ARTICLES

Unintentional Domain shifts and the Ravens’ Paradox

elinepa

By Dimitrios Chaniotis

A contribution to the ongoing discussion on the ravens’ paradox by offering a novel perspective is the aim of the paper. I will defend an approach which takes into account the impact brought about by shifting between the domains of logical propositions and I will argue that the paradox can be seen as the result of such an unintentional shift. Throughout the text I use  →, ~, &, ∨, (x), ∃x for the material conditional, negation, conjunction, disjunction, universal and existential quantifiers respectively.

1.         The ravens paradox in its general context

Let us think of A and B as two propositions. The logic formalisation of their constant conjunction could be expressed as a law (or rule) suggesting the truth of the relation A → B. By accepting that A → B holds and knowing that A is true, the certainty of B is guaranteed. This form of reasoning is called deduction. So given that when (or if) it rains there are clouds and the fact that it is raining now, we can safely deduce that now there are clouds.

Deduction is not our only way to draw conclusions. If given B and A → B, we examine whether A holds, we use abductive reasoning. The street in front of my house gets wet when it rains, and now is wet. I can therefore assume that it has rained. Certainty is not guaranteed, but still, abduction is a legitimate form of reasoning extensively used by detectives or other professionals.

Evidently, an inference has not to be certain in order to be legitimate or commonly employed. Establishing laws based primarily on experience that eventually take the form A → B, comes under this category. Since for example we see clouds every time it rains, it has been established that if it rains there are clouds. More generally induction, or the logical passage from partial to general, supports the scientific practice of establishing a variety of natural laws such as metals conduct electricity. Nobody has ever examined all metals in the universe. By experimenting on some samples on Earth, the scientific community felt confident that their inference was more or less valid, and very close to certainty. But are we justified in using induction? David Hume argued that induction justification is nothing but the result of our conviction that natural laws are uniform and eternal, something that is not at all guaranteed. Yes, we foresee that we will enjoy sunrise tomorrow but our expectation rests on nothing more than the repetition of the same phenomena. If natural laws are uniform and eternal, such a repetition is anticipated. But we only infer the uniformity and eternity of natural laws just because of this repetition. Like being trapped inside a vicious logical circle, we are constantly assuming what we are trying to prove. This is the problem that Hume sees.

Despite that, by following Francis Bacon’s methodology of analyzing conditions and circumstances together with the use of experiments, we have seen great scientific progress. Nonetheless, Bacon’s method of scientific induction is not what induction is all about. As Jean Nicod puts it (Nicod, 2014[1], p. 203-204), there are two types of induction. The first proceeds by ‘simple enumeration of instances’. The second (scientific induction) ‘is resolved into a complex of enumerative inductions’ since, each time a scientific theory is verified by favourable experimental results, enumeration of their instances is realized. It is therefore enumerative induction that supports the whole edifice of inductive reasoning and as a consequence, any scientific advances we make.

Challenging Nicod’s thoughts, Carl Hempel in 1945, tried to annihilate the power of enumerative induction and the way it was understood by Nicod. He reached the conclusion (see section 3) that a white shoe confirms the hypothesis ‘All ravens are black’. Clearly then, since this paradoxical result casts doubt on the concept of enumerative induction, it calls into question not only inductive reasoning but also our confidence to scientific methodology in general.

2.         Confirmation and probability in accordance with Nicod.

But how does Nicod understand confirmation and probability? According to him (Nicod, 2014, p. 219), upon considering the law A → B[2], the presence of B in a case of A is favourable to it (increases the law’s probability) whereas the absence of B is unfavourable and invalidates the law completely. The presence of L in the event of K, can be relevant to A → B only if the law K → L is somehow related to A → B. Otherwise the presence of K (or non-A in general) in the place of A, is irrelevant. The above summarize what is known as Nicod’s criterion (NC).

Furthermore, certainty can only be sought through invalidation. So finding black ravens is nothing more than favourable to ‘All ravens are black’ whereas the finding of just one non-black raven invalidates this law and leads with certainty to the conclusion that it is not the case that ‘All ravens are black’. Moreover, when aiming at establishing a law, it could be the case that other candidate laws would have to be rejected. Invalidation of them could then turn out to be the preferred strategy. In a black and white world for example (see subsection 5d), I could choose to eliminate the possibility of finding white ravens in order to establish that ‘All ravens are black’.

But to what extent does the presence of B in the case of A increase the probability that a law holds? Nicod produced the following result (Nicod, 2014, p. 267):

(p/h)/(p/hq)=(q/h)/1

Here p denotes the law under consideration, h the general background information and q the proposition that the law is verified in a new instance. With a/b we designate the conditional probability of a when b has already been occurred. So, ‘the probability of the law before a verification is to the probability of the law after a verification p/h:p/hq as the probability itself q/h is to certainty.’ Evidently, if verification is to render a law more probable, we are in demand of two requirements: Firstly, given h, the probability of the law to hold should not be equal to zero. Secondly, the probability q/h should be < 1. In other words, repeated identical samples offer no increase to the law probability.

