The (Relevant) Logic of Scientific Discovery
aa r X i v : . [ m a t h . L O ] J a n The (Relevant) Logic of ScientificDiscovery
Timothy Childers , Ondrej Majer , and Peter Milne Czech Academy of Sciences, Institute of Philosophy, Prague,childers@flu.cas.cz, majer@flu.cas.cz Dept. of Philosophy, University of Stirling, Stirling, [email protected]
Part I: Semantics
This paper presents a novel, thorough-going interpretation of some rel-evant logics. The interpretation employs an idealized modelling of thesearch for regularities in scientific inquiry. Laboratories (research teams)set up, carry out, and assess experiments, and thereby arrive at regularitiesor their absence. The modelling captures the distinction between confir-mation and refutation, a distinction which motivates a rejection of contra-position.We found our interpretation on relevant logics built over Dunn–Belnapfour-valued semantics for negation, conjunction and disjunction. We em-ploy two accessibility relations: one describing confirmation of a regular-ity, the second disconfirmation. It is this fine-grained approach that neces-sitates the inadmissibility of contraposition.While novel our interpretation is not without anticipations of someof the details. There have been many nods to interpretations similar toours—for example, in [9, 3, 4, 5]—and formally our account falls underwhat Richard Sylvan called “the American plan completed” [25]. How-ever, while Sylvan also employs two accessibility relations, his system issignificantly more complicated, in the main because he sets out to pre-serve contraposition. There are only a few other examples of a four-valued1ogic explicated in terms of two accessibility relations. Recently TakuroOnishi [20] has employed the second relation to provide a semantics forco-implication.Finally, data is messy. We begin to tackle this in the second part ofthis paper by adding probabilities to the logical framework. We beginwith probabilities over Dunn-Belnap logics, and show how they can beinterpreted as relative frequencies and betting quotients. We consider anumber of updating strategies, including updating for groups of labs.
We begin with laboratories, or, more broadly, teams or groups of researchers,that learn, and then share what they’ve learned with other laboratories orteams. To keep things brief, we’ll talk about laboratories but the model haswider application. The laboratories’ endeavours can be divided into fourstages. First, and obviously, there is a planning stage. This is followed bythe initiating event, for example, the sending out of a signal, the adminis-tration of a treatment regime, the delivery of questionnaires. Then there’sa follow-up event, possibly distant in space and/or time, for example, thereceipt of the signal, the collection of data on the effect of the treatmentregime, the gathering in of completed questionnaires. Finally, the previousstages are evaluated. All four stages may be carried out by the same indi-vidual or team, or there may be some division of labour, perhaps widelydistributed so that, say, multiple signals are sent out and received at vari-ous sites, groups of patients with different medical histories are subject tothe same treatment regime, questionnaires are distributed to a variety ofsample populations in diverse locales.We can reduce the four stages to three by ignoring the planning stage.This seems reasonable: while, on the one hand, an experiment may be pro-posed and only decades later carried out, the team evaluating the resultshas to know what the initiating and follow-up teams report, how they areco ¨ordinated, and how to evaluate those reports; it does not need to knowwhat prompted the initiating and follow-up teams to do what they did. We can then describe our labs at the various stages with a ternary rela-tion. We will say that R xyz iff Teams need not be identified with particular sets of individuals. Certainly, theyshould not be in long-term longitudinal studies such as the UK’s on-going National ChildDevelopment Study which focuses on over 17,000 people born in a single week in 1958. collates and evaluates the data obtained from a test set-upinitiated by y and followed up by z .In searching for regularities the laboratories look for perfect correla-tions between the initiating events of the experiment and the outcomes ofthat experiment. The labs seek to determine if a given experimental set-up always leads to a particular outcome, and never leads to an alternativeoutcome. This is naturally represented by a conditional:the conditional A → B is true from the point of view of a lab s , if, for every pair of labs h t , u i in s ’s purview testing the regu-larity, t initiating and u following up, when lab t reports that A is true, lab u reports that B is true.In our framework, the search for regularities supervenes on tests for trueconditionals. But we need to say a little more about what ‘true’ amounts to in thissetting. Lab reports may be positive and negative of course: that the ex-periment was set up in the proper way or not, or that an outcome was ob-served or not. But they can also be null: experiments might not be set upat all, or outcomes might not be observed at all (equipment breaks down,subjects aren’t compliant, etc.). And the same labs can also report contra-dictory findings (equipment is on the blink, a subject is schizophrenic). This leads to four possible values, true , false , neither true nor false , both trueand false . This, we should emphasise, is our take on scientific practice . It can be thought of asparasitic upon the the widespread but also widely derided reading of scientific laws andregularities as quantified conditionals of which the paradigm in the literature seems to be‘All ravens are black’. In [12] Toby Friend ‘offer[s] some argument for the view that lawsdo indeed have a quantified conditional form.’ It’s important to note that Friend’s argu-ment ‘for the conditional feature of the schema of laws aims to show that a conditional isimplicit in our very understanding of laws, even when they do not appear to explicitlyhave that form’ [12, p. 127]. A claim such as that chocolate consumption is inversely as-sociated with prevalent coronary heart disease (which may be the case [8]) can be turnedinto a generalization—a “law”—concerning cross-sectional studies. “In some cases, and perhaps in all, observations of two sufficiently similar entitiesare inconsistent when the same comparison is repeated several times.” [14, p. 3] Readers of a nervous disposition may substitute ‘reported true’ or ‘adjudged true’ or‘accepted as true’ for our use of ‘true’ and likewise, mutatis mutandis , for our use of ‘false’.
Cf.
Nuel Belnap’s ‘told True’ and ‘told False’ in [3] and Michael Dunn’s disclaimer:Do not get me wrong—I am not claiming that there are sentences which arein fact both true and false. I am merely pointing out that there are plenty ofsituations where we suppose, assert, believe, etc., contradictory sentences
3f special significance for our story will be a subset of S of well-behavedlabs, picked out by simple criteria. Firstly, labs work on multiple projects.A lab that has oversight where one set of experiments is concerned maybe supplying data to another testing other regularities. Experimentationis organized and regulated activity. For pairs of labs (or research groupsor experimental teams), one initiating the other completing diverse ex-periments, possibly overseen by diverse others, the completing lab has toknow what the initiating lab has done—for only so will it know it has workto do, data to collect and communicate to the relevant overseeing lab. Thuswhen x is a well-behaved lab making a good job of its co ¨ordinating role,and R xyz , either by making the same determinations itself or by havingthem reported to it, z must have made the same acceptances as true andfalse as y . We denote the fact that, for some well-behaved lab x , R xyz by y ≤ z . In effect, this is an information ordering on labs: z has all theinformation y has (and possibly more since z has to complete any experi-ment y has initiated under x ’s oversight). So read, we should expect it tobe reflexive and transitive—a pre-order. Secondly, some regularities— e.g. anything of the form A → A —are laws of logic. Any well-behaved labshould adjudge those regularities as true. L designates the (non-empty!)subset of S comprising the well-behaved labs.All labs conduct ”equipment checks” – calibrating instruments, clean-ing test tubes, approving questionnaires. We assume that a well-behavedlab has oversight of these activities. That is(a) for all x ∈ S , ( ∃ u ∈ L ) R uxx .Suppose that u is a well-behaved lab, that R uxy and that y ≤ z . Thenthe z team, having made all the determinations, whether off its own bat orby having them reported to it, that y has, can stand in for y in supplying u with the data it needs about the test initiated by x . That is(b) if u ∈ L , R uxy and y ≤ z then R uxz .Somewhat analogously, if R xyz and w ≤ x then w having made nodeterminations that x has not, knows nothing that stands against y ’s and z ’s reports that x does not and hence can use their determinations in itsown evaluations of regularities. That is(c) R xyz and w ≤ x then R wyz . to be true, and we therefore need a semantics which expresses the truthconditions of contradictions in terms of the truth values that the ingredientsentences would have to take for the contradictions to be true. [9, p. 157] u and v are both well-behaved and R uvx then u and v ’s good habitsrub off/are imposed on x . That is(d) if u , v ∈ L and R uvx then x ∈ L .It follows already from what was said above that x must acknowledge allregularities that are logical truths.These are our basic modelling assumptions. Other constraints may beadopted. For example, we might make explicit that the division of labourthat sees y initiating and z completing the test of a regularity under x ’soversight is accidental, not a necessary feature of the experiment. If R xyz ,a single research group w could play the y role — R xwz — and play the z role — R xyw . That is(e) if R xyz then ( ∃ w ∈ S )[ R xwz and R xyw ] .We trust that our basic assumptions (a)–(d) are plausible descriptionsof how no doubt idealised, well-organised, co ¨ordinated research activitygoes. It would, of course, be disingenuous of us to claim that we have fash-ioned them without some thought as to the behaviour of one of the ternaryrelations in Routley’s “American plan” semantics for relevance logic butwe do think that they capture an aspect of (idealised) scientific practice—far from the full story, of course, but the basics.The obvious next step is to say that the overseeing lab x reports theregularity A → B to be true when, for each pair h y , z i , y initiating, z fol-lowing up, in its purview, z reports that B is true when x reports that A is true. As far as it goes that’s just dandy. When we turn to think of a labdeclaring a putative regularity to be false, we need to keep two consider-ations in mind. First, gathering positive evidence for a hypothesis can bea very different activity from gathering evidence to disconfirm a hypoth-esis. (This comes to the fore in our later treatment of probability.) Thesecond is that hypothesised regularities are not to be given up too read-ily. Scientific experiments are, in principle, reproducible and—ideally,at least—only an in principle reproducible negative result gets established “Kant was perhaps the first to realise that the objectivity of scientific statements isclosely connected with the construction of scientific theories—with the use of hypothesesand universal statements. Only when certain events recur in accordance with rules orregularities, as is the case with repeatable experiments, can our observations be tested—in principal—by anyone. We do not take our own observations quite seriously, or acceptthem as scientific observations, until we have repeated and tested them.” [22, §8]
5s counting against the regularity under investigation. (In practice, ofcourse, reproducibility is an ideal which cannot always be met, neither in“hard” sciences such as cosmology (by the nature of the case) and parti-cle physics (due to the cost of the equipment) nor in many biological andsocial sciences (again by the nature of the case but also because attemptsat strictly reproducing may well violate ethical standards). ) So it’s notenough for a lab to count the regularity A → B as false—as falsified— thatone pair in its purview have reported A true and B false. The experimentalfinding has to be reproducible: as we shall formulate it, the negative resulthas to remain available —this is our way of discounting Popper’s “stray” re-sults. For these reasons we employ a second accessibility relation, R xyz ,governing refutations. The availability constraint takes the form: if x ≤ w and R xyz then R wyz . That the relations R and R are largely indepen-dent reflects the fact that the decision to count a putative regularity asfalsified may be a methodological decision but it isn’t purely formal—aswe might say, provocatively, it isn’t a logical decision.In diverse ways, these considerations guide our formal definitions offrames, models, valuations, logical consequence and logical truth—to whichwe now turn. At this point we have enough of the pieces in place to proceed with for-mulating our logic. We begin with the definition of a frame, adapted fromthe Routley–Meyer semantics for relevant logic. Definition 3.1. A frame is a quintuple F = h S , L , ≤ , R , R i , where S is anon-empty set, L a subset of S, ≤ is an ordering on S, and R and R are ternaryrelations on S. We stipulate that x ≤ y iff ( ∃ u ∈ L ) R uxy. “We say that a theory is falsified only if we have accepted basic statements thatcontradict it. This condition is necessary, but not sufficient; for we have seen that non-reproducible single occurrences are of no significance to science. Thus a few stray basicstatements contradicting a theory will hardly induce us to reject it as falsified. We shalltake it as falsified only if we discover a reproducible effect which refutes the theory.” [22,§22, emphasis in the original] See further [15, §IV] on how scientists react to anomaly anddiscrepancy. Note too that ‘replication can even be hazardous. The German scientist Georg Wil-helm Reichmann was fatally electrocuted during an attempt to reproduce Ben Franklin’sfamous experiment with lightning’ [7, p. 4972]. Textbook accounts are to be found in [24, Ch. 5] and, very briefly, in [17, AppendixB]. , R , L and ≤ have the properties i) - v) and possibly one or more of theother five. In particular for any x , y , z , w ∈ S:i) x ≤ xii) if x ≤ y and y ≤ z then x ≤ z;iii) L is an upwards ≤ -closed subset of ( S , ≤ ) ;iv) if w ≤ x and R xyz then R wyz;v) if x ≤ w and R xyz then R wyz;vi) R xxx;vii) R xxx;viii) if R xyz then ( ∃ w ∈ S )( R xyw and R xwz ) ;ix) if R xyz then ( ∃ w ∈ S )( R xyw and R xwz ) ;x) if R xyz then ( ∃ w ∈ S )( R xyw and R xzw ) . We take L to be the set of formulas of a standard propositional lan-guage; i.e. , formulas of finite length generated from a fixed stock, At ( L ) ,of atomic propositions using the connectives ¬ , ∧ , ∨ and → .Models are defined more or less as usual but with two atomic persis-tence/heredity contraints. Definition 3.2. A model M is a pair h F , v i where F is a frame and v is a valuation , i.e. a function from labs and atomic formulas to subsets of the set { T , F } of truth-values. That is, v : ( S × At ( L )) → P ( { T , F } ) . A valuation vsatisfies the pair of constraintsi) if x ≤ y and T ∈ v ( x , p ) then T ∈ v ( y , p ) ;ii) if x ≤ y and F ∈ v ( x , p ) then F ∈ v ( y , p ) . We extend v to a function v M from labs and formulas to subsets of { T , F } . We evaluate negation, conjunction and disjunction according tothis augmented Hasse diagram, familiar from Belnap’s “useful four-valuedlogic” ([
3, 4, 5 ]) . 7 { T }{ F } { T , F }¬¬ ¬ ¬ In essence we are retaining what may, following [27, p. 385], be called“truisms about truth”: e.g. , ‘the truism that a conjunction is true if and onlyif its conjuncts are true’. Equally we have the truisms that a disjunction istrue if and only at least one disjunct is true and that a disjunction is false ifand only if both disjuncts are false. Consider the disjunction A ∨ B when A is assigned ∅ and B is assigned { T , F } : as B is true, A ∨ B is true; as A is not false, A ∨ B is not at the same time false, and so it is assigned { T } asthe Hasse diagram indicates. Definition 3.3.
