved black. This paraphrase is not good. Here, logical formulisation may help to see things more clearly. All unobserved ravens 14

15 are unobserved black may be formulised as x( ORx OBx). Clearly, this is incorrect. This formula means for all x, if x is not observed-raven, then x is not observed-black. A white shoe is neither observed-raven nor observed-black. However, a white shoe is not a raven. Let us try an alternative interpretation. Let UR and UB be universals. UR means unobserved-raven and UB means unobserved-black. Then, all unobserved ravens are unobserved black may be formulised as x(urx UBx). This new grammar continues to be problematic. Again: a white shoe is neither unobserved-raven nor unobserved-black; however, a white shoe is not a raven. Finally, let us try other formulisation for our paraphrase: x(( Ox ORx) OBx). This formula means for all x, if x is unobserved and x is observed-raven, then x is observed-black. Clearly, this sentence is not equivalent to all unobserved ravens are unobserved black. I conclude that the predicates OR_:_is observed-raven and OB_:_is observed-black are spurious in this discussion. The predicates O_:_is observed, R_:_is raven and B_:_is black are enough for our job. All unobserved ravens are black is simply formulised as x(( Ox Rx) Bx). 5 Second objection Smart s second objection is an analogy: if Humean s pattern of inference is irrational, then Armstrong s pattern of inference is also irrational. This is the passage: Armstrong s initial objection that the Humean s pattern of inference was reducible to the irrational e h [observed instances entails claims about unobserved instances] but if we look at Armstrong s own pattern of inference we can clearly see that he himself is committed to this pattern, as ultimately Armstrong s inference is an inductive inference, and all inductive inferences 15

16 take this form. ( ) In this case, the inference is not to the observed instances plus the unobserved instances (e + h), but from the observed instances to the unobservable (!) natural necessitation relation between the respective universals. If the Humean s observed to unobserved pattern of inference is irrational, then so is Armstrong s. (Smart 2013: 325). 6 Reply to the second objection In the above passage, there is a misunderstanding of Armstrong s proposal. The Humean jump from the observed to the unobserved per se is not irrational. The Humean justification of induction is irrational because his conception of laws of nature is flawed. The rationality of an inductive inference does not depend on the observational nature of the inference. We make rational inferences from the observed to the observable. For example, I may say that my car does not work because it is cold this morning. We make rational inferences from the observed to the unobservable. Scientific realists posit unobservable concrete theoretical entities to explain observed phenomena. Cloud chamber tracks are explained by the existence of unobservable particles. Mathematical realists posit unobservable abstract entities (e.g., prime numbers) to explain physical observed phenomena. The cicada life-cycle period is explained by the properties of prime numbers of mathematical number theory (Baker 2005). We make rational inferences from the unobserved to the observable. Maxwell inferred a distribution law for molecular velocities from the unobserved molecular nature of a gas that was posited by kinetic theory (Achinstein 1971: 118). We make rational inferences from the unobservable to the unobservable. Within mathematical applications, mathematical explanations are used to explain mathematical facts. The mathematical 16

17 fact that S(n) = (n+1)/n is proved by mathematical induction, but other proofs are considered to be more explanatory (Steiner 1978). Humeans and necessitarians have different conceptions of laws of nature. For the Humean, laws are taken to be universally quantified truths. That is, it is a law that all Rs are Bs means that all Rs are Bs, x(rx Bx). x(rx Bx) is logically equivalent to x((ox Rx) Bx) x(( Ox Rx) Bx). Therefore, for the Humean, the explanans it is a law that all Rs are Bs is the conjunction x((ox Rx) Bx) x(( Ox Rx) Bx). However, neither conjunct of the explanans explains the explanandum, x(ox Rx) Bx). On the one hand, the first conjunct x((ox Rx) Bx) does not explain why x((ox Rx) Bx) because it is circular. On the other hand, the second conjunct x(( Ox Rx) Bx) can hardly explain why x((ox Rx) Bx), as unobserved phenomena strictly concerning ravens, does not have explanatory power. Thus, the Humean does not explain why x((ox Rx) Bx). His conception of laws of nature is flawed. This, and only this, justification is irrational (Armstrong 1983: 52-59). Smart objects that a regularity can explain observations. He gives an example of political polls: the proportions in the total population do seem to explain the results of the polls (Smart 2013: 324). The total proportions do not explain why people choose to vote in the candidates but the total proportions explain why polls provide good predictions of election results. This example is flawed. It simply seems to me that there is no regularity here. Clearly, the proportions in the total population are not a regularity. The proportions in the total population are not fixed in time. Yesterday, I was Liberal; today, I am Socialist. This ideological fluctuation in time is valid for many. From the side of the necessitarian s proposal, there is also a jump from the observed to the unobserved (more precisely, a jump from the observed instances to the 17