Last but not least, probability differs from certainty in its nature. It is not absolute but only relative to the general background information available[3]. Any new piece of information cannot affect a proposition that is certain whereas can render a probable proposition more (or less) probable. Confirming a law by simple enumeration relies upon that. Moreover, an event contrary to a hypothesis cannot be seen as unexpected since in the first place a hypothesis can only be probable, by no means certain.   

3.         How was the paradox reached?

The ravens paradox emerges as the outcome of three seemingly plausible claims. The first is NC presented in section 2. The second is ‘The Equivalence Condition’ (EC) which states that whatever confirms or disconfirms one sentence (or hypothesis), also confirms (or disconfirms) any logically equivalent sentence (or hypothesis). The third claim that Swinburne (Swinburne, 1971, p. 318) calls ‘The scientific laws condition’ (SLC), states that ‘All X are Y’ is logically equivalent to ‘All Y are X’. How then, from these claims did we arrive at the paradoxical conclusion? At first, Hempel considered the sentences (Hempel, 1945, p. 11):

S1: (χ) (Rχ → Bχ)
S2: (χ) (~Bχ → ~Rχ )

where  Rχ stands for ‘χ is a raven’, and  Bχ ‘χ  is black’. According to him the two sentences are logically equivalent: ‘they are different formulations of the same hypothesis’. He then considered four objects. The first (α) is a raven and black, the second (β) a raven but not black, the third (γ) a non-raven but black, and the last one (δ) a non-raven non-black object. Given NC, the instance of α confirms S1 and the instance of δ confirms S2 whereas α is irrelevant to S2 and δ is irrelevant to S1. But now due to EC and the equivalence of S1, S2, we would expect both α and δ to confirm both S1 and S2. However, NC tells us that this is not the case and we are thus forced to conclude that confirmation depends ‘not only on the content of the hypothesis, but also on its formulation’ (Hempel, 1945, p. 11). As a consequence, for every universal conditional there can be a formulation ‘for which there cannot possibly exist any confirming instances’. In order to support his last assertion, Hempel constructed another equivalent to S1 and S2 that I will denote S3: (χ) [(Rχ & ~Bχ ) → (Rχ & ~Rχ )], ‘yet no object whatever can confirm this sentence’. That is correct, there is nothing for which  holds. So by accepting S1 and S3 to be equivalent and adhering to EC, since there is nothing to confirm S3 there can be nothing to confirm S1 as well. But, can anyone maintain that it is impossible to find a black raven that confirms the hypothesis ‘All ravens are black’? So at first, what Hempel sees is a major problem with NC. The second problem (Hempel, 1945, p. 14) is that by accepting EC and the logical equivalence of S1 and S2, a confirming instance of the latter e.g. any red pencil, a white shoe, etc., becomes a confirming instance of the former, i.e. ‘All ravens are black’. This last assertion lies at the heart of the paradox. There seems to be no rationality in accepting that a white shoe confirms the proposition ‘All ravens are black’. We have thus reached a position where NC seems problematic and the paradox unavoidable[4].

4.         Approaches to solving the paradox

The current situation is as follows: Three plausible claims (NC, EC, SLC) seem mutually incompatible. Can one or more of them be refuted? If not, we will be left with one last option i.e. whatever seems as a paradoxical conclusion must after all be accepted. I will start with this last option which is the one that Hempel supported.