For every model M = h F , v i we define the function v M : ( S × L ) → P ( { T , F } ) as follows (we omit the superscript when it is clear fromthe context):i) for atomic p, v M ( x , p ) = v ( x , p ) ;ii) T ∈ v M ( x , ¬ A ) iff F ∈ v M ( x , A ) , F ∈ v M ( x , ¬ A ) iff T ∈ v M ( x , A ) ;iii) T ∈ v M ( x , A ∧ B ) iff T ∈ v M ( x , A ) and T ∈ v M ( x , B ) ,F ∈ v M ( x , A ∧ B ) iff F ∈ v M ( x , A ) or F ∈ v M ( x , B ) ;iv) T ∈ v M ( x , A ∨ B ) iff T ∈ v M ( x , A ) or T ∈ v M ( x , B ) ,F ∈ v M ( x , A ∨ B ) iff F ∈ v M ( x , A ) and F ∈ v M ( x , B ) .We modify the standard Routley–Meyer evaluation conditions for the conditional(and adapt to the four-valued setting):v) T ∈ v M ( x , A → B ) iff ( ∀ y , z ∈ S )[ if R xyz and T ∈ v M ( y , A ) then T ∈ v M ( z , B )] ; i) F ∈ v M ( x , A → B ) iff ( ∃ y , z ∈ S )[ R xyz and T ∈ v M ( y , A ) and F ∈ v M ( z , B )] . In §2 we said that ≤ is an information ordering. We justify that remarkwith the Persistence Lemma (or Hereditary Condition). Lemma 3.1 (Persistence Lemma) . In any model M and for any formula A,if x ≤ y and T ∈ v M ( x , A ) then T ∈ v M ( y , A ) ;if x ≤ y and F ∈ v M ( x , A ) then F ∈ v M ( y , A ) .Proof. Base case This holds for atomic formulas by the definition of model . Induction hypothesis
Suppose that the lemma holds for all formulas of length ≤ k . Inductive step
Let A be of length k +
1. There are four cases to considerin each of which B and, where present, C is of length ≤ k . (We omit thesubscript of the valuation function v M ).(i) A = ¬ B : T ∈ v ( x , ¬ B ) only if F ∈ v ( x , B ) only if, by the inductionhypothesis, F ∈ v ( y , B ) only if T ∈ v ( y , ¬ B ) ; F ∈ v ( x , ¬ B ) only if T ∈ v ( x , B ) only if, by the induction hypothesis, T ∈ v ( y , B ) only if F ∈ v ( y , ¬ B ) ;(ii) A = B ∧ C : T ∈ v ( x , B ∧ C ) only if T ∈ v ( x , B ) and T ∈ v ( x , C ) onlyif, by the induction hypothesis, T ∈ v ( y , B ) and T ∈ v ( y , C ) only if T ∈ v ( y , B ∧ C ) ; F ∈ v ( x , B ∧ C ) only if F ∈ v ( x , B ) or F ∈ v ( x , C ) only if, by theinduction hypothesis, F ∈ v ( y , B ) or F ∈ v ( y , C ) only if v ∈ ( y , B ∧ C ) ;(iii) A = B ∨ C : T ∈ v ( x , B ∨ C ) only if T ∈ v ( x , B ) or T ∈ v ( x , C ) onlyif, by the induction hypothesis, T ∈ v ( y , B ) or T ∈ v ( y , C ) only if T ∈ v ( y , B ∨ C ) ; F ∈ v ( x , B ∨ C ) only if F ∈ v ( x , B ) and F ∈ v ( x , C ) only if, bythe induction hypothesis, F ∈ v ( y , B ) and F ∈ v ( y , C ) only if F ∈ v ( y , B ∨ C ) ; Technically, this is our major departure from Routley’s completion of the Americanplan. In our notation, Routley adopts this evaluation clause: F ∈ v M ( x , A → B ) iff ( ∃ y , z ∈ S )[ R xyz and F ∈ v M ( y , B ) and F / ∈ v M ( z , A )] .Routley has in place some additional constraints that have no parallel here. A = B → C : T ∈ v ( x , B → C ) only if ∀ z , w [if R xzw and T ∈ v ( z , B ) then T ∈ v ( w , C )] ; if R yzw then, by clause iv ) in Definition 3.1, R xzw ; consequently, if T ∈ v ( z , B ) then T ∈ v ( w , C )] ; and so, z and w being arbitrary, we have that ∀ z , w [if R yzw and T ∈ v ( z , B ) then T ∈ v ( w , C )] , i.e. , T ∈ v ( y , B → C ) ; F ∈ v ( x , B → C ) only if ∃ z , w [ R xzw and T ∈ v ( z , B ) and F ∈ v ( w , C )] ; by clause v ) of Definition 3.1, ∃ z , w [ R yzw and T ∈ v ( z , B ) and F ∈ v ( w , C )] ; that is, F ∈ v ( y , B → C ) .Our definitions of logical consequence and logical truth are commonin the literature on relevant logic. Here we make good on the thought thatthe well-behaved labs, not necessarily all labs, acknowledge the truth oflaws of logic. Definition 3.4 (Logical Consequence and
Logical Truth) . Given a model M determined by the frame F = h S , L , ≤ , R , R i and valuation v and a non-empty set of formulas X, we write X (cid:15) M A iff, at every x ∈ S, T ∈ v M ( x , A ) when, for all B ∈ X, T ∈ v M ( x , B ) . We write (cid:15) M A iff, at every x ∈ L,T ∈ v M ( x , A ) .We then define logical consequence and logical truth as follows:X (cid:15) A iff, for all models M , X (cid:15) M A; (cid:15) A iff, for all models M , (cid:15) M A. As this definition makes clear, in a model M the nodes in the distin-guished subset L of S are states in which all logical truths hold; outside L some logical truths may fail to be true. (Letting t be the collection ofall logical truths, we have, obviously and uninformatively, that (cid:15) A iff t (cid:15) A . )As is standard, the definitions of logical consequence and logical truthpermit proof of a weak deduction theorem. Theorem 3.1 ((Weak) Deduction Theorem) . For all formulas A and B in L ,A (cid:15) B iff (cid:15) A → B.Proof.
It is enough to show that for an arbitrary model M , Alternatively, following and adapting [24], we could introduce a sentential constant t with the constraint on valuations that v ( x , t ) = { T } if x ∈ L , v ( x , t ) = ∅ if x / ∈ L .By appeal to the ≤ -closure of L , we would find that t satisfies the first clause of thePersistence Lemma; it satisfies the second clause trivially. We would then have Read’sProposition 5.3 (p. 85), i.e. , t (cid:15) A iff (cid:15) A , and could say with Read, “ t is the logic.” (cid:15) M B iff (cid:15) M A → B .Suppose first that (cid:15) M A → B . Let x ∈ S and T ∈ v M ( x , A ) . Since, for all x ∈ S , ( ∃ u ∈ L ) R uxx , we have that T ∈ v M ( u , A → B ) , since (cid:15) M A → B ,and therefore that T ∈ v M ( x , B ) . Thus A (cid:15) M B .Now suppose that M A → B so that, for some u ∈ L , there exist x and y in S such that R uxy , T ∈ v M ( x , A ) and T / ∈ v M ( y , B ) . Now, since u ∈ L , x ≤ y and so, by the Persistence Lemma, T ∈ v M ( y , A ) . Hence A M B .The definitions of Section 3 which build in the constraints on the rela-tions R and R motivated in section 2 yield what we are calling the Logicof Scientific Discovery (LScD).In Definition 3.4 we have, in effect, taken { T } and { T , F } to be thedesignated values in the Hasse diagram (as foreshadowed in the shadingabove). The → -free fragment of our logic is the familiar Dunn–Belnapfour-valued logic, arguably what remains of the classical logic of nega-tion, conjunction and disjunction when one gives up on the principles thattruth and falsity are exhaustive (bivalence) and mutually exclusive (non-contradiction). While some may hold with Wittgenstein ([29, §290]) that literally noth-ing follows from a contradiction and so there may be exceptions to A (cid:15) A ,relevantists do not demur. As follows from Definition 3.4 and Theorem3.1, A → A is a theorem for every formula A of L . On the other hand,the logic of first-degree entailment gives us no → -free theorems. In thisregard, our logic really is a logic of regularities. LScD logics have axiom schemata A1 - A11 and possibly other of the fol-lowing, depending on which constraints other than i ) – v ) are adoptedfrom Definition 3.1: It’s worth noting that, as a moment’s reflection on the symmetry properties of theHasse diagram reveals, nothing would change with regard to which inference patternsare sound if, instead of { T } and { T , F } , we were to take { T } and ∅ to be the designatedvalues—avoidance of falsity rather than pursuit of truth then being the goal. Classicallogic does not distinguish between these goals, of course, and, in a sense that may appealto inferentialists, neither does what remains of the classical logic of negation, conjunctionand disjunction when one gives up on the principles that truth and falsity are exhaustiveand mutually exclusive.This observation provides the basis for an easy proof that in Dunn–Belnap logic, the → -free fragment of our logic, B (cid:15) ¬ A when A (cid:15) ¬ B . A → A ;A2 A → ( A ∨ B ) , B → ( A ∨ B ) ;A3 ( A ∧ B ) → A , ( A ∧ B ) → B ;A4 (( A → C ) ∧ ( B → C )) → (( A ∨ B ) → C )) ;A5 (( A → B ) ∧ ( A → C )) → ( A → ( B ∧ C )) ;A6 ( A ∧ ( B ∨ C )) → (( A ∧ B ) ∨ C ) ;A7 ¬ ( A ∨ B ) → ( ¬ A ∧ ¬ B ) , ( ¬ A ∧ ¬ B ) → ¬ ( A ∨ B ) ;A8 ¬ ( A ∧ B ) → ( ¬ A ∨ ¬ B ) , ( ¬ A ∨ ¬ B ) → ¬ ( A ∧ B ) ;A9 A → ¬¬ A , ¬¬ A → A ;A10 ¬ (( A ∨ B ) → C ) → ( ¬ ( A → C ) ∨ ¬ ( B → C )) ;A11 ¬ ( A → ( B ∧ C )) → ( ¬ ( A → B ) ∨ ¬ ( A → C )) ;A12 ( A ∧ ( A → B )) → B ;A13 ( A ∧ ¬ B ) → ¬ ( A → B ) ;A14 (( A → B ) ∧ ( B → C )) → ( A → C ) ;A15 (( A → B ) ∧ ¬ ( A → C )) → ¬ ( B → C ) ;A16 (( ¬ A → ¬ B ) ∧ ¬ ( C → A )) → ¬ ( C → B ) and these rules of proof:R1 A , A → B / B (modus ponens);R2 A , B / A ∧ B (adjunction);R3 A → B / ( C → A ) → ( C → B ) (prefixing);R4 A → B / ( B → C ) → ( A → C ) (suffixing);R5 A → B / ¬ ( A → C ) → ¬ ( B → C ) (negated suffixing);R6 ¬ A → ¬ B / ¬ ( C → A ) → ¬ ( C → B ) (negated prefixing).Of the axioms, only A4, A5 and A10 – A16 require any work, the restfollowing via the Deduction Theorem from Dunn–Belnap logic.12 oundness of A4. Suppose that T ∈ v ( x , ( A → C ) ∧ ( B → C )) . Then T ∈ v ( x , A → C ) and T ∈ v ( x , B → C ) ; thus for all y , z ∈ S ,if R xyz and T ∈ v ( y , A ) then T ∈ v ( z , C ) and if R xyz and T ∈ v ( y , B ) then T ∈ v ( z , C ) .But then, if R xyz and T ∈ v ( y , A ) or T ∈ v ( y , B ) then T ∈ v ( z , C ) , henceif R xyz and T ∈ v ( y , A ∨ B ) then T ∈ v ( z , C ) , i.e. , T ∈ v ( x , ( A ∨ B ) → C ) .Thus, for arbitrary M , ( A → C ) ∧ ( B → C ) (cid:15) M ( A ∨ B ) → C and hence ( A → C ) ∧ ( B → C ) (cid:15) ( A ∨ B ) → C . By the Deduction Theorem, (cid:15) (( A → C ) ∧ ( B → C )) → (( A ∨ B ) → C ) . Soundness of A5.
Suppose that T ∈ v ( x , ( A → B ) ∧ ( A → C )) . Then T ∈ v ( x , A → B ) and T ∈ v ( x , A → C ) ; thus ( ∀ y , z ∈ S )[ if R xyz and T ∈ v ( y , A ) then T ∈ v ( z , B )] and ( ∀ y , z ∈ S )[ if R xyz and T ∈ v ( y , A ) then T ∈ v ( z , C )] . But then, ( ∀ y , z ∈ S )[ if R xyz and T ∈ v ( y , A ) then T ∈ v ( y , B ) and T ∈ v ( z , C )] , hence ( ∀ y , z ∈ S )[ if R xyz and T ∈ v ( y , A ) then T ∈ v ( z , B ∧ C )] , i.e. , T ∈ v ( x , A → ( B ∧ C )) . Thus for arbitrary M , ( A → B ) ∧ ( A → C ) (cid:15) M A → ( B ∧ C ) and hence ( A → B ) ∧ ( A → C ) (cid:15) A → ( B ∧ C ) . By the Deduction Theorem, (cid:15) (( A → B ) ∧ ( A → C )) → ( A → ( B ∧ C )) . Soundness of A10.