18 observed instances plus the unobserved instances). However, contrary to the Humean s proposal, that jump is mediated by an extra thing a law of nature, N(R,B). 9 According to the necessitarian, laws of nature are not simply the universal quantification all Rs are Bs ; laws are first-order universals; they are dyadic relations of necessitation between first-order universals. Laws of nature are the tertium quid that mediates the observed and the unobserved. The inference from all observed Rs are Bs to it is a law that all Rs are Bs, N(R,B), is an instance of inference to the best explanation. That is, induction is a case of inference to the best explanation. N(R,B) is what best explains our observations. Thus, Armstrong s own pattern of inference is rational because it is a case of inference to the best explanation. Rhetorically, à la Strawson (1952), if inference to the best explanation is not rational, what is then rational? 10 7 Third objection In this objection, Smart puts aside the artificial universals OR and OB. Instead of these universals, he considers the traditional universals F and G. Smart insists that logical equivalence does not entail explanatory equivalence. That is, x(fx Gx) is logically equivalent to x((ox Fx) Gx) x(( Ox Fx) Gx) but it does not follow that x(fx Gx) has the same explanatory force of x((ox Fx) Gx) x(( Ox Fx) Gx). Let R(F,G) be the universal regularity between the universals Fness and G-ness. R(F,G) means all Fs are Gs. He thinks that he has shown that, 9 Contrary to Smart, the natural necessitation relation between universals, N(R,B), is not an unobservable entity. The natural necessitation relation is an object of perception. The notion of contingent necessity is postulated by our experience of singular causation relations that are an object of direct perception. This point is beyond the scope of this paper. 10 Smart takes his explanation, R(F,G), an instance of inference to the best explanation, too. 18

19 frequently, R(F,G) is the best explanation of our observations of Fs being Gs. To sum up, Smart s patterns of inference is the following: all observed Fs are Gs (via IBE) R(F,G); R(F,G) all observed Fs are Gs and all unobserved Fs are Gs. 8 Reply to the third objection Given my replies above, I do not have much to say about the third objection. As far I can see, Smart s claim (i.e., frequently, R(F,G) is the best explanation of our observations of Fs being Gs) is supported in two points: (1) all ravens are black explains all observed-ravens are observed-black ; (2) regarding political polls, the proportions in the total population explain the results of the polls. I replied against these two points. At my reply to the first objection, I argued that all ravens are black does not explain all observed-ravens are observed-black. At my reply to the second objection, I argued that the proportions in the total population are not a regularity. Thus, it is not often the case that R(F,G) is the best explanation of our observations of Fs being Gs. Moreover, at my reply to the second objection, I argued that N(F,G) is the best explanation of our observations of Fs being Gs, as the law of nature, N(F,G), is not simply an universal regularity, R(F,G) Conclusion This paper replied to Smart s objections against Armstrong s solution of the problem of induction. I have shown that the cooked-up predicates observed-raven and observedblack do not support Smart s desiderata. Armstrong s solution to the problem of induction does not require universally quantified conditionals to have explanatory value. 11 Armstrong (1983) makes a long discussion that demonstrates why the regularity theory fails to succeed in this point. It is redundant to follow this point. 19

20 Humean s pattern of inference is irrational, because his conception of laws of nature is flawed. Instead of universal regularities, the natural necessitation between universals is the best explanation for our observed regularities. References Achinstein, Peter Law and Explanation. Oxford: Oxford University Press. Armstrong, David A Theory of Universals: Volume 2: Universals and Scientific Realism. Cambridge: Cambridge University Press What is a Law of ature? Cambridge: Cambridge University Press Reply to van Fraassen. Australasian Journal of Philosophy 66 (2): doi: / What Makes Induction Rational? Dialogue 30 (04): doi: /s The Identification Problem and the Inference Problem. Philosophy and Phenomenological Research 53 (2): doi: / Baker, Alan Are There Genuine Mathematical Explanations of Physical Phenomena? Mind 114 (454): doi: /mind/fzi223. Beebee, Helen Necessary Connections and the Problem of Induction. oûs 45 (3): doi: /j x. Bird, Alexander The Ultimate Argument against Armstrong s Contingent Necessitation View of Laws. Analysis 65 (286): doi: /j x. Castro, Eduardo On Induction: Time-limited Necessity vs. Timeless Necessity. Teorema 33 (3):

21 Harman, Gilbert The Inference to the Best Explanation. The Philosophical Review 74 (1): doi: / Lipton, Peter Inference to the Best Explanation. London: Routledge. Mumford, Stephen Laws in ature. London: Routledge. Smart, Benjamin Is the Humean Defeated by Induction? Philosophical Studies 162 (2): doi: /s Steiner, M Mathematical Explanation. Philosophical Studies 34 (2): Strawson, Peter Introduction to Logical Theory. London: Methuen & Co Ltd. 21