a.         There is nothing wrong with the seemingly paradoxical conclusion

Hempel attempts to dissolve the paradox by claiming that it is the result of a false impression. A similar strategy is followed by Howson and Urbach who claim (Howson and Urbach, 2006, p. 100) that when we recognize that ‘confirmation is a matter of degree, the conclusion ceases to be counter-intuitive’. Or by Fitelson and Hawthorne, who suggest that (Fitelson and Hawthorne, 2010, p. 266) quantitative results that show that a black raven favors ‘All ravens are black’ more than a non-black non-raven, demonstrate that ‘a careful Bayesian[5] analysis puts the paradox of the ravens to rest’. Coming back to Hempel, he states (Hempel, 1945, p. 18): ‘The impression of a paradoxical situation is not objectively founded; it is a psychological illusion’. To support his view, he describes a situation where somebody examines the hypothesis ‘All sodium salts burn yellow’. He correctly notes that a piece of ice that produces a colourless flame confirms the assertion ‘Whatever does not burn yellow is no sodium salt’ (I will call this step two) and continues: ‘consequently, by virtue of the equivalence condition, it would confirm the original formulation’. Can this be a safe conclusion? The implied equivalence of contrapositives in step two plays a crucial role to Hempel’s argumentation but is accepted without second thought. Even so, since it seemed paradoxical to hold that a piece of ice confirms that ‘All sodium salts burn yellow’ he tried to remedy the situation by adding an extra assumption viz. ‘the sample is of unknown chemical constitution’. The subsequent chemical analysis showed that the sample contained no sodium salt and thus Hempel reached his final conclusion (Hempel, 1945, p. 19): ‘is what was to be expected on the basis of the hypothesis that all sodium salts burn yellow – no matter in which of its various equivalent formulations it may be expressed’. Two observations are important here. Firstly, if the equivalence of contrapositives does not hold (as it will be shown in subsection 5c to be the case), we only have a confirmation of ‘whatever does not burn yellow is not sodium salt’. We cannot reach the conclusion that the burning of ice confirms the hypothesis ‘All sodium salts burn yellow’ even if EC holds. Secondly, whenever we consider a proposition such as ‘All sodium salts burn yellow’ we need to discern the way we employ it. If we use it with confidence we can draw safe conclusions e.g. if we try to exclude substances, the non-yellow flame provides us with the certainty that no sodium salts exist in a given chemical composition. On the contrary, if we are trying to confirm the law, the burning of non-sodium salts proves irrelevant. Because the burning of ice, also confirms the assertion ‘Whatever does not burn red is no sodium salt’ (pick any non-yellow colour). So by accepting Hempel’s rationale, it can well be said to confirm the competing hypothesis ‘All sodium salts burn red’. But that means that we are unable to determine which law we are in fact attempting to confirm. So in my opinion, because of these two observations, the option that there is no paradox has to be rejected.

b.         Rejecting Nicod’s criterion

Rejecting NC is perhaps ‘the most popular response’ (Sainsbury, 2009, p. 98). In section 3 we saw how Hempel reached the conclusion that confirmation (in Nicod’s sense) not only does it depend on formulation but there are formulations for which there can be no confirming instances at all. The standard Bayesian solution (Vranas, 2004, p. 547), (Clarke, 2010, p. 430) accepts that a white object confirms the hypothesis ‘All ravens are black’ but much less[6] than a black raven. And that creates the wrong intuition that white objects are irrelevant. However, this solution employs two assumptions. Firstly, is accepted that the fact of a non-black confirming instance given K background knowledge ( α / K), does not affect the probability P( α / h & K) of the h hypothesis. This assumption was attacked by Vranas. Secondly, there are many more non-black objects than ravens. This assertion seems plausible but the ravens paradox is about confirmation in general, not about a specific flock of birds in my backyard. Instead of ‘ravens’ and ‘black’ we could have used the words ‘fish’ and ‘trunk’ in ‘No fish has an elephant trunk’ (χ) (Fχ → ~Tχ ),  where Fχ denotes ‘χ  is a fish’ and Tχ  ‘χ  has an elephant trunk’. The analogue of our second assumption is that there are many more creatures with elephant trunks than there are fish, which does not hold. So, I concur with Clark in that the standard Bayesian solution is ‘insufficiently general’ (Clark, 2010, p. 427). The same applies to any other solution which puts forward specific presuppositions concerning ratios of ravens or black objects.

Another example aiming at invalidating NC was proposed by Sainsbury (Sainsbury, 2009, p. 98). Finding snakes outside a particular region, confirms the hypothesis (as per NC) that all snakes inhabit regions other than this particular one. But each finding rather suggests that the region isn’t snake-free. A similar example is given by Swinburne (Swinburne, 1971, p. 326) ‘All grasshoppers are located in parts of the world other than Pitcairn Island’. Initially, Swinburne rejects Hempel’s claim that NC is false (Swinburne, 1971, p. 322). He then argues in order to show that both  and  confirm ‘All ’s are ’s’. Since the first will confirm strongly our hypothesis and the second not that much, we arrive at the apparent plausibility of NC (this is a similar result to the standard Bayesian solution). Swinburne utilizes the principle (Swinburne, 1971, p. 323):  ‘a hypothesis h is confirmed by an observation report b in relation to background knowledge k if and only if P(b/k & h) > P(b/k)’.  As background knowledge Ko, he assumes the ratios of ’s to ’s to be x:1-x and of ’s to ’s to be y:1-y. These ratios always apply even if the values of x and y are unknown. He then constructs four different observations that names respectively: b1-observation the , b2-observation the , b3-observation the  and b4-observation the . Next, he calculates the probabilities of both b/Koand b/Ko&h for all four. They are respectively: for b1, xy and x, for b2, x(1-y) and  0, for b3, y(1-x) and y-x, for b4,(1-x)(1-y) and 1-y. After this result (that I will call A), Swinburne makes additional assumptions concerning the actual values of x and y and reaches diverse results. However these assumptions do not hold in every possible case. By accepting for example that x << 1-y (or that there are far less ravens than non-black objects), we are bound to reach another insufficiently general result since as has been mentioned, the paradox is about confirmation in general. For that reason I will not accept beforehand any specific values for x or y. In that way the results (if any) will be as generic as possible. So from (A) and by subtracting P(b/Ko & h) -P(b/Ko), we find for b1 and b4 exactly the same values i.e. x-xy  which is > 0 and for b2 and b3, xy-x which is < 0. Can these symmetrical results be of any value? In my opinion the most plausible interpretation – which also applies to Sainsbury’s and Swinburne’s examples, more to be said in section 6 – is that b1 confirms, b2 disconfirms and b3,b4 are irrelevant when examining the initial domain of discourse, b4 confirms, b2 disconfirms and b1, b3 are irrelevant when examining (what I will call) the p-complementary (to the initial) domain. This I will call Result 0 (R0). Note that the initial domain refers to ’s whereas the p-complementary to ’s.  So in the case of ‘All ravens are black’ the initial domain is the set of ravens, whereas the p-complementary domain is the union of sets of non-black objects where each set is defined by the unique colour of its elements.