Suppose that T ∈ v ( x , ¬ (( A ∨ B ) → C )) . Then ( ∃ y , z ∈ S )[ R xyz and T ∈ v ( y , A ∨ B ) and F ∈ v ( z , C )] . But then, ( T ∈ v ( y , A ) or T ∈ v ( y , B )) and F ∈ v ( z , C ) , hence T ∈ v ( y , A ) and F ∈ v ( z , C ) or T ∈ v ( y , B ) and F ∈ v ( z , C ) , i.e. , T ∈ v ( x , ¬ ( A → C )) or T ∈ v ( x , ¬ ( B → C )) ,so T ∈ v ( x , ¬ ( A → C ) ∨ ¬ ( B → C )) . Thus for arbitrary M , ¬ (( A ∨ B ) → C ) (cid:15) M ¬ ( A → C ) ∨ ¬ ( B → C ) and hence ¬ (( A ∨ B ) → C ) (cid:15) ¬ ( A → C ) ∨¬ ( B → C ) . By the Deduction Theorem, (cid:15) ¬ (( A ∨ B ) → C ) → ( ¬ ( A → C ) ∨ ¬ ( B → C )) . Soundness of A11.
Suppose that T ∈ v ( x , ¬ ( A → ( B ∧ C ))) . Then ( ∃ y , z ∈ S )[ R xyz and T ∈ v ( y , A ) and F ∈ v ( z , B ∧ C )] . But then, T ∈ v ( y , A ) and F ∈ v ( z , B ) or F ∈ v ( z , C ) , hence T ∈ v ( y , A ) and F ∈ v ( z , B )] or T ∈ v ( y , A ) and F ∈ v ( z , C )] , i.e. , T ∈ v ( x , ¬ ( A → B )) or T ∈ v ( x , ¬ ( A → C )) , so T ∈ v ( x , ¬ ( A → B ) ∨ ¬ ( A → C )) . Thus for arbitrary M , ¬ ( A → ( B ∧ C )) (cid:15) M ¬ ( A → B ) ∨ ¬ ( A → C ) and hence ¬ ( A → ( B ∧ C )) (cid:15) ¬ ( A → B ) ∨ ¬ ( A → C ) . By the Deduction Theorem, (cid:15) ¬ ( A → ( B ∨ C )) → ( ¬ ( A → B ) ∨ ¬ ( A → C )) . Soundness of A12.
Here we appeal to constraint vi) on R . Suppose that T ∈ v ( x , A ) and T ∈ v ( x , A → B ) . Since R xxx , v ( x , B ) . Thus for arbitrary M , A ∧ ( A → B ) (cid:15) M B . Hence A ∧ ( A → B ) (cid:15) B . By the DeductionTheorem, (cid:15) ( A ∧ ( A → B )) → B . 13 oundness of A13. Here we appeal to constraint vii) on R . Suppose that T ∈ v ( x , A ∧ ¬ B ) . Then T ∈ v ( x , A ) and F ∈ v ( x , B ) . As R xxx wehave that ( ∃ y , z )[ R xyz and T ∈ v M ( y , A ) and F ∈ v ( z , B )] . Hence F ∈ v ( x , A → B ) and so T ∈ v ( x , ¬ ( A → B )) . Thus for arbitrary M , A ∧¬ B (cid:15) M ¬ ( A → B ) . Hence A ∧ ¬ B (cid:15) ¬ ( A → B ) and so, by the DeductionTheorem, (cid:15) ( A ∧ ¬ B ) → ¬ ( A → B ) . Soundness of A14.
Here we appeal to constraint viii) on R . Suppose that T ∈ v ( x , A → B ) and T ∈ v ( x , B → C ) . Let Rxyz . By viii), there’s a node w such that R xyw and R xwz . So now, if T ∈ v ( y , A ) then T ∈ v ( w , B ) andthus T ∈ v ( z , C ) . So ( ∀ y , z )[ if R xyz and T ∈ v M ( y , A ) then T ∈ v ( z , C )] , i.e. , T ∈ v ( x , A → C ) . Hence for arbitrary M , ( A → B ) ∧ ( B → C ) (cid:15) M A → C and so ( A → B ) ∧ ( B → C ) (cid:15) A → C . By the Deduction Theorem, (cid:15) (( A → B ) ∧ ( B → C )) → ( A → C ) . Soundness of A15.
Here we appeal to constraint ix) on R and R . Supposethat T ∈ v ( x , A → B ) and T ∈ v ( x , ¬ ( A → C )) . Let y and z be such that R xyz , T ∈ v ( y , A ) and F ∈ v ( z , C ) . By ix), there’s a node w such that R xyw and R xwz . As T ∈ v ( y , A ) , T ∈ v ( w , B ) . So w and z are suchthat R xwz , T ∈ v ( w , B ) and F ∈ v ( z , C ) . Thus T ∈ v ( x , ¬ ( B → C )) .Hence for arbitrary M , ( A → B ) ∧ ¬ ( A → C ) (cid:15) M ¬ ( B → C ) and so M , ( A → B ) ∧ ¬ ( A → C ) (cid:15) ¬ ( B → C ) . By the Deduction Theorem, (cid:15) (( A → B ) ∧ ¬ ( A → C )) → ¬ ( B → C ) . Soundness of A16.
Here we appeal to constraint x) on R and R . Supposethat T ∈ v ( x , ¬ A → ¬ B ) and T ∈ v ( x , ¬ ( C → A )) . Let y and z be suchthat R xyz , T ∈ v ( y , C ) and F ∈ v ( z , A ) , i.e. , T ∈ v ( z , ¬ A ) . By x), there’s anode w such that R xyw and R xzw . As T ∈ v ( z , ¬ A ) , T ∈ v ( w , ¬ B ) , i.e. , F ∈ v ( w , B ) . So y and w are such that R xyw , T ∈ v ( y , C ) and F ∈ v ( w , B ) .Thus T ∈ v ( x , ¬ ( C → B )) . Hence for arbitrary M , ( ¬ A → ¬ B ) ∧ ¬ ( C → A ) (cid:15) M ¬ ( C → B ) and so ( ¬ A → ¬ B ) ∧ ¬ ( C → A ) (cid:15) ¬ ( C → B ) . By theDeduction Theorem, (cid:15) (( ¬ A → ¬ B ) ∧ ¬ ( C → A )) → ¬ ( C → B ) .For the rules we proceed as follows making heavy use of the Persis-tence Lemma (Lemma 3.1) and the Deduction Theorem (Theorem 3.1). Soundness of R1.
Suppose that, in some model M , (cid:15) M A and (cid:15) M A → B .By (the proof of) the Deduction Theorem, A (cid:15) M B , i.e. , for all ∀ x ∈ S [ if T ∈ v M ( x , A ) then T ∈ v M ( x , B )] . Let u ∈ L . Then T ∈ v M ( u , A ) , hence T ∈ v M ( u , B ) . As this holds for all u ∈ L , (cid:15) M B . Thus (cid:15) B when (cid:15) A and (cid:15) A → B . 14 oundness of R2. Suppose that (cid:15) M A and (cid:15) M B . Let u ∈ L . Then T ∈ v M ( u , A ) and T ∈ v M ( u , B ) . Hence T ∈ v M ( u , A ∧ B ) . As u is an arbitrarymember of L , (cid:15) M A ∧ B . Thus (cid:15) A ∧ B when (cid:15) A and (cid:15) B . Soundness of R3.
Suppose that (cid:15) M A → B . By (the proof of) the DeductionTheorem, A (cid:15) M B . Now, let u ∈ L and R uxy , so that x ≤ y . If T ∈ v M ( x , C → A ) , by the Persistence Lemma, T ∈ v M ( y , C → A ) so that forall z , w ∈ S , if R yzw and T ∈ v M ( z , C ) then T ∈ v M ( w , A ) . As A (cid:15) M B , T ∈ v M ( w , B ) . And thus T ∈ v M ( y , C → B ) . Putting the pieces together,for all u in L , T ∈ v M ( u , ( C → A ) → ( C → B )) , i.e. , (cid:15) M ( C → A ) → ( C → B ) . Thus (cid:15) ( C → A ) → ( C → B ) when (cid:15) A → B . Soundness of R4.
Suppose that (cid:15) M A → B . By (the proof of) the DeductionTheorem, A (cid:15) M B . Now, let u ∈ L and R uxy , so that x ≤ y . If T ∈ v M ( x , B → C ) , by the Persistence Lemma, T ∈ v M ( y , B → C ) . Now, forall z , w ∈ S , if R yzw and T ∈ v M ( z , A ) then, since A (cid:15) M B , T ∈ v M ( z , B ) ;since v M ( y , B → C ) , T ∈ v M ( w , C ) . And thus T ∈ v M ( y , A → C ) . Puttingthe pieces together, for all u in L , T ∈ v M ( u , ( B → C ) → ( A → C )) , i.e. , (cid:15) M ( B → C ) → ( A → C ) . Thus (cid:15) ( B → C ) → ( A → C ) when (cid:15) A → B . Soundness of R5.
Suppose that (cid:15) M A → B . By (the proof of) the DeductionTheorem, A (cid:15) M B . Now, let u ∈ L and R uxy , so that x ≤ y . If T ∈ v M ( x , ¬ ( A → C )) then, by the Persistence Lemma, T ∈ v M ( y , ¬ ( A → C )) so that there exists z and w such that R yzw , T ∈ v M ( z , A ) and F ∈ v M ( w , C ) . Now, as A (cid:15) M B , there exists z and w such that R yzw , T ∈ v M ( z , B ) and F ∈ v M ( w , C ) . Hence T ∈ v M ( y , ¬ ( B → C )) . Putting thepieces together, for all u in L , T ∈ v M ( u , ¬ ( A → C ) → ¬ ( B → C )) , i.e. , (cid:15) M ¬ ( A → C ) → ¬ ( B → C ) . Thus (cid:15) ¬ ( A → C ) → ¬ ( B → C ) when (cid:15) A → B . Soundness of R6.
Suppose that (cid:15) M ¬ A → ¬ B . By (the proof of) the De-duction Theorem, ¬ A (cid:15) M ¬ B . Now, let u ∈ L and R uxy , so that x ≤ y . If T ∈ v M ( x , ¬ ( C → A )) then, by the Persistence Lemma, T ∈ v M ( y , ¬ ( C → A )) so that there exists z and w such that R yzw , T ∈ v M ( z , C ) and F ∈ v M ( w , A ) , i.e. , T ∈ v M ( w , ¬ A ) . Now, as ¬ A (cid:15) M ¬ B , there exists z and w such that R yzw , T ∈ v M ( z , C ) and T ∈ v M ( w , ¬ B ) , i.e. , F ∈ v M ( w , B ) .Hence T ∈ v M ( y , ¬ ( C → B )) . Putting the pieces together, for all u in L , T ∈ v M ( u , ¬ ( C → A ) → ¬ ( C → B )) , i.e. , (cid:15) M ¬ ( C → A ) → ¬ ( C → B ) .Thus (cid:15) ¬ ( C → A ) → ¬ ( C → B ) when (cid:15) ¬ A → ¬ B .15 .2 Completeness Our basic system consists of the axioms A1 – A11 and rules R1 – R6. Thissystem can be extended by some (or all) of A12 to A16.As usual a proof is a finite sequence of formulas of L such that everyformula in the sequence is either an instance of one of the axiom schematain play or is obtained from previous members of the sequence by applica-tion of rules R1 – R6. We write ⊢ A if there is a proof whose last memberis A . Lemma 3.2.
In the basic system, hence in all considered here, ⊢ A → ( A ∧ A ) ; ⊢ ( A ∧ A ) → A; ⊢ A → ( A ∨ A ) ; ⊢ ( A ∨ A ) → A; ⊢ ( A ∧ B ) → ( B ∧ A ) ; ⊢ ( A ∧ ( B ∧ C )) → (( A ∧ B ) ∧ C ) ; ⊢ (( A ∧ B ) ∧ C ) → ( A ∧ ( B ∧ C )) ; ⊢ ( A ∨ B ) → ( B ∨ A ) ; ⊢ ( A ∨ ( B ∨ C )) → (( A ∨ B ) ∨ C ) ; ⊢ (( A ∨ B ) ∨ C ) → ( A ∨ ( B ∨ C )) .Proof. Axioms A1, A3 and A5 and rules R1, R2, and R3 ensure the idem-potency, associativity and commutativity of conjunction; axioms A1, A2and A4 and rules R1, R2, and R4 ensure the idempotency, associativityand commutativity of disjunction.