c.         Rejecting the equivalence condition

Hempel offers strong arguments that are in favour of EC (Hempel, 1945, p. 12-13). First of all, in case that certain data confirmed a hypothesis and EC did not hold, we would be obliged to investigate if some other equivalent formulation does or does not confirm this hypothesis. This is not a scientific practice at all. Secondly, in deductive arguments, when a law is employed as a premise for another hypothesis, the replacement of this premise with another equivalent proposition does not affect the validity of the argument. So if we trust deductive logic, the validity of EC seems indisputable. The same arguments were also utilized by Swinburne (Swinburne, 1971, p. 321) at the end of his critique to Scheffler. The latter claimed that EC does not hold (Scheffler, 2014[7], p. 412-418) resting his conviction on the phrase: ‘the outcome of induction would reduce itself to the invalidation of possible laws by contrary cases’ (Nicod, 2014, p. 221). By contrary to ‘All A are B’ we mean ‘No A is B’ or ‘All A are not B’ whereas contradictory to ‘All A are B’ is ‘Some A are not B’. Swinburne correctly notes that Scheffler’s idea that confirmation has to do with contrary statements is ‘undoubtedly wrong’. However, Swinburne seems to overlook the major problem with Scheffler’s reasoning which is the equivalence of contrapositives. Scheffler’s argumentation unfolds as follows: The contrary of h1: ‘All R’s are B’s’ is ‘All R’s are B’s’. Since Ra & Ba disconfirms the contrary of h1, it confirms h1 whereas Ra & Ba is irrelevant. The contrary of h2: ‘All B’s are R’s’ is ‘All B’s are R’s’. Since Ra & Ba disconfirms the contrary of h2, it confirms h2. Ra & Ba is irrelevant. So the instance Ra & Ba confirms h1 but not h2 whereas the instance Ra & Ba confirms h2 but not h1. Since h1 is logically equivalent to h2, we would expect (due to EC) both instances to confirm both h1 and h2 which is not the case. Therefore Scheffler concludes that EC is wrong. Both Scheffler and Swinburne take for granted the equivalence of h1 and h2. In 5c this will be shown not to be the case.

The possibility of simply demanding that ‘anything that confirms a generalization must be an instance of it’ is also discussed (Sainsbury, 2009, p. 97). Had such a thought been adopted, EC would have become irrelevant in the process of confirming a hypothesis. Sainsbury’s rejects this idea by presenting an outbreak of legionnaires’ disease at St. George’s school. Some students were free from the disease but they did not attend school. According to Sainsbury, they do confirm the hypothesis ‘all students who attended school contacted legionnaires disease’ despite the fact that they are not an instance of it. Sainsbury implicitly invokes the equivalence of contrapositives in order to point out that it would be absurd to conclude that EC does not hold. The example is further discussed by Clarke. He asserts (Clarke, 2010, p. 435): ‘the hypothesis in Sainsbury’s example ought to be held equivalent to its contrapositive, this is so only within a certain relevant domain.’ More in section 6 and 5d as Result 3.

d.         Denying the equivalence of contrapositives

In order to deal with the paradox Hempel examines the possibility of denying the equivalence of contrapositives. So he first modifies the formulation of ‘All ravens are black’ (Hempel, 1945, p. 15):

S1a: (χ) (Rχ → Bχ) & ∃χ (Rχ)
S2a: (χ) (~Bχ → ~Rχ ) & ∃χ (~Bχ)