Definition 3.5 (Theories) . Let X , Y be a non-empty sets of formulas, thena) we write X ⊢ B iff there are formulas A , A , . . . , A n ∈ X , n > , suchthat ⊢ ( A ∧ A ∧ . . . ∧ A n ) → B;b) we write X ⊢ Y iff ( ∃ B , B , . . . , B n ∈ Y ) X ⊢ B ∨ B ∨ . . . ∨ B n ;c) X is a theory iff, for all A ∈ L , A ∈ X when X ⊢ A;d) a theory X is prime iff, for all A , B ∈ L , A ∨ B ∈ X iff A ∈ X or B ∈ X;e) a theory X is proper iff X = L ;f) a theory X is logical iff, for all of A1 – A11 and whichever of A12 –A16 wemay choose to add, every instance in L belongs to X. (This makes ‘logical’a relative notion.) Lemma 3.3 (Properties of theories) . Let X be a theory. Theni) X ⊢ A iff A ∈ X;ii) if Y is a set of formulas, then X ⊢ Y if X ∩ Y = ∅ ;iii) if A ∈ X and ⊢ A → B then B ∈ X; v) X is closed under conjunction;v) the deductive closure of a set of sentences is a theory, i.e., where Y ⊆ L , { A ∈ L : Y ⊢ A } is a theory.Proof. i) ‘If’ from c) in the definition above. ‘Only if’ from axiom schemaA1.ii) and iii) follow trivially.iv) Let A , B ∈ X . By (A1), ⊢ ( A ∧ B ) → ( A ∧ B ) ; as A , B ∈ X , X ⊢ A ∧ B by definition. As X a theory, A ∧ B ∈ X .v) Let Z = { A ∈ L : Y ⊢ A } . If Z ⊢ A then ( ∃ B , . . . , B n ∈ Z ) ⊢ ( B ∧ . . . ∧ B n ) → A . As Y ⊢ B i for each B i , there are C i , . . . , C ii m ∈ Y such that ⊢ C i ∧ . . . ∧ C ii m → B i . By A3 and R4 (suffixing), andappealing heavily to Lemma 3.2, for each i , 1 ≤ i ≤ n , ⊢ ( C ∧ . . . ∧ C i ∧ C ∧ . . . ∧ C i ∧ . . . ∧ C n ∧ . . . ∧ C ni n ) → B i ; by R2, A5, andR1, ⊢ ( C ∧ . . . ∧ C i ∧ . . . ∧ C n ∧ . . . ∧ C ni n ) → ( B ∧ . . . ∧ B n ) and byR4 (suffixing), ⊢ ( C ∧ . . . ∧ C i ∧ . . . ∧ C n ∧ . . . ∧ C ni n ) → A . Hence Y ⊢ A and A ∈ Z . Lemma 3.4. If ⊢ A → ( B ∨ C ) and ⊢ ( D ∧ C ) → E then ⊢ ( A ∧ D ) → ( B ∨ E ) .Proof. By A3, R1 and R4, ⊢ ( A ∧ D ) → ( B ∨ C ) ; by A3, A5, R1 and R2, ⊢ ( A ∧ D ) → (( B ∨ C ) ∧ D ) . By A2, A3, A4, A5, R1 and R2, ⊢ (( B ∨ C ) ∧ D ) → ( D ∧ ( C ∨ B )) . By R1 and R3, ⊢ ( A ∧ D ) → ( D ∧ ( C ∨ B )) . By A6,R1 and R3, ⊢ ( A ∧ D ) → (( D ∧ C ) ∨ B ) . By A2 and R3, ⊢ ( D ∧ C ) → ( B ∨ E ) . By A2, A4, R1 and R2, ⊢ (( D ∧ C ) ∨ B ) → ( B ∨ E ) . By R1 and R3, ⊢ ( A ∧ D ) → ( B ∨ E ) . Corollary 3.1 (Cut) . If X ∪ { C } ⊢ Y and X ′ ⊢ Y ′ ∪ { C } then X ∪ X ′ ⊢ Y ∪ Y ′ .Proof. Suppose that X ∪ { C } ⊢ Y and X ′ ⊢ Y ′ ∪ { C } . Then there are A , . . . , A m ∈ X and B , . . . , B n ∈ Y such that ⊢ ( A ∧ . . . ∧ A m ∧ C ) → ( B ∨ . . . ∨ B n ) . Similarly there are A ′ , , . . . , A ′ k ∈ X ′ and B ′ , . . . , B ′ l ∈ Y such that ⊢ ( A ′ ∧ . . . ∧ A ′ k ) → ( B ′ ∨ . . . ∨ B ′ l ∨ C ) . By the previous lemma, ⊢ ( A ∧ . . . ∧ A m ∧ A ′ ∧ . . . ∧ A ′ k ) → ( B ∨ . . . ∨ B n ∨ B ′ ∨ . . . ∨ B ′ l ) . Hence X ∪ X ′ ⊢ Y ∪ Y ′ . 17 heorem 3.2 (Lindenbaum’s Lemma) . If we have two sets of formulas X , Ysuch that X Y then we can extend them to X ′ , Y ′ such that X ⊆ X ′ , Y ⊆ Y ′ , X ′ Y ′ and their union is the whole language X ′ ∪ Y ′ = L , . Moreover X ′ isa prime theory and Y ′ is closed under disjunction.Proof. Let A , . . . , A n . . . be an enumeration of L . Let X = X , Y = YX n + = X n ∪ { A n } , Y n + = Y n i f X n ∪ { A n } Y n X n + = X n , Y n + = Y n ∪ { A n } i f X n ∪ { A n } ⊢ Y n X ′ = [ n ∈ N X n Y ′ = [ n ∈ N Y n Obviously, X ⊆ X ′ , Y ⊆ Y ′ and X ′ ∪ Y ′ = L . We show by induction that X n Y n for all n . By definition X Y . Suppose that X n Y n and X n + ⊢ Y n + . If X n ∪ { A n } Y n then by the construction X n + = X n ∪ { A n } and Y n + = Y n . Then X n + ⊢ Y n + implies X n ∪ { A n } ⊢ Y n contrary towhat we assumed. So it must be the case that X n ∪ { A n } ⊢ Y n and byconstruction X n + = X n , Y n + = Y n ∪ { A n } . Hence X n ∪ { A n } ⊢ Y n and X n ⊢ Y n ∪ { A n } . By Cut, X n ⊢ Y n , a contradiction. Hence for all n ∈ N , X n Y n . But now, if X ′ ⊢ Y ′ then, by the finiteness of proof, for some n ∈ N , X n + ⊢ Y n + , which we have just shown is not possible. Hence X ′ Y ′ .Suppose that X ′ ⊢ A and A / ∈ X ′ . As X ′ ∪ Y ′ = L , A ∈ Y ′ . But then X ′ ⊢ Y ′ which we have just shown not to be the case. Thus A ∈ X ′ and X ′ is a theory.Suppose that A ∨ B ∈ X ′ , A / ∈ X ′ and B / ∈ X ′ . As X ′ ∪ Y ′ = L , A ∈ Y ′ and B ∈ Y ′ . So X ′ ⊢ Y ′ which we have just shown not to be the case.Thus A ∈ X ′ or B ∈ X ′ . X ′ is prime. (Notice that since X ′ ∪ Y ′ = L and, by Lemma 3.3 ii), X ′ ∩ Y ′ = ∅ , X ′ is prime iff Y ′ is closed underdisjunctions.)(Thanks to the appeal to Cut, in deriving Lindenbaum’s Lemma we haveused axiom schemata A1, A2, A3, A4, A5 and A6 and rules R1, R2, R3, andR4.) Corollary 3.2.
Let X be a theory and Y a set of formulas disjoint from X andclosed under disjunction. Then there is a prime theory X ′ such that X ⊆ X ′ andX ′ is disjoint from Y.Proof. If X ⊢ Y then, for some B , B , . . . , B n ∈ Y , X ⊢ B ∨ B ∨ . . . ∨ B n .But then B ∨ B ∨ . . . ∨ B n ∈ X , as X is a theory and B ∨ B ∨ . . . ∨ B n ∈ Y Y is closed under disjunction so, contrary to hypothesis, X ∩ Y = ∅ .So X Y and, by Lindenbaum’s Lemma, ∃ X ′ , Y ′ such that X ⊆ X ′ , Y ⊆ Y ′ , X ′ ∪ Y ′ = L , X ′ Y ′ and X ′ is a prime theory. By Lemma 3.3 ii), X ′ and Y ′ are disjoint, hence X ′ is disjoint from Y . Definition 3.6 (Operations on sets of sentences) . Let X and Y be non-emptysets of formulas. Thena) For subsets X and Y of L , X = Y = { C ∈ L : ( ∃ A ∈ L )[ A → C ∈ X and A ∈ Y ] } .b) For subsets X and Y of L , X > Y = {¬ ( A → C ) : A ∈ X , ¬ C ∈ Y } . Lemma 3.5 (A fact about = ) . When X and Y are theories, X = Y is a theory.Proof.
We first show that when X and Y are theories, X = Y is closed underconjunction.If C , D ∈ X = Y then ( ∃ E , F ∈ Y )[ E → C and F → D ∈ X ] . By A3 andR4 (suffixing), X ⊢ ( E ∧ F ) → C and X ⊢ ( E ∧ F ) → D . As X is a theory, ( E ∧ F ) → C ∈ X and ( E ∧ F ) → D ∈ X ; by A5, X ⊢ ( E ∧ F ) → ( C ∧ D ) ;as X is a theory, ( E ∧ F ) → ( C ∧ D ) ∈ X . By Lemma iv), E ∧ F ∈ Y . Hence C ∧ D ∈ X = Y .We now show that when X and Y are theories, X = Y is deductivelyclosed.If X = Y ⊢ C then, making use of what we have just shown, ( ∃ E ∈ X = Y ) ⊢ E → C . As E ∈ X = Y , ( ∃ B ∈ Y ) B → E ∈ X . By R3 (prefixing), ⊢ ( B → E ) → ( B → C ) , hence X ⊢ B → C and so B → C ∈ X . But then C ∈ X = Y .By Lemma 3.3 v), X = Y is a theory. Now we are ready to build our canonical model. As usual, the domain S ofthe canonical model comprises all proper, prime theories in L , theoryhoodbeing relative to the logic in play. Definition 3.7.
Let L be a logic in a language L axiomatized by the axiomschemata A1-A11 and the rules R1-R6, possibly with some of the axiom schemataA12-A16. We denote by ⊢ L the corresponding provability relation. We define ourcanonical frame F L = h S , L , ≤ , R , R i over the domain S of all proper primetheories with the canonical relations defined as follows:i) L = { X ∈ S : { A ∈ L : ⊢ L A } ⊆ X } . i) For X , Y ∈ S, Y ≤ Z iff Y ⊆ Z.iii) For theories X , Y , Z, R XYZ iff X = Y ⊆ Z.iv) For theories X , Y , Z, R XYZ iff Y > Z ⊆ X. We should check that the canonical frame is indeed a frame, i.e. , it sat-isfies the conditions of Definition 3.1.
Lemma 3.6 (Canonical frame) . A canonical frame F L satisfies the conditionsof Definition 3.1 relevant to the logic L .Proof. We must first check that, for all X , Y in S , X ⊆ Y iff ( ∃ U ∈ L ) R UXY .Suppose, first, that X ⊆ Y . Let t = { C ∈ L : ⊢ L C } . If A → B ∈ t and A ∈ X then ⊢ L A → B ; by Lemma 3.3 iii), B ∈ X hence B ∈ Y . Thus t = X ⊆ Y , i.e. , R tXY .Now we show that when X , Y and Z are proper theories, R XYZ and Z is prime, there is a proper, prime theory X ′ such that X ⊆ X ′ and R X ′ YZ .To begin, let W = { A ∈ L : for some B ∈ Y , C / ∈ Z , A ⊢ L B → C } . As Z is proper, W is non-empty. Let A , B ∈ W , so, for C , D ∈ Y , E , F / ∈ Z , A ⊢ L C → E and B ⊢ L D → F . By prefixing and suffixing and appeals toaxioms A2 and A3, ⊢ L A → (( C ∧ D ) → ( E ∨ F )) and ⊢ L B → (( C ∧ D ) → ( E ∨ F )) ; by R A R A ∨ B ⊢ L ( C ∧ D ) → ( E ∨ F ) . C ∧ D ∈ Y as Y is a theory; E ∨ F / ∈ Z as Z is prime. Hence A ∨ B ∈ W . — W is closedunder disjunction.Suppose that A ∈ X ∩ W . For some B ∈ Y , C / ∈ Z , A ⊢ L B → C . As A ⊢ L B → C , X ⊢ L B → C . As X is a theory, B → C ∈ X . But then C ∈ Z as R XYZ . — Contradiction. Thus X ∩ W = ∅ . By Corollary 3.2, there isa prime theory X ′ such that X ⊆ X ′ and X ′ is disjoint from W . As W = ∅ , X ′ is proper. Let A → B ∈ X ′ . If A ∈ Y then A → B / ∈ W , hence B ∈ Z .Thus X ′ = Y ⊆ Z and R X ′ YZ .And so there’s a proper, prime theory U such that t ⊆ U and R UXY ;as t ⊆ U , U ∈ L .To show the converse, i.e. , that X ⊆ Y if ( ∃ U ∈ L ) R UXY , we note thatif U ∈ L then, for all A ∈ L , A → A ∈ U , hence A ∈ U = X and thus A ∈ Y since R UXY when A ∈ X .i) and ii) (reflexivity and transitivity of ≤ ) are trivial.iii) That L is upward closed subset of h S , ≤i is immediate from the defi-nition.iv) If W ⊆ X then W = Y ⊆ X = Y , hence W = Y ⊆ Z when X = Y ⊆ Z .20) If X ⊆ W then Y > Z ⊆ W when Y > Z ⊆ X .vi) If for some A ∈ X , A → C ∈ X then, by Lemma 3.3 iv), A ∧ ( A → C ) ∈ X . According to A10, ⊢ L ( A ∧ ( A → C )) → C , hence X ⊢ L C ;as X is a theory, C ∈ X . Thus R XXX by definition, given axiom A10 .vii) Suppose that A ∈ X and ¬ C ∈ X . By Lemma 3.3 iv), A ∧ ¬ C ∈ X .According to A11, ⊢ L ( A ∧ ¬ C ) → ¬ ( A → C ) hence X ⊢ L ¬ ( A → C ) ; as X is a theory, ¬ ( A → C ) ∈ X . Thus R XXX by definition, given axiom A11 .viii) Suppose that R XYZ . Then R XY ( X = Y ) , as follows from the def-inition of R , and X = Y is proper as Z is. Now let B ∈ X = Y , B → C ∈ X . Then, for some A ∈ Y , A → B ∈ X . By Lemma 3.3 iv), ( A → B ) ∧ ( B → C ) ∈ X . According to A12, ⊢ L (( A → B ) ∧ ( B → C )) → ( A → C ) hence X ⊢ L A → C ; as X is a theory, A → C ∈ X .As A ∈ Y , C ∈ X = Y . Thus R X ( X = Y )( X = Y ) by definition. As R XYZ , X = Y ⊆ Z and so R X ( X = Y ) Z ) .We show next that, for any proper theories X , Y and Z , if R XYZ and Z is prime then there is a proper, prime theory Y ′ such that Y ⊆ Y ′ and R XY ′ Z . To begin, let W = { A ∈ L : ( ∃ B ∈ L )[ A → B ∈ X and B / ∈ Z ] } . As R XYZ , Y and W are disjoint. Let A , B ∈ W , so, forsome C , D / ∈ Z , A → C , B → D ∈ X . As Z is prime, C ∨ D / ∈ Z . Also,by R3 (prefixing), A → ( C ∨ D ) , B → ( C ∨ D ) ∈ X ; by A3, R4, R2,A4, and R1, ( A ∨ B ) → ( C ∨ D ) ∈ X , thus A ∨ B ∈ W . — W is closedunder disjunction. If W = ∅ , we may take Y ′ to be any proper, primeextension of Y ; as W = ∅ , R XY ′ Z . If W = ∅ , then, by Corollary 3.2,there is a (proper) prime theory Y ′ such that Y ⊆ Y ′ and Y ′ is disjointfrom W . Let A → B ∈ X : if A ∈ Y ′ then A / ∈ W , hence B ∈ Z ; thus X = Y ′ ⊆ Z and R XY ′ Z .We have shown, given axiom A12 , that when R XYZ there is a proper,prime theory W such that R XWZ and R XYW .ix) Suppose that ¬ ( B → C ) ∈ ( X = Y ) > Z . Then B ∈ X = Y and ¬ C ∈ Z . As B ∈ X = Y , for some A ∈ Y , A → B ∈ X and ¬ ( A → C ) ∈ Y > Z . As R XYZ , ¬ ( A → C ) ∈ X . By Lemma 3.3 iv), ( A → B ) ∧ ¬ ( A → C ) ∈ X . According to A13, ⊢ (( A → B ) ∧ ¬ ( A → C )) → ¬ ( B → C ) hence X ⊢ ¬ ( B → C ) ; as X is a theory, ¬ ( B → C ) ∈ X . Thus ( X = Y ) > Z ⊆ X , i.e. , R X ( X = Y ) Z .We show next that, for any proper theories X , Y and Z , if R XYZ and X is prime, there is a proper, prime theory Y ′ such that Y ⊆ Y ′ and21 XY ′ Z . To begin, let W = { A ∈ L : ( ∃ B ∈ L )[ ¬ B ∈ Z and ¬ ( A → B ) / ∈ X ] } . As R XYZ , Y and W are disjoint. Let A , C ∈ W , so, forsome ¬ B ∈ Z , ¬ ( A → B ) / ∈ X and, for some ¬ D ∈ Z , ¬ ( C → D ) / ∈ X . ¬ B ∧ ¬ D ∈ Z , as Z is a theory; by A7, ¬ ( B ∨ D ) ∈ Z . As ⊢ ¬ ( B ∨ D ) → ¬ B , ⊢ ¬ ( A → ( B ∨ D )) → ¬ ( A → B ) by R6. Hence ¬ ( A → ( B ∨ D )) / ∈ X . Likewise ¬ ( C → ( B ∨ D )) / ∈ X . As X isprime, ¬ ( A → ( B ∨ D )) ∨ ¬ ( C → ( B ∨ D )) / ∈ X . By axiom A10, ⊢ ¬ (( A ∨ C ) → ( B ∨ D )) → ( ¬ ( A → ( B ∨ D )) ∨ ¬ ( C → ( B ∨ D ))) ,hence ¬ (( A ∨ C ) → ( B ∨ D )) / ∈ X . Thus A ∨ C ∈ W . — W is closedunder disjunction. If W = ∅ , we may take Y ′ to be any proper, primeextension of Y . As W = ∅ , R XY ′ Z . If W = ∅ , then, by Corollary3.2, there is a prime theory Y ′ such that Y ⊆ Y ′ and Y ′ ∩ W = ∅ . If A ∈ Y ′ then A / ∈ W , hence, for all ¬ B ∈ Z , ¬ ( A → B ) ∈ X . Thus Y ′ > Z ⊆ X and so R XY ′ Z .We have shown, given axiom A13 , that when R XYZ there is a proper,prime theory W such that R XYW and R XWZ .x) Suppose that R XYZ . Let A ∈ Y , ¬ B ∈ X = Z . So, for some C ∈ Z , C → ¬ B ∈ X . By A9 and R3, ¬¬ C → ¬ B ∈ X and, by A9, ¬¬ C ∈ Z (as Z is a theory). As R XYZ , ¬ ( A → ¬ C ) ∈ X . ByLemma 3.3 iv), ( ¬¬ C → ¬ B ) ∧ ¬ ( A → ¬ C ) ∈ X . According to A14, ⊢ (( ¬¬ C → ¬ B ) ∧ ¬ ( A → ¬ C )) → ¬ ( A → B ) , hence ¬ ( A → B ) ∈ X . Thus Y > ( X = Z ) ⊆ X , i.e. , R XY ( X = Z ) .We show next that, for any proper theories X , Y and Z , if R XYZ and X is prime, there is a proper, prime theory Y ′ such that Y ⊆ Y ′ and R XY ′ Z . To begin, let W = { A ∈ L : ( ∃ B ∈ L )[ ¬ B ∈ Z and ¬ ( A → B ) / ∈ X ] } . As R XYZ , Y and W are disjoint. Let A , C ∈ W , so, forsome ¬ B ∈ Z , ¬ ( A → B ) / ∈ X and, for some ¬ D ∈ Z , ¬ ( C → D ) / ∈ X . ¬ B ∧ ¬ D ∈ Z , as Z is a theory; by A7, ¬ ( B ∨ D ) ∈ Z . As ⊢ ¬ ( B ∨ D ) → ¬ B , ⊢ ¬ ( A → ( B ∨ D )) → ¬ ( A → B ) by R6. Hence ¬ ( A → ( B ∨ D )) / ∈ X . Likewise ¬ ( C → ( B ∨ D )) / ∈ X . As X isprime, ¬ ( A → ( B ∨ D )) ∨ ¬ ( C → ( B ∨ D )) / ∈ X . By axiom A10, ⊢ ¬ (( A ∨ C ) → ( B ∨ D )) → ( ¬ ( A → ( B ∨ D )) ∨ ¬ ( C → ( B ∨ D ))) ,hence ¬ (( A ∨ C ) → ( B ∨ D )) / ∈ X . Thus A ∨ C ∈ W . — W is closedunder disjunction. If W = ∅ , we may take Y ′ to be any proper, primeextension of Y . As W = ∅ , R XY ′ Z . If W = ∅ , then, by Corollary3.2, there is a prime theory Y ′ such that Y ⊆ Y ′ and Y ′ ∩ W = ∅ . If A ∈ Y ′ then A / ∈ W , hence, for all ¬ B ∈ Z , ¬ ( A → B ) ∈ X . Thus Y ′ > Z ⊆ X and so R XY ′ Z .We have shown, given axiom A14 , that when R XYZ there is a proper,22rime theory W such that R XYW and R XZW .Adding the canonical valuation defined for atomic p in L as T ∈ v ( X , p ) iff p ∈ X , F ∈ v ( X , p ) iff ¬ p ∈ X we obtain the canonical LScD model M . So defined, v automatically sat-isfies the pair of constraintsi) if X ≤ Y and T ∈ v ( X , p ) then T ∈ v ( Y , p ) ;ii) if X ≤ Y and F ∈ v ( X , p ) then F ∈ v ( Y , p ) .So h F , v i is a model in the sense of Definition 3.2. Lemma 3.7 (Valuation lemma) . Given a canonical model M C = h F C , v i , forall X ∈ S and A ∈ L :T ∈ v M ( X , A ) iff A ∈ X andF ∈ v M ( X , A ) iff ¬ A ∈ X. In the inductive proof that T ∈ v ( X , A ) iff A ∈ X , we skip the easycases and attend only to conditionals and their negations. We have:• If A → B ∈ X then, for any Y , Z ∈ S such that R XYZ , if A ∈ Y then B ∈ Z . By the induction hypothesis this means that T ∈ v ( Y , A ) and T ∈ v ( Z , B ) , hence T ∈ v ( X , A → B ) .• Suppose that A → B / ∈ X . Let Y = { B ∈ L : A ⊢ L B } ; let Z = X = Y .By lemmata 3.3 v) and 3.5, Y and Z are theories. Obviously, A ∈ Y .If B ∈ Z = X = Y , then A → B ∈ X ; as this is contrary to hypothesis, B / ∈ Z . By Lindenbaum’s Lemma, there is a proper, prime theory Z ′ such that Z ⊆ Z ′ and B / ∈ Z ′ so Z ′ is proper. X and Z ′ are proper, prime theories. As R XYZ , R XYZ ′ and so, asin the proof of Lemma 3.6 viii), there is a proper, prime theory Y ′ extending Y such R XY ′ Z ′ , where A ∈ Y ′ , B / ∈ Z ′ . By the inductionhypothesis, T ∈ v ( Y ′ , A ) and T / ∈ v ( Z ′ , B ) , hence T / ∈ v ( X , A → B ) .23 Let ¬ ( A → B ) ∈ X . Let Y = { C ∈ L : A ⊢ L C } and Z = { C ∈ L : ¬ B ⊢ L C } . By Lemmata 3.3 v), Y and Z are theories. Let C ∈ Y and ¬ D ∈ Z . As ⊢ A → C , ¬ ( C → B ) ∈ X , by R5 (negated suffixing);as ⊢ ¬ B → ¬ D , ¬ ( C → D ) ∈ X , by R6 (negated prefixing). Thus Y > Z ⊆ X , i.e. , R XYZ . As in the proof of Lemma 3.6 ix) and x),there are proper, prime theories Y ′ and Z ′ such that Y ⊆ Y ′ , Z ⊆ Z ′ and R XY ′ Z ′ , where A ∈ Y ′ , ¬ B ∈ Z ′ . By the IH we have R XY ′ Z ′ and T ∈ v ( Y ′ , A ) and F ∈ v ( Z ′ , B ) , hence T ∈ v ( X , ¬ ( A → B )) .• if ¬ ( A → B ) / ∈ X then, if R XYZ , ¬ ( A → B ) / ∈ Y > Z so A / ∈ Y or ¬ B / ∈ Z . By the IH we have ( ∀ Y , Z ∈ S )[ if R XYZ and T ∈ v ( Y , A ) then T / ∈ v ( Z , ¬ B )] , i.e. , it is not the case that ( ∃ Y , Z ∈ S )[ R XYZ and T ∈ v ( Y , A ) and F ∈ v ( Z , B )] . Thus T / ∈ v ( X , ¬ ( A → B )) .This completes the proof. If X L A , then the canonical model is a counter-model: X = M C A , i.e. , there is a node Y ∈ S C such that T ∈ v M C ( Y , B ) foreach B ∈ X and T / ∈ v M C ( Y , A ) . Some comments are now in order about the interpretation of our logics ofscientific discovery. While they all share the same rules, the minimal
LScD has axioms A1-A11 only. Other logics will be justified by their capturinghow laboratories share data (and not by logical convenience). For exam-ple, while axiom A12 would be logically convenient, it also correspondsto the strong requirement that the R accessibility relation be reflexive, R xxx . Thus for every lab there would be at least one test on which it un-dertakes all the work: intiating the test, completing the test, collating theresults of the tests, and evaluating them.It is also worth noting that none of the LScD logics have contraposition,not even in rule form. That is, A → ¬ B / B → ¬ A is not sound. The following model M = h F , v i demonstrates this: we set S = { u , x , y } , L = { u } , R = {h u , u , u i , h u , x , x i , h u , x , y i , h u , y , y i , h x , x , x i , h x , x , y i , h x , y , y i , h y , y , y i} , R = {h u , u , u i , h x , x , x i , h y , y , y i , h y , x , x i} ; v u ( A ) = v x ( A ) = v y ( A ) = ∅ ; v u ( B ) = v x ( B ) = v y ( B ) = { T , F } . Taking z ≤ w toobtain when ( ∃ u ∈ L ) R uzw , we have ≤ = {h u , u i , h x , x i , h x , y i , h y , y i} .We then set F = h S , L , ≤ , R , R i . This model satisfies all of conditions i )– x ) in Definition 3.1. Moreover, T ∈ v u ( A → ¬ B ) , T / ∈ v u ( B → ¬ A ) .24e can motivate the failure of contraposition with the hoary exam-ple of Eddington’s expedition, set up to determine whether its positionappears to shift when a star’s light passes near a massive object. It’s adifferent matter to test whether a star’s position does not appear to shiftwhen its light does not pass near a massive object. Even sillier: if you petyour cat it will purr – this is easily tested. But checking whether your cat’snot purring when you’re not petting it is not so simple. The former canbe checked from your couch, the latter may require significant mobilityand stealth. The absence of contraposition goes to explain the differencebetween our approach and Routley’s, a number of constraints he placeson the second accessibility relation being designed to deliver contraposi-tion. Likewise Oshini places constraints on the second accessibility rela-tion with the aim of providing semantics for a co-implication connective.The failure of contraposition in our system is mitigated to some extent bythe holding of the rules for negated prefixing and negated suffixing (R5and R6). It’s failure in general, though, makes comparison with extantsystems of relevance logic difficult.The LScD logics are very flexible: but scientific practice requires evenmore flexibility. We have operated on the assumption that lab reports areunambiguous, even when reporting their ambiguity. In the following partwe remove this assumption by incorporating probabilities into our frame-work. 25 art II: Probabilities
In Part I we developed Logics of Scientific Discovery that describe certainaspects of scientific practice. In this part we take note of another aspectof scientific practice: lab resorts almost always involve probabilities. Wedefine probabilities – or, more accurately, appropriately generalized prob-ability functions – with our Logic(s) of Scientific Discovery, not classicallogic, as the underlying logic. We provide relative frequency and bettingquotient interpretations.We begin with probabilities at the level of the individual laboratory, i.e.probabilities for propositions in the → -free fragment of our logics (Dunn-Belnap logic). Later in this part we give an analysis of the interaction ofprobabilities over networks of laboratories, so defining probabilities overthe full vocabulary. We are led to develop analogues of Bayesian condi-tionalization, Jeffrey conditionalization and Adams conditioning. Proba-bilities of conditionals are dealt with in the last section. We begin with a generalized version of the Kolmogorov axioms.
Definition 5.1 (Probabilities) . A probability space is a pair hL , p i , where L isthe set of all → -free formulas generated by the set At ( L ) of atomic formulas in L (see §3), (cid:15) is the relation of logical consequence specified in 3.4, and p is afunction from L into the real numbers satisfying:i) for all A ∈ L , ≤ p ( A ) ≤ ,ii) for all A , B ∈ L , if A (cid:15) B then p ( A ) ≤ p ( B ) ,iii) for all A , B ∈ L , p ( A ∧ B ) + p ( A ∨ B ) = p ( A ) + p ( B ) ,iv) for all A , B ∈ L , if p ( B ) > then p ( A | B ) = p ( A ∧ B ) p ( B ) . As they stand, the axioms admit a trivialising interpretation: a functionwhich uniformly assigns the value 0 to all members of L , leaving p ( A | B ) undefined for all pairs A , B . We could exclude it by adding this principleas a further axiom: 26or some A ∈ L , 0 < p ( A ) .Axiom 5.1 iii) is written to account for the non-Boolean structure of thelanguage. As we employ Dunn-Belnap four-valued logic, negation doesnot determine partitions; as negation and partitions come apart, we are nolonger guaranteed that p ( A ∧ ¬ A ) =
0. Indeed the usual statement of theadditivity axiom for propositions is devoid of application. We replace itwith what is in classical probability theory an easily derived consequence.Suppose we hold it possible that A be both (reported) true and (re-ported) false. Then we may assign a non-zero probability to A ∧ ¬ A andthus, from Axiom 5.1 iii), it follows that p ( A ) + p ( ¬ A ) > p ( A ∨ ¬ A ) . Sup-pose, next, that we hold it possible that A be neither (reported) true nor(reported) false. In close analogy to the previous case, taking the uncer-tainty u ( B ) assigned a proposition B to be 1 − p ( B ) , we should assign anon-zero uncertainty to A ∨ ¬ A and thus, from Axiom 5.1 iii), we find that u ( A ) + u ( ¬ A ) > u ( A ∧ ¬ A ) . Axiom 5.1 ii), too, is, in the classical setting, derived from the sameadditivity axiom and the constraint—not sound in our setting! but often,classically, adopted as an axiom—that p ( A ) + p ( ¬ A ) = From Axiom 5.1 iv) we find, thanks to the resources of Dunn–Belnaplogic, that when p ( B ) > p ( . | B ) satisfies Definition 5.1 i) – iii) with the up-per bound in Axiom 5.1 i) attained by p ( B | B ) ; moreover, when p ( C | B ) > p ( A ∧ C | B ) p ( C | B ) = p ( A | B ∧ C ) .We now turn to providing relative frequency and betting quotient in-terpretations of the axioms. The orthodox relative frequency interpretation is readily adapted to ourframework, the only necessary modification needed being separate defi-nitions of the frequency of a proposition and its negation. With outcomesof the n th trial as stipulated, we have: Cf. [23, pp. 107–108]. For related analyses see, e.g. , [16, 18, 30]. By which we mean: if (cid:15) ¬ ( A ∧ B ) then p ( A ∨ B ) = p ( A ) + p ( B ) . We take this notion of uncertainty from [1, 11]. Classically, if A (cid:15) CL B then (cid:15) CL ¬ ( A ∧ ¬ B ) , and hence, from i), respected classically,and the (classical) additivity axiom, we have that 1 ≥ p ( A ∨ ¬ B ) = p ( A ) + p ( ¬ B ) = p ( A ) + ( − p ( B )) whence p ( A ) ≤ p ( B ) . efinition 5.2 (Relative Frequencies) . f req n ( A ) = ( f req n − ( A ) + if T ∈ v ( A ) , f req n − ( A ) if T / ∈ v ( A ) . (1) f req n ( ¬ A ) = ( f req n − ( ¬ A ) if F / ∈ v ( A ) , f req n − ( ¬ A ) + if F ∈ v ( A ) . (2)Relative frequency is r f req n ( A ) = f req n ( A ) n . Lemma 5.1 ( rfreq is a probability) . rfreq satisfies the axioms of Definition 5.1.Proof. Axiom 5.1 i): Obvious.Axiom 5.1 ii): By Definition 3.4, if A (cid:15) B , then whenever T ∈ v ( A ) , T ∈ v ( B ) and hence by Definition 5.2, for any n , f req n ( B ) ≥ f req n ( A ) .Axiom 5.1 iii): By induction, for any n , f req n ( A ∨ B ) + f req n ( A ∧ B ) = f req n ( A ) + f req n ( B ) .Axiom 5.1 iv): The conditional probability p ( A | B ) is the relative frequencyof A restricted to trials in which B is the outcome, that is, f req n ( A ∧ B ) f req n ( B ) , i.e. r f req n ( A ∧ B ) r f req n ( B ) , assuming that B has occurred, i.e., f req n ( B ) > The betting quotient interpretation of probability is also readily adaptedto our framework.
Definition 5.3 (Bet) . A bet on (proposition) A with (positive or negative) stake S at betting quotient p pays ( − p ) S to the bettor if A takes a designated value( { T } , { T , F } ) and pays pS to the bookmaker if it doesn’t. Definition 5.4 (Conditional bet) . A (conditional) bet on (proposition) B con-ditional on (proposition) A with (positive or negative) stake S at betting quo-tient p pays nothing to either bettor or bookmaker if A does not take a designatedvalue and otherwise pays ( − p ) S to the bettor if B takes a designated value andpays pS to the bookmaker if it doesn’t.
Notice that in these definitions the amounts paid to bettor and to book-maker may be negative; equivalently, stakes are always positive but theroles of bettor and bookmaker 28 efinition 5.5 (Dutch book) . A bettor faces a phDutch book on a family ofbets if, given the chosen betting quotients and stakes, she faces certain loss, i.e. , on all assignments of sets of truth-values to atomic propositions, thebettor suffers a net loss (which is paid to the bookmaker).
Theorem 5.1 (Dutch Book Argument) . A bettor may face a Dutch Book on afinite family of bets, through an unfortunate choice of stakes, if her betting quo-tients do not satisfy Axioms i) – iv).Proof.
Axiom 5.1 i).
Firstly, − pS and ( − p ) S are both negative if, andonly if, either ( i ) S < p < ii ) S > p >
1. — Our bettorfaces a Dutch Book on a single bet if, and only if, p < p > A at betting quotient p with stake S and at bet-ting quotient q with stake S . From the immediately preceding, the bettorimmediately faces a Dutch book if any of these are the case: p < p > q < q > ≤ p ≤ ≤ q ≤ G = − pS − qS T / ∈ v ( A ) G = ( − p ) S + ( − q ) S T ∈ v ( A ) .If G < G < ( − p ) G + pG <
0, hence ( p − q ) S < p = q .Now, supposing that p = q , choose S of the same sign as p − q and set S = S , S = − S . We then have G = − pS + qS = ( q − p ) S < G = ( − p ) S − ( − q ) S = ( q − p ) S < ≤ p ≤ ≤ q ≤
1, our bettor faces a Dutch Bookon the pair of bets if, and only if, p = q . Axiom 5.1 ii).
There are two cases to consider. Firstly if, in addition to A (cid:15) B , B (cid:15) A , there are just two possibilities when we consider a pairof bets on A and B at betting quotients p and q and stakes S and S ,respectively: neither A nor B takes a designated value or both do.From above, the bettor immediately faces a Dutch book if any of theseare the case: p < p > q < q > ≤ p ≤ ≤ q ≤
1. Algebraically, the argument now proceeds exactly asabove for there are just these two cases to consider:29 = − pS − qS T / ∈ v ( A ) , T / ∈ v ( B ) G = ( − p ) S + ( − q ) S T ∈ v ( A ) , T ∈ v ( B ) .Consequently, granted that 0 ≤ p ≤ ≤ q ≤
1, our bettor facesa Dutch Book on the pair of bets if, and only if, p = q .The second case: A (cid:15) B but B A . There are three possibilities whenwe consider a pair of bets on A and B at betting quotients p and q andstakes S and S , respectively: neither A nor B takes a designated value, B takes a designated value but A does not, both take a designated value. Asbefore, the bettor immediately faces a Dutch book if any of these are thecase: p < p > q < q > ≤ p ≤ ≤ q ≤ G = − pS − qS T / ∈ v ( A ) , T / ∈ v ( B ) G = − pS + ( − q ) S T / ∈ v ( A ) , T ∈ v ( B ) G = ( − p ) S + ( − q ) S T ∈ v ( A ) , T ∈ v ( B ) .If G < G < G < ( − q ) G + qG < ( − p ) G + pG <
0, hence − pS < ( − q ) S < S > S <
0. Now, ( − q ) G + qG <
0, hence ( q − p ) S <
0. And so p > q .Now, supposing that p > q , choose S > S = − S . We findthat G = − pS − qS = ( q − p ) S < G = − pS + ( − q ) S = − pS − ( − q ) S < G = ( − p ) S + ( − q ) S = ( q − p ) S < ≤ p ≤ ≤ q ≤
1, our bettor faces a Dutch Bookon the triple of bets if, and only if, p > q . Axiom 5.1 iii).
Four bets are to be made: on A at betting quotient p andstake S , on B at betting quotient q and stake S , on A ∧ B at betting quo-tient r and stake S and on A ∨ B at betting quotient s and stake S . As A ∧ B (cid:15) A , A ∧ B (cid:15) B , A (cid:15) A ∨ B and B (cid:15) A ∨ B , from above the bettorimmediately faces a Dutch book if any of these are the case: p < p > < q > r < r > s < s > r > p , r > q , p > s , q > s so wesuppose that 0 ≤ r ≤ p ≤ s ≤ ≤ r ≤ q ≤ s ≤ A (cid:15) B then A (cid:15) A ∧ B and A ∨ B (cid:15) B , hence, if the the bettor is not immediately to face a Dutchbook, p = r and q = s , whence p + q = r + s . Secondly, if B (cid:15) A then B (cid:15) A ∧ B and A ∨ B (cid:15) A , hence, if the the bettor is not immediately toface a Dutch book, p = s and q = r , whence p + q = r + s .(If A (cid:15) B and B (cid:15) A then, if the the bettor is not immediately to face aDutch book, p = q = r = s , whence p + q = r + s .)We now suppose that A B and B A .Let G = − pS − qS − rS − sS T / ∈ v ( A ) , T / ∈ v ( B ) G = − pS + ( − q ) S − rS + ( − s ) S T / ∈ v ( A ) , T ∈ v ( B ) G = ( − p ) S − qS − rS + ( − s ) S T ∈ v ( A ) , T / ∈ v ( B ) G = ( − p ) S + ( − q ) S + ( − r ) S + ( − s ) S T ∈ v ( A ) , T ∈ v ( B ) .If G < G < G < G < G = ( − s ) G + sG < G = ( − s ) G + sG < G = ( − s ) G + sG < i.e. , G = − pS + ( s − q ) S − rS < G = ( s − p ) S − qS − rS < G = ( s − p ) S + ( s − q ) S + ( s − r ) S < G < G < G < G < s =
0, hence s > p or p > G = ( s − p ) G + pG < G = ( s − p ) G + pG < i.e. , G = s [( s − p − q ) S − rS ] < G = s [( s − q ) S + ( p − r ) S ] < G < G < G < G < p > r or r > ( p − r )[( s − p − q ) S − rS ] + r [( s − q ) S + ( p − r ) S ] < i.e. , p [( r + s ) − ( p + q )] S < p = r = ( s − q ) S <
0. Either way, p + q = r + s .Now, supposing that p + q = r + s , we choose S to be of the same signas ( p + q ) − ( r + s ) and set S = S = − S = − S = S . We find that G = − pS − qS + rS + sS = − [( p + q ) − ( r + s )] S < G = − pS + ( − q ) S + rS − ( − s ) S = − [( p + q ) − ( r + s )] S < G = ( − p ) S − qS + rS − ( − s ) S = − [( p + q ) − ( r + s )] S < G = ( − p ) S + ( − q ) S − ( − r ) S − ( − s ) S = − [( p + q ) − ( r + s )] S < ≤ r ≤ p ≤ s ≤ ≤ r ≤ q ≤ s ≤
1, our bettorfaces a Dutch Book on the family of four bets if, and only if, p + q = r + s . Axiom 5.1 iv).
Three bets are to be made: on B at betting quotient p andstake S , on A ∧ B at betting quotient q and stake S , and on A conditionalon B at betting quotient r and stake S . As A ∧ B (cid:15) B , from above thebettor immediately faces a Dutch book if any of these are the case: p < p > q < q > q > p . We suppose that 0 ≤ q ≤ p ≤ G = − pS − qS T / ∈ v ( B ) G = ( − p ) S − qS − rS T / ∈ v ( A ) , T ∈ v ( B ) G = ( − p ) S + ( − q ) S + ( − r ) S T ∈ v ( A ) , T ∈ v ( B ) .If G < G < G = ( − r ) G + rG < i.e. , ( − p ) S +( r − q ) S <
0. If, in addition, G <
0, then pG + ( − p ) G < i.e. , ( pr − q ) S <
0. And so pr = q .Now, supposing that pr = q , choose S to be of the same sign as pr − q ,set S = rS , S = − S , S = S and we find that G = − prS + qS = − ( pr − q ) S < G = ( − p ) rS + qS − rS = − ( pr − q ) S < G = ( − p ) rS − ( − q ) S + ( − r ) S = − ( pr − q ) S < ≤ q ≤ p ≤
1, our bettor faces a Dutch Book on the tripleof bets if, and only if, pr = q . Theorem 5.2 (Converse Dutch Book Argument) . A bettor cannot, throughan unfortunate choice of stakes, face a Dutch Book on a finite family of bets if herbetting quotients satisfy Axioms 5.1 i) – iv).Proof.