This formulation is rejected since a law does not demand the actual existence of objects that is concerned with. We have laws for ideal gases for example. No ideal gas exists though. As a second alternative, Hempel examines the acceptance of the ‘class of ravens’ as the field (domain) of application. In order to reject this second option he argues as follows (Hempel, 1945, p. 17-19): ‘In particular, for a scientific hypothesis to the effect that all P’s are Q’s, the field of application cannot simply be said to be the class of all P’s; for a hypothesis such as that all sodium salts burn yellow finds important applications in tests with negative results’ and: ‘Every P is a Q … restricts all objects whatsoever to the class of those which either lack the property P or also have the property Q. Now, every objecteither belongs to this class or falls outside it, … there is no object which is not implicitly “referred to” by a hypothesis of this type’. Some remarks are important. In order to invalidate a substance (test it and get negative results with certainty), it is compulsory as we have seen in 4a, to have established beforehand that the law holds. On the contrary, a hypothesis is simply confirmed each time we have P & Q and we seek to establish that all P’s are Q’s. But Hempel in order to reject the need for a domain, does not differentiate between the concepts of an established law and that of a hypothesis (first quote above) and extends the domain of all P’s are Q’s to ‘every object’ (second quote). But is he entitled to do so? Tomassi writes (Tomassi, 2006, p. 206): ‘whether a formula of QL[8] is true or false depends upon what it is that we take that formula to be about. In QL truth and falsity are relativised to the set of things we understand ourselves to be talking about, i.e. to the domain.’ So by the phrase ‘All sodium salts’ we first choose which elements from the whole universe ‘live’ in our domain and then deal with them only. Moreover, the fact that truth or falsity depends on the domain of a formula, poses constraints to our capacity of classifying sentences as equivalent which otherwise would have been equivalent in propositional logic. Hempel considers as equivalent all of (Hempel, 1945, p. 11, p. 14):

S1: (χ) (Rχ → Bχ)
S2: (χ) (~Bχ → ~Rχ )
S3:  (χ) (Rχ & ~Bχ) → (Rχ & ~Rχ )
S4: (χ) (Rχ v ~Rχ) → (~Rχ v Bχ )  

But if S1 holds i.e if all ravens are black, the domain of S3 is the empty set since nothing is raven and non-black. The domain of S4 is the whole universe. We are thus invited to accept the equivalence of sentences that talk about different things: possibly nothing at all (S3), every object in the universe (S4), ravens (S1), non-black objects (S2).

5.         The impact of domain shifting to the ravens paradox

So far we have reached the following results: 1. Hempel’s view that there is no paradox has to be rejected 2. NC exhibits symmetry concerning the change of probability values after a confirming instance, in both the initial and the p-complementary domain 3. We cannot reject EC 4. The role that domains play in the equivalence of propositions is possibly overlooked. In this section I will examine the constraints that domains pose for equivalence to hold and I will defend the view that the paradox is the result of unintentional domain shifting.

a.         Some contrapositives are indeed equivalent.

What is logical equivalence? ‘We say that the formula φ is logically equivalent to the formula ψ if φ ⊨ ψ and ψ ⊨ φ’ (Hodges, 2001, p. 109) where the symbol ⊨ denotes semantic entailment. So for φ to be logically equivalent to ψ, φ and ψ must be either both true or both false. From propositional logic, we know that A→B↔~”B”→~”A” where A and B can represent any proposition. ‘It rains, therefore there are clouds’ is equivalent to ‘there are no clouds, therefore it doesn’t rain’. Domains do not exist in propositional logic. In predicate logic, we use quantifiers and variables that range over a set of objects. This set that we call domain, is the set of objects that we care of. Other objects are irrelevant. Sometimes this set is perceived as the widest possible in order that its elements make sense for a given proposition. As it is noted (Tomassi, 2006, p. 194), the ‘domain may well be unspecified. In fact, it is a common practice among formal logicians to leave the domain in just such an indefinite state’. For example, the expression ‘if x is a multiple of 4 then x is divisible by 2’ is equivalent to ‘if x is not divisible by 2 then x is not a multiple of 4’. The domain in that case ranges over all possible values for which both expressions make sense i.e. over all integers. So I could write with confidence (x)(Mx→Dx)↔(x)(~Dx→~Mx), where Mx denotes ‘x is a multiple of 4’, and Dx ‘x is divisible by 2’. However, as will be shown in the next subsection, when domains are altered the equivalence between contrapositives does not hold in general.

 b.        Domain shifts and p-complementary domains

Let us consider the following contrapositive propositions:

Pr1:(z)(z is a number→z+5=0 has a solution)

Pr2:(z)(z+5=0 has no solution→z is not a number)

When one sees Pr1,Pr2 will probably guess that -5 is the solution of z+5=0 rendering Pr1 a true proposition. In that case, Pr2 is true as well, since the assertion ‘z+5=0 has no solution’ is false. So both propositions Pr1,Pr2 are equivalent when I seek for solutions within the set of integers (Z). However, if my intention was to seek for solutions in the set of natural numbers (N), there is no z for which z+5=0. In that case, Pr1 becomes false. Moreover Pr2 becomes false as well, since now ‘z+5=0 has no solution’ is true, whereas ‘z is not a number’ is false. From this example we see that if the domain of one of the propositions was shifted from N to Z (or vice versa), the proposition’s truth value would change and thus no equivalence could be possible. I will call this Result 1 (R1). It will be useful to examine whether contrapositives could in some cases preserve truth values after a domain shift. So let’s consider Pr1 as above, and Pr3:(z)(z+5=0 has no solution→z is a number).[9]