Given the language L , let the betting quotients q ( A ) , A ∈ L satisfyAxioms 5.1 i) – iv).A classical probability distribution satisfies these axioms ( cf. [21]):i) p is a real-valued function such that for all A ∈ L , 0 ≤ p ( A ) ≤ A , B ∈ L , if A (cid:15) B then p ( A ) ≤ p ( B ) ,iii) for all A , B ∈ L , p ( A ∧ B ) + p ( A ∨ B ) = p ( A ) + p ( B ) ,iv) for all A , B ∈ L , if p ( B ) > p ( A | B ) = p ( A ∧ B ) p ( B ) ,32v) for all A ∈ L , if (cid:15) A then P ( A ) = A ∈ L , if (cid:15) ¬ A then P ( A ) = (cid:15) ’ stands for classical consequence. In fact in the application we areabout to make of this, we can happily strengthen it to mean classical con-sequence given the semantic account of Dunn–Belnap logic in §3. We’llindicate this by ‘ (cid:15) ST ’. (As is common practice, our meta-language is clas-sical.)Define a function P on the algebra generated by the (classical) meta-linguistic propositions T ∈ v ( A ) , A ∈ L by setting P ( T ∈ v ( A )) = q ( A ) .As follows from the axioms above, P ( T / ∈ v ( A )) = − q ( A ) for all A ∈ L .We need to show that in making this assignment there is no conflictbetween the axioms governing P and the axioms governing q . This we doas follows:i) As 0 ≤ q ( A ) ≤
1, 0 ≤ P ( T ∈ v ( A )) ≤ A , B ∈ L , T ∈ v ( A ) (cid:15) ST T ∈ v ( B ) iff, for all valuations v , T ∈ v ( B ) if T ∈ v ( A ) iff A (cid:15) B .iii) For all A , B ∈ L , P ( T ∈ v ( A ) and T ∈ v ( B )) + P ( T ∈ v ( A ) or T ∈ v ( B )) = P ( T ∈ v ( A ∧ B )) + P ( T ∈ v ( A ∨ B )) = q ( A ∧ B ) + q ( A ∨ B ) = q ( A ) + q ( B ) = P ( T ∈ v ( A )) + P ( T ∈ v ( B )) .iv) For all A , B ∈ L , if P ( T ∈ v ( B )) >
0, equivalently, if q ( B ) >
0, then P ( T ∈ v ( A ) | T ∈ v ( B )) = P ( T ∈ v ( A ) and T ∈ v ( B )) P ( T ∈ v ( B )) = P ( T ∈ v ( A ∧ B )) P ( T ∈ v ( B )) = q ( A ∧ B ) q ( B ) = q ( A | B ) .Notice too that, say, P ( T ∈ v ( A ) and T / ∈ v ( B )) = P ( T ∈ v ( A )) − P ( T ∈ v ( A ) and T ∈ v ( B )) = q ( A ) − q ( A ∧ B ) and P ( T / ∈ v ( A ) and T / ∈ v ( B )) = − P ( T ∈ v ( A ) or T ∈ v ( B )) = − q ( A ∨ B ) . Just as in classical logic,Dunn–Belnap logic has DeMorgan’s Laws, Laws of distributivity of ‘ ∧ ’over ‘ ∨ ’ and vice versa , and Double Negation equivalence, so any formulacan be expressed in disjunctive normal form as a disjunction of conjunc-tions of literals. What we rely on here is the pseudo-classical behaviour of ∧ and ∨ which arises from their satisfying what, above, we called truismsabout truth.The expected value, relative to P , of a bet on A at betting quotient p with stake S is − P ( T / ∈ v ( A )) pS + P ( T ∈ v ( A ))( − p ) S = ( P ( T ∈ v ( A )) − p ) S .33learly, this is zero if, and only if, P ( T ∈ v ( A )) = p . And so, when P is setup as above, i.e. by setting P ( T ∈ v ( A )) = q ( A ) , the expectation is zerofor a bet on A at betting quotient q ( A ) , no matter the size and sign of thestake.The expected value, relative to P , of a (conditional) bet on A condi-tional on B at betting quotient p with stake S is P ( T / ∈ v ( B )) · − P ( T / ∈ v ( A ) T ∈ v ( B )) pS + P ( T ∈ v ( A ) and T ∈ v ( B ))( − p ) S = [ P ( T ∈ v ( A ) and T ∈ v ( B )) − pP ( T ∈ v ( B ))] S .And clearly this is zero just if p × P ( T ∈ v ( B )) = P ( T ∈ v ( A ) and T ∈ v ( B )) , that is, just if p × P ( T ∈ v ( B )) = P ( T ∈ v ( A ∧ B )) . And so, when P is set up by setting P ( T ∈ v ( A )) = q ( A ) , and so on for the members of L , the expectation is zero for a (conditional) bet on A conditional on B atbetting quotient q ( A | B ) , no matter the size and sign of the stake.Consider a family of n bets on the propositions A , A , . . . , A n at bettingquotients q ( A ) , q ( A ) , . . . , q ( A n ) and stakes S , S , . . . S n , respectively.Given any assignment of truth-values— ∅ , { T } , { F } , { T , F } — to literals,we can work out the gain/loss on each bet. The net gain/loss is the sumof the gains/losses on the n bets. Consequently we can work out the ex-pected net gain/loss relative to P . But classically the expected value of asum is the sum of the expected values of the summands and, as we haveseen, for each bet this is zero (including conditional bets). The expectedvalue is negative if each possible value is negative, as the net gains/losseswould be if sure loss was faced. Hence sure loss is not faced: the bettordoes not face a Dutch book. (Reversed bet) . A reversed bet on (proposition) A with (pos-itive or negative) stake S at betting quotient p pays pS to the bookmaker if Atakes a designated value and pays ( − p ) S to the bettor if it doesn’t.
Definition 5.7 (Reversed Conditional bet) . A reversed (conditional) bet on(proposition) B conditional on (proposition) A with (positive or negative) stake S at betting quotient p pays nothing to either bettor or bookmaker if A doesnot take a designated value and otherwise pays pS to the bookmaker if B takes adesignated value and pays ( − p ) S to the bettor if it doesn’t. u ( A ) stand for the betting quotient for a reversed bet, we canrun analogues of the Dutch book arguments above, switching − x and 1 − x , for x = p , q , r , s , in the characterization of pay-offs, to find that Theorem 5.3 (Dutch Book Argument for reversed bets) . A bettor may face aDutch Book on a finite family of reversed bets, through an unfortunate choice ofstakes, if her betting quotients do not satisfy these axioms:v) u is a real-valued function such that for all A ∈ L , 0 ≤ u ( A ) ≤ ,vi) for all A , B ∈ L , if A (cid:15) B then u ( B ) ≤ u ( A ) ,vii) for all A , B ∈ L , u ( A ∧ B ) + u ( A ∨ B ) = u ( A ) + u ( B ) ,viii) for all A , B ∈ L , if u ( B ) < then u ( A | B ) = − − u ( A ∧ B ) − u ( B ) . From the part of the proof of the Dutch Book Argument concerningAxiom 5.1 i), we see that in order to avoid a Dutch book on a bet on A atbetting quotient p , 0 ≤ p ≤
1, and a reversed bet on A at betting quotient q , 0 ≤ q ≤
1, we must set q = − p .Combining bets and reversed bets, including conditional bets, we havethis Dutch Book Argument: Corollary 5.1 (Combined Dutch Book Argument) . A bettor may face a DutchBook on a finite family of bets and reversed bets, through an unfortunate choice ofstakes, if her betting quotients do not satisfy Axioms 5.1 i) – viii) and this furtheraxiom:ix) for all A ∈ L , u ( A ) = − p ( A ) . This shows that the betting quotients for reversed bets stand to §5’s un-certainties as the betting quotients for bets stand to probabilities. As thatassociation might make one suspect, Axiom ix) shows that there is reallyno need to introduce reversed bets in addition to ordinary bets. (Noticethat the argument for Corollary 5.1 applies in the classical case as well.)
The definition of conditional probability used above is the same as theclassical one but, as the setting has changed, we must devote some atten-tion to how it is to be employed in adaptations of classical updating rules.Classical updating rules depend on the Theorem of Total Probability, towhich we now turn. 35n the classical case we prove the Theorem by relying on the equiva-lence A = A ∧ W i B i , where the B i form a partition, i.e., W i B i = ⊤ and for i = j , B i ∧ B j = ⊥ . In the classical case partitions exist as a matter of logic.In the present setting there is no such guarantee. When we have the ef-fect of one, relative to the probability distribution in play (see below), weobtain an analogous Theorem: Theorem 5.4 (Theorem of Total Probability) . p ( A ) = ∑ i p ( A | B i ) p ( B i ) whenp ( B i ∧ B j ) = , i = j, and p ( W i B i ) = . We first state and prove
Lemma 5.2. p ( n W i = C i ) = n ∑ i = p ( C i ) when p ( C i ∧ C j ) = , ≤ i < j ≤ n.Proof. Trivially true for n = n = k . Then, by axioms ii) and iii) inDefinition 5.1, p ( k + W i = C i ) = p (( k W i = C i ) ∨ C k + ) = p ( k W i = C i ) + p ( C k + ) − p (( k W i = C i ) ∧ C k + ) . By the induction hypothesis, this is k + ∑ i = p ( C i ) − p ( k W i = ( C i ∧ C k + )) . Since, by axioms i)–iii), 0 ≤ p (( C i ∧ C k + ) ∧ ( C j ∧ C k + )) ≤ p ( C i ∧ C j ) ≤
0, by the induction hypothesis again, p ( k W i = ( C i ∧ C k + )) = k ∑ i = p ( C i ∧ C k + ) ; by hypothesis, p ( C i ∧ C k + ) =
0, 1 ≤ i ≤ k , hence p ( k + W i = C i ) = k + ∑ i = p ( C i ) .The result now follows by induction. Proof of Theorem of Total Probability.
By axioms i)–iii) in Definition 5.1, p ( A ∧ W i B i ) = p ( A ) , since 1 = p ( W i B i ) ≤ p ( A ∨ W i B i ) ≤
1. By axiom ii), p ( A ∧ W i B i ) = p ( W i ( A ∧ B i )) and since, by axioms i)–iii) , 0 ≤ p (( A ∧ B i ) ∧ ( A ∧ B j )) ≤ p ( B i ∧ B j ) , the result now follows by the preceding lemma andaxiom iv) in Definition 5.1.This Theorem of Total Probability differs from the classical case only inthat the probability distribution determines the applicability of the theo-rem, and, in particular, determines which sets behave enough like parti-tions. That said, the mechanics of the proof are almost identical to those inthe classical case. 36 bservation 5.1. That a set of propositions behaves like a partition under oneprobability distribution may well entail that it does so under a related distribution.For example, if { B i : 1 ≤ i ≤ n } behaves like a partition under the probabilitydistribution p, i.e., p ( B i ∧ B j ) = , i = j, and p ( W i B i ) = , then, for any Csuch that p ( C ) > , { B i ∧ C : 1 ≤ i ≤ n } behaves like a partition under theprobability distribution p ( ·| C ) . Definition 5.8 (Bayesian Conditionalization) . A probability distribution p isupdated to the distribution p ∗ by Bayesian conditionalization on B if p ( B ) > and for all propositions A, p ∗ ( A ) = p ( A | B ) . As pointed out by van Fraassen [28], diachronic Dutch Book argumentsrequire the assumption of an announced updating strategy (and hencevulnerability can be avoided by not announcing such a strategy). In ourframework, we could simple take having previously announced—or con-ventionally instituted—updating strategies to be a feature of well-behavedlabs. If we do, then there is a Dutch Book argument for adhering to thestrategy.
Theorem 5.5 (Diachronic Dutch Book for Bayesian Conditionalization) . Abettor may face a Dutch Book on a finite family of bets, through an unfortunatechoice of stakes, if, having announced an updating strategy, her betting quotientsdo not satisfy Definition 5.8.Proof.
Three bets are to be made: on B at betting quotient p and stake S ,on A ∧ B at betting quotient q and stake S and on A updated on B atbetting quotient r and stake S . Assume p > q . Let G = pS T / ∈ v ( B ) G = ( − p ) S − qS − rS T / ∈ v ( A ) , T ∈ v ( B ) G = ( − p ) S + ( − q ) S + ( − r ) S T ∈ v ( A ) , T ∈ v ( B ) .The argument concerning Axiom iv) in the proof of Theorem 5.1 nowapplies, given some trivial modifications, as it does to the case q < p .The following converse Diachronic Dutch Book argument guaranteesthe consistency of the strategy: 37 heorem 5.6 (Converse Dutch Book Argument for Conditionalization) . Having announced the she will follow the updating stratgey of Bayesian con-ditonalization (Definition 5.8), a bettor cannot, through an unfortunate choice ofstakes, face a Dutch Book on a finite family of (diachronic) bets.Proof.
The proof is essentially that of [26]. Given that the bettor condi-tionalizes, we can translate her bets at different times to bets at one time.For any bet on a proposition A offered at a later time we substitute a con-ditional bet at the earlier time on A , where the condition is some truthknown at that earlier time. Theorem 5.2 ensures that the bettor’s expecta-tion of loss on this (synchronic) family of bets is 0. We now turn to a more general form of updating, where beliefs over a par-tition { B i : 1 ≤ i ≤ n } change exogenously, using Howson and Urbach’s[13] felicitous term, while degrees of belief conditional on the members ofthe partition remain the same. When this happens the probabilities overother propositions need to be redistributed, that is, we need to go from aprobability function p over propositions to a new probability function p ∗ .In the classical case we use the Theorem of Total Probability with the givenequalities p ( A | B i ) = p ∗ ( A | B i ) to obtain p ∗ ( A ) = ∑ i p ( A | B i ) p ∗ ( B i ) , i.e. , Jeffrey conditionalization.We can use Theorem 5.4 to obtain an analogous form: Definition 5.9 (Jeffrey conditionalization) . p ∗ ( A ) = ∑ i p ( A | B i ) p ∗ ( B i ) , when n ∑ i = p ∗ ( B i ) = and p ∗ ( B i ∧ B j ) = for i = j. Notice that, by stipulation, the B i ’s behave as a partition with respect tothe distribution p ∗ but not necessarily the distribution p . It’s not where you’recoming from, it’s where you’re going to that matters.38 .5.1 Dutch Book arguments regarding Jeffrey conditionalization Even a brief perusal of the Dutch Book arguments for probability kinemat-ics provided by Brad Armendt [2] and Bryan Skyrms [26] shows them tobe far too long to replicate, let alone adapt to the current setting, here. Wemay return to this topic on another occasion.