For  the domain is assumed now to be Z– i.e. the set of all negative integers and for  the set of all positive integers Z+. The consequent of  is the negation of the consequent of . In that way each  in  remains an element of Z+. Even though we cannot speak of equivalence due to the different domains used,  and  are both true. The property that matters here, is that Z– and Z+ can be seen as the initial and p-complementary domains respectively (see 4b), since: 1. The initial (Z–) is the domain of ’s antecedent 2. The p-complementary (Z+) is the complement to Z– (the latter being the consequent’s domain) and effectively the domain of ’s antecedent. So there are cases where contrapositives (with domains the initial and p-complementary), preserve truth values despite a domain shift. This I will call Result 2 (R2).

Domain shifts occur naturally during ordinary language communication. John heard his wife Sarah saying ‘Only my dog loves me’. What actually Sarah had in mind was the set of their pets which also included 2 cats. In this domain, the proposition is equivalent to ‘Any creature that is not my dog, doesn’t love me’ and is pointing to her cats. But John assumed that the domain for both propositions was the set of all living beings in the house. In that case the proposition was pointing to him as well. Within John’s assumed domain, equivalence also holds. But the equivalence is destroyed when the domain of the first proposition is the set of pets and for the second the set of all living beings in the house. In such a case the first proposition is true whereas the second is false.

c.         A safe criterion for equivalence to hold between contrapositives

From various logic books (Tomassi, 2006, p. 215), (Hodges, 2001, p. 174), (Guttenplan, 1997, p. 176), we infer that (x)(Rx→Bx) ↔ ~∃x(~”[” Rx →Bx])↔~∃x(Rx & ~Bx). I will call this schema ‘proper equivalence’ since it preserves domains. Why is that? Simply because our mind focuses on Rx for all 3 equivalents and is not distracted from elements of other sets. In that sense, it is a safe criterion for equivalence to hold. So instead of accepting that ‘All ravens are black’ is equivalent to ‘Anything non-black is non-raven’, we could have safely said ‘All ravens are black’ is equivalent to ‘There is nothing that is raven and is not black’. In both latter expressions my domain is constrained to ravens only and thus the paradox cannot emerge. Another example concerning domain shifting and the equivalence of contrapositives is the following: ‘All living beings are mortal beings’. Is this expression equivalent to ‘All immortal beings are non-living beings’? Here, a domain shift from the set of living beings to the set of immortal beings is effected. More importantly, the first expression makes perfect sense and we consider it as true. What about the second? Is it the case that there are immortal beings? If yes, how is it possible to be non-living beings? Perhaps then, there are no immortal beings at all. But the point is, can we effortlessly consider the second expression as either true or false? Despite the clear meaning of the first and our readiness to decide on its truth or falsity, one has to think hard even to figure out what is meant by the second. This example demonstrates that we cannot unconditionally consider contrapositives as equivalent. On the other hand if we had used the expression ‘There is no living being that is immortal’ following the proper equivalence schema, no domain shift is effected, no ambiguity appears and no doubt concerning the assignment of truth values. Equivalence now is evident.

From the above we can conclude that since domains are not preserved and the proper equivalence schema is not followed, the equivalence cannot hold for any of S1, S2, S3, S4 (of 4d and section 3). The same applies to h1, h2 (of 4c) which are identical to S1, S2. ‘All sodium salts burn yellow’ and its contrapositive ‘Whatever does not burn yellow is no sodium salt’ of 4a is another case that domains are not preserved. The domain of the first includes sodium salts whereas the domain of the second includes whatever doesn’t burn yellow. No equivalence is possible then.   

d.         What if the safe criterion is not met?

Going back to the paradox, it is noted (Clarke, 2010, p. 433): ‘We standardly formalize universal generalizations like R with sentences of the form “(x)(Rx →Bx)”, which is classically equivalent to the parallel formalization of ~B, “(x)(~Bx →~Rx)”. With … it is no surprise that we should be inclined to apply those same methods to the hypothesis R.’ And ‘Despite the strong prima facie case for representing R in the standard way, I think this move is the misstep that leads to paradox.’ Moreover, (Clarke, 2010, p. 435): ‘I claim, that an acceptable solution to the ravens paradox must hold R inequivalent to its contrapositive in the context of the paradox, but that we must allow the equivalence to hold in other contexts [10].’ So, as expected from the discussion in 5b and 5c, Clarke holds that R should be held inequivalent to its contrapositive. Still, both expressions ‘All ravens are black’ and ‘Anything non-black is non-raven’ have the same truth values. But now it is R2 that tells us that despite the domain shift and the effected inequivalence, two contrapositives can produce identical truth values (and therefore the one can entail the other). Even so, in order to understand what creates the feeling of paradox, we need to investigate further. Let’s imagine a black and white world and an island inhabited by birds the vast majority of which are black. Among the black birds there are some ravens and a limited number of unidentified white objects. Upon realizing that only a small number of white objects exist, we take the rational decision to examine everything white instead of focusing on black objects. So in order to confirm Rx →Bx we examine successively white shoes, pigeons etc. In doing so, each time we find a white non-raven, we are in effect (as per R0) confirming the contrapositive ~Bx →~Rx. In that case, we have intentionally shifted from the initial domain of ravens into the p-complementary of white objects. However, it could be the case that we do not realize: a) The equivalence to the original hypothesis does not hold, exactly because we chose to shift domains b) The truth of the contrapositive entails the truth of the initial hypothesis (due to R2) c) Confirming the contrapositive can offer us the feeling of confirming the initial as well. How is (c) explained? Each time we recognize a white object as a non-raven, the probability of finding a white raven is dropped proportionally to the ratio 1/N where N is the total number of white objects on the island. The belief is thus subconsciously introduced that the initial hypothesis each time is confirmed. When all N white objects have been examined and none is identified as raven, then whichever raven left has to be black. Therefore both the contrapositive and the initial hypothesis have been established with certainty. But even if my world consisted of a finite number of colours other than 2, finite to the extent that is easily perceived, similar results could be anticipated. Now the p-complementary domain is the union of those sets that each one includes objects of one specific colour. Having this in mind, we start by examining white objects, then red, yellow etc, until the whole union of sets of non-black objects is examined. Each time a coloured object is found not to be a raven, the probability of finding a raven of this colour is dropped proportionally to the ratio 1/m where m is the total number of objects of this colour. When all sets of non-black colour objects have been checked and no raven is found, again, both the contrapositive and the initial hypothesis have been established with certainty. So, each time the probability of finding a non-black raven is lessened, the feeling that the initial hypothesis is confirmed seems inevitable. As a conclusion, even if the safe criterion is not met, when examining a union of a finite number of sets p-complementary to the initial, by confirming the contrapositive we can gain the feeling of confirming the original hypothesis, despite the fact that the equivalence between them does not hold. This I will call Result 3 (R3).[11]

e.         Do unintentional domain shifts embody the core of the problem?

In 5d we examined the case of objects within a union of a finite number of sets, where each set was defined by the single non-black colour of its elements. If a practically infinite number of such sets exist in our p-complementary domain, this strategy is ineffective. Inside an infinite domain, the finding of a white shoe for example, has practically zero effect on the probability of not finding non-black ravens. This subconscious realization creates the paradoxical feeling that nothing (zero confirmation in the p-complementary domain) counts for something (confirmation in the initial domain). Therefore, the shifting from the domain of ravens to a union of an infinite number of sets of non-black colour objects, cannot be handled properly and is in that sense an unintentional move. But even if we try to construct a union of a large (but still finite) number of sets (intentional move) that would be rendered p-complementary to our initial set of ravens, there is not always a safe number of colours in order for this to happen. It then comes as a natural consequence both the inequivalence of contrapositives and our inability to comprehend how the truth of a contrapositive instance could be a confirming instance of our initial hypothesis. The situation thus described, is in harmony with Koshy’s redefinition of the notion of instance (Koshy, 2017, p. 108). He proposes that ‘If I is an instance of H, then it cannot be an instance to any other competing hypothesis’. So a white shoe could be seen (as per traditional definition) as a confirming instance of both ‘All non-blue objects are non-ravens’ and ‘All non-black objects are non-ravens’ which however are competing hypothesis. Therefore as per his new definition, a white shoe can be a confirming instance of neither of them. Koshy leaves as an open question ‘what the contrapositive instance of the hypothesis all ravens are black is’ (Koshy, 2017, p. 109). It is now the union of a limited number of sets that form the p-complementary domain that comes into play. In our black and white world, there is no competing hypothesis to ‘All white objects are non-ravens’. So a white object confirms the hypothesis ‘All ravens are black’. In a three-coloured world, e.g. white, blue and black, we have to look at the union of sets of blue and white objects so as not to encounter any competing hypothesis. In such a case the only contrapositive hypothesis is ‘All blue or white objects are non-ravens’. Therefore either a white shoe or a blue pencil, confirm the initial hypothesis. So the contrapositive takes now the form of a disjunction. The same applies to a union of a finite number of sets where we shape our contrapositive hypothesis in a disjunctive form so as each member of the p-complementary domain is examined. If on the other hand we encounter a practically unlimited number of colours, there can be no disjunction embodying all of them and thus no hypothesis thus to be tested.

6.         Scenarios under the light of domain shifts.

It has been shown that the paradox can be seen as the result of an unintentional domain shift from the initial to the p-complementary domain which is a union of an infinite number of sets.  Can this outcome assist us with the examples so far encountered? I will start with Swinburne and Sainsbury (subsection 4b). Since the two examples are of the same kind, I will speak of grasshoppers only. Firstly, I will consider what Nicod writes (Nicod, 2014, p. 216): ‘a prediction which is known to be erroneous is not rendered probable by arguments which would make it such in the absence of such knowledge’. As has been mentioned in section 2, probability is relative to its general context. We know that people die. Thus, a confirming instance of a living human is rendered null and void in relation to a hypothesis of eternal life. We similarly know that grasshoppers are not static. Therefore, the discovery of a grasshopper in area A give us good reasons to infer that it could have been found in a nearby area B. Next, I will exploit the concept of domain shifts. Taking the contrapositive so as to read: ‘Any object located on Pitcairn is a non-grasshopper’, we shift from the initial domain (set of grasshoppers) to the p-complementary defined by the property: location Pitcairn. So, each time we find a non-grasshopper (insect or even object) on Pitcairn, we get the feeling of confirming the original hypothesis (as of R3).  Moreover by implementing NC in this domain in accordance with R0, doubts (as of 4b) concerning NC credibility, are remedied despite the fact that our hypothesis will eventually prove wrong due to the background information available.

The outbreak of legionnaires’ disease in 4c can also be seen under the light of domain shifts. Even though the equivalence of contrapositives does not hold (we are in different domains), it has been shown (R2) that contrapositives can have the same truth values. Moreover from the p-complementary domain we can confirm the contrapositive (R0) and also gain the feeling of confirming the original hypothesis (R3). The contrapositive to ‘all students who attended school contacted legionnaires disease’ is ‘students who did not contact legionnaires disease did not attend school’. So assuming the totality of students at St. George’s school as our universe, the p-complementary domain is ‘students who did not contact legionnaires disease’. Therefore, these students confirm both the contrapositive and the initial hypothesis, and that is independent of EC’s validity that is discussed in Sainsbury’s example.

This is also a reply to Clarke’s assertion at the end of 4c. Confirmation of the initial hypothesis can be explained without the need of invoking equivalence within a certain relevant domain.

7.         Conclusions

1.  The ravens paradox can be seen as the result of an unintentional shift from the initial domain of the proposition ‘All ravens are black’, to a domain which is the union of a practically infinite number of sets.

2.  Contrapositives are not always equivalent (R1).

3.  Within the initial and the p-complementary domain:

a.  Nicod’s Criterion is symmetrically valid (R0).

b.  Truth values can be preserved between contrapositives (R2).

c.  When the p-complementary is a union of a finite number of sets, by confirming the contrapositive, the feeling of confirming the initial hypothesis is gained even if equivalence does not hold (R3).

Reading List:

Clarke, R. 2010: “The Raven’s Paradox” is a misnomer. Synthese, 175(3), 427–440.

Fitelson, B. and Hawthorne, J. 2010:How Bayesian Confirmation Theory Handles the Paradox of the Ravens. In E. Eells &J. Fetzer (eds.), The Place of Probability in Science. Dordrecht Heidelberg London New York: Springer.

Guttenplan, S. 1997: The Languages of Logic. Blackwell Publishing.

Hempel, C. 1945: Studies in the logic of confirmation (I.). Mind, 54(213), 1–26.

Hodges, W. 2001: Logic. London: Penguin Books.

Howson, C. and Urbach, P. 2006: Scientific Reasoning, the Bayesian Approach. Chicago and La Salle, Illinois: Open Court.

Koshy, P. 2017: A Solution to the Raven Paradox: A Redefinition of the Notion of Instance. Journal of Indian Council of Philosophical Research, 34(1), 99–109.

Nicod, J. 2014: Foundations of Geometry and Induction. New York: Routledge.

Sainsbury, R. M. 2009: Paradoxes. New York: Cambridge University Press.

Scheffler, I. 2014: The anatomy of inquiry. New York: Routledge.

Swinburne, R. G. 1971: The paradoxes of confirmation: A survey. American Philosophical Quarterly, 8(4), 318-330.

Tomassi, P. 2006: Logic. New York: Routledge

Vranas, P. 2004: Hempel’s Raven paradox: A lacuna in the standard Bayesian solution. British Journal for the Philosophy of Science, 55(3), 545–560.

[1] First published in 1930

[2] The same applies to ‘All A are B’.

[3] In section 6 we’ll make use of this property.

[4] Both problems could be remedied if the equivalence of S1, S2, S3 did not hold. The inequivalence of S1, S2, S3 will be shown in subsection 5c.

[5] Bayesian results will be discussed in 4b.

[6] NC asserts that a white object is irrelevant.

[7] First published in 1964

[8] Quantificational logic

[9] Despite the fact that I call Pr3 contrapositive to Pr1, I do not implement it in the form “ Pr3:(z)(z+5=0 has no solution→z is not a number”. That is because I have now shifted the domain from Z- to Z+ Z– to Z+

[10] We have seen in 5a when contrapositives are indeed equivalent.

[11] The form of the contrapositive in the case of a union of a finite number of sets will be discussed in 5e.