As far as we know, we are the first to have introduced a relative frequencyinterpretation in a four-valued framework.Edwin Mares [19] offers a Dutch Book argument for a set of axiomsintended to be applied to a broad class of structures. His approach is se-mantic: where we assign probabilities to sentences of the language, Maresgoes via models in which the sentences are interpreted. Furthermore, wehave no analogue of his axiomIf W ∈ A then P ( W ) = ∅ ∈ A then P ( ∅ ) = A is an algebra of subsets of W . There is also a stylistic differencein the way we set out the Dutch Book argument. Mares works in termsof expected values; we provide arguments in the style of de Finetti andonly introduce expectations in our Converse Dutch Book arguments. Weoffer these for the synchronic and diachronic cases; Mares offers neither.J. Michael Dunn [10] defines probabilities within a four-valued frame-work where the probabilities are classical. Dunn does not offer a frequencyinterpretation, although §5.1 could be rewritten in his terms. Dunn’s ap-proach takes the four values to be independent, so that p ( v ( A ) = T ) + p ( v ( A ) = F ) + p ( v ( A ) = { T , F } ) + p ( v ( A ) = ∅ ) =
1. The primary dis-tinction for us, in the case of both bets and frequencies is whether a propo-sition takes a designated value or not: we have a two-way split, whileDunn has a four-way split. Dunn’s approach is more granular, capturingdistinctions between sequences of values such as h ∅ , { T , F }i and h F , T i . Our approach ties probability more closely to the entailment relation ofDunn–Belnap logic. One might wonder what exactly an expected value is when taken relative to a func-tion that, ex hypothesi , does not satisfy the probability axioms. This was pointed out to us by Mike Dunn. Co ¨ordinated Updates and Conditionals
It is now time to return to the full language of the first part. So far wehave equipped each laboratory with a probability function. We now indexthose probabilities by laboratory and turn to how they may co ¨ordinatetheir results. Indexing the probability functions leads to a modal versionof probability, allowing us to examine analogues of Bayesian conditional-ization, Jeffery conditionalization, and Adams’ conditioning (as RichardBradley calls it [6]); we also say a little on the probability of conditionals.
Returning to our fundamental motivation: we have a laboratory x thatcollects and evaluates data obtained from a laboratory y that initiates atest set-up and which is then followed up by a laboratory z . We will nowfocus on labs that are in regular close contact: labs that hold meetingsat designated times to co ¨ordinate their research. During these meetings,the labs may choose to follow the advice of the others by adopting theirprobabilities in their domain of expertise, provided doing so is coherent.We shall confine attention to the updates of the lab x which has oversight.Co ¨oordination might look like this: lab x begins with some prior proba-bility p x ( A ) . x turns to lab y ’s expertise for a relevant likelihood p y ( B | A ) ,and to z ’s expertise for p z ( B ) . The net effect of this updating process isgiven by Definition 6.1 (Co ¨ordinated Conditionalization) . Having obtained values forp y ( B | A ) and p z ( B ) , p x ( A ) is updated by co ¨ordinated conditionalization on B to p ∗ x ( A ) , where p ∗ x ( A ) = p y ( B | A ) p x ( A ) p z ( B ) .We can think of Definition 6.1 as a consistency constraint on group ac-tivity. We should also note that there may be many pairs of labs y and z reporting to x : we take x as updating piecemeal as labs report their re-sults. Theorem 6.1.
When p x ( B ) · p y ( B ) · p z ( B ) > , p ∗ x ( · ) is obtained by co¨ordinatedconditionalization on B if, and only if, We could adopt a different updating strategy. We could wait until all information isin, and then update by aggregating laboratory probabilities via opinion pooling. Indeed,laboratory x could lay out a protocol for classical statistical methods. ) p ∗ x ( B ) = p x ( B ) p z ( B ) ;ii) for all propositions A, p ∗ x ( A ) = if, and only if, p x ( A ) = or p y ( B | A ) = ;iii) for all propositions A and C, if p ∗ x ( A ) · p ∗ x ( C ) > then p x ( A ) p ∗ x ( A ) · p y ( B | A ) = p x ( C ) p ∗ x ( C ) · p y ( B | C ) .Proof. If p ∗ x ( · ) is obtained by co ¨ordinated conditioning on B then, since p y ( B ) > P y ( B | B ) = p x ( A ) = p y ( B | A ) = p ∗ x ( A ) =
0, so suppose that neither obtains. Then, since p x ( B ) p z ( B ) = p ∗ x ( B ) = p x ( A ) p ∗ x ( A ) · p y ( B | A ) = p x ( B ) p ∗ x ( B ) · p y ( B | B ) = p x ( B ) p ∗ x ( B ) = p z ( B ) , hence p ∗ x ( A ) = p y ( B | A ) p x ( A ) p z ( B ) .Theorem 6.1 characterizes a particular form of promiscuous adoption ofothers’ opinions motivated by our interpretation. Of particular interestare the following special cases. Definition 6.2 (Co ¨ordinated Bayesian Conditionalization) . Let { B , B } be-have as a partition under the probability distribution p z , i.e. p z ( B ∧ B ) = and p z ( B ∨ B ) = , and suppose that p y ( B ) > and that p y ( B ∧ B ) = p z ( B ∧ B ) . Then p ∗ x is said to be obtained from p y and p z by co ¨ordinatedBayesian conditionalization on B just in case, for all propositions A,p ∗ x ( A ) = p y ( A | B ) . Theorem 6.2.
When { B , B } behaves as a partition under the probability dis-tribution p z and p y ( B ) > and p y ( B ∧ B ) = p z ( B ∧ B ) then p ∗ x ( . ) isobtained from p y by co¨ordinated Bayesian conditionalization on B if, and only if,i) Extremal: p ∗ x ( B ) = and p ∗ x ( B ) = ;ii) x-y rigidity: for all propositions A, p ∗ x ( A | B ) = p y ( A | B ) ;iii) Partition: { B , B } behaves as a partition under p ∗ x . roof. Suppose that { B , B } behaves as a partition under p z , that p y ( B ) >
0, that p y ( B ∧ B ) = p z ( B ∧ B ) , and that p ∗ x ( . ) is obtained from p y and P z by co ¨ordinated Bayesian conditionalization on B .As p y ( B ) > p y ( B | B ) and p y ( B | B ) are well defined; moreover, p ∗ x ( B ) = p y ( B | B ) = p y ( B ∧ B ) p y ( B ) = p y ( B ) p y ( B ) = p ∗ x ( B ) = p y ( B | B ) = p y ( B ∧ B ) p y ( B ) = p z ( B ∧ B ) p y ( B ) = p y ( B ) = ≤ p ∗ x ( B ∧ B ) ≤ p ∗ x ( B ) = = p ∗ x ( B ) ≤ p ∗ x ( B ∨ B ) ≤ { B , B } behaves as a partition under p ∗ x . p ∗ x ( A | B ) = p ∗ x ( A ∧ B ) P ∗ x ( B ) = p ∗ x ( A ∧ B ) = p y ( A ∧ B | B ) = p y ( A | B ) .Conversely, suppose that i), ii) and iii) obtain. By the Theorem of TotalProbability (Theorem 5.4) p ∗ x ( A ) = p ∗ x ( A | B ) p ∗ x ( B ) + p ∗ x ( A | B ) p ∗ x ( B ) = p ∗ x ( A | B ) = p y ( A | B ) . Definition 6.3 (Co ¨ordinated Jeffrey conditionalization) . When { B i : 1 ≤ i ≤ n } behaves as a partition under the probability distribution p ∗ z , i.e. p ∗ z ( B i ∧ B j ) = , i = j, and p ∗ z ( W i B i ) = , p y ( B i ) > , ≤ i ≤ n, and p y ( B i ∧ B j ) = p ∗ z ( B i ∧ B j ) , i = j, then p ∗ x ( · ) is obtained from p y and p ∗ z by co ¨ordinated Jeffreyconditionalization on { B i : 1 ≤ i ≤ n } just in case, for all propositions A,p ∗ x ( A ) = ∑ i p y ( A | B i ) p ∗ z ( B i ) . Theorem 6.3.
When { B i : 1 ≤ i ≤ n } behaves as a partition under the proba-bility distribution p ∗ z , p y ( B i ) > , ≤ i ≤ n, and p y ( B i ∧ B j ) = p ∗ z ( B i ∧ B j ) ,i = j, then p ∗ x ( · ) is obtained from p y and p ∗ z by co¨ordinated Jeffrey conditional-ization on { B i : 1 ≤ i ≤ n } , if, and only if,i) x-z rigidity: p ∗ x ( B i ) = p ∗ z ( B i ) ;ii) x-y rigidity: for all propositions A, p ∗ x ( A | B i ) = p y ( A | B i ) ;iii) Partition: the B i ’s behave as a partition with respect to the distributions p ∗ x . roof. Suppose that { B i : 1 ≤ i ≤ n } behaves as a partition under theprobability distribution p ∗ z , p y ( B i ) >
0, 1 ≤ i ≤ n , and that p y ( B i ∧ B j ) = p ∗ z ( B i ∧ B j ) , i = j .As p y ( B i ) > p y ( B i | B i ) and p y ( B j | B i ) are well defined for all i , j , 1 ≤ i , j ≤ n . p y ( B i | B i ) = p y ( B i ∧ B i ) p y ( B i ) = p y ( B i ) p y ( B i ) = p y ( B i | B j ) = p y ( B i ∧ B j ) p y ( B j ) = p ∗ z ( B i ∧ B j ) p y ( B j ) = p y ( B j ) = i = j . Consequently, p ∗ x ( B i ) = ∑ j p y ( B i | B j ) p ∗ z ( B j ) = p ∗ z ( B i ) .When i = j , 0 ≤ p y ( B i ∧ B j | B k ) ≤ min { p y ( B i | B k ) , p y ( B j | B k ) } = i = k or j = k . Hence p ∗ x ( B i ∧ B j ) = ∑ k p y ( B i ∧ B j | B k ) p ∗ z ( B k ) = p ∗ x ( _ i B i ) = ∑ j p y ( _ i B i | B j ) p ∗ z ( B j ) = ∑ j p y (( W i B i ) ∧ B j ) p ( B j ) p ∗ z ( B j )= ∑ j p ( B j ) p ( B j ) p ∗ z ( B j ) = ∑ j p ∗ z ( B j ) = { B i : 1 ≤ i ≤ n } behaves as a partition under p ∗ x . p ∗ x ( A | B i ) = p ∗ x ( A ∧ B i ) P ∗ x ( B i ) = ∑ j p y ( A ∧ B i | B j ) p ∗ z ( B j ) . P ∗ z ( B i ) = p y ( A ∧ B i | B i ) = p y ( A | B i ) since 0 ≤ p y ( A ∧ B i ∧ B j ) ≤ p y ( B i ∧ B j ) = p ∗ z ( B i ∧ B j ) = i = j .Conversely, suppose that i), ii) and iii) obtain. By the Theorem of TotalProbability (Theorem 5.4) p ∗ x ( A ) = ∑ j p ∗ x ( A | B i ) p ∗ x ( B i ) = ∑ j p y ( A | B i ) p ∗ z ( B i ) . Richard Bradley [6] introduces (and names)
Adams conditioning as a coun-terpart to Jeffrey conditionalization, where likelihoods change but the prob-abilities of certain propositions remain the same. That is, we look for a43ew probability function where, for some salient A , the probability of A remains unchanged, i.e. p ( A ) = p ∗ ( A ) , while the likelihoods change from p ( ·|· ) to p ∗ ( ·|· ) . We adapt this idea to the present setting. Definition 6.4 (Co ¨ordinated Adams conditioning) . Let { A , A } and { B , B } behave as partitions under the probability distribution p z , i.e. p z ( A ∧ A ) = p z ( B ∧ B ) = and p z ( A ∨ A ) = p z ( B ∨ B ) = . Suppose that > p y ( B | A ) > and that lab y is caused to change its conditional probabilities forB given A from p y ( B | A ) to p ∗ y ( B | A ) and for B given A from p y ( B | A ) to p ∗ y ( B | A ) where p ∗ y ( B | A ) + p ∗ y ( B | A ) = . Then x’s new probabilitiesp ∗ x are said to be obtained by co ¨ordinated Adams conditioning on this changein conditional probabilities, just in case:p ∗ x ( C ) = p ∗ y ( B | A ) p y ( B | A ) · p z ( A ∧ B ∧ C )+ p ∗ y ( B | A ) p y ( B | A ) · p z ( A ∧ B ∧ C ) + p z ( A ∧ C ) .We prove this analogue of Bradley’s Theorem 1 [6, p. 352] Theorem 6.4.
Where { A , A } and { B , B } behave as partitions under theprobability distribution p z and p z ( B | A ) = p y ( B | A ) and p z ( B | A ) = p y ( B | A ) , x’s new probabilities p ∗ x are obtained by Adams conditioning on achange in y’s conditional probabilities of B given A and B given A iff:i) x-z Rigidity: p ∗ x ( A ) = p z ( A ) and p ∗ x ( A ) = p z ( A ) ;ii) x-y Rigidity: p ∗ x ( B | A ) = p ∗ y ( B | A ) and p ∗ x ( B | A ) = p ∗ y ( B | A ) ;iii) x-z Rigidity: for all propositions C, p ∗ x ( C | A ∧ B ) = p z ( C | A ∧ B ) ,p ∗ x ( C | A ∧ B ) = p z ( C | A ∧ B ) , and p ∗ x ( C | A ) = p z ( C | A ) ;iv) Partition: { A ∧ B , A ∧ B , A } behaves as a partition under p ∗ x .Proof. (After [6, p. 362].) If p ∗ is obtained by Adams conditioning on achange in y ’s conditional probabilities of B given A and B given A then by Definition 6.4 and the assumption that p z ( B | A ) = p y ( B | A ) and p z ( B | A ) = p y ( B | A ) , 44 ∗ x ( A ) = p ∗ y ( B | A ) p y ( B | A ) · p z ( A ∧ B ) + p ∗ y ( B | A ) p y ( B | A ) · p z ( A ∧ B ) + p z ( A ∧ A )= p ∗ y ( B | A ) p z ( B | A ) · p z ( A ∧ B ) + p ∗ y ( B | A ) p z ( B | A ) · p z ( A ∧ B )= p ∗ y ( B | A ) · p z ( A ) + p ∗ y ( B | A ) · p ( A ) z = [ p ∗ y ( B | A ) + p ∗ y ( B | A )] · p z ( A )= · p z ( A ) = p z ( A ) .Noticing that 0 ≤ p z ( A ∧ B ∧ A ) ≤ p z ( A ∧ A ) = ≤ p z ( A ∧ B ∧ A ) ≤ p z ( A ∧ A ) =
0, we have p ∗ x ( A ) = p ∗ y ( B | A ) p y ( B | A ) · p z ( A ∧ B ∧ A )+ p ∗ y ( B | A ) p y ( B | A ) · p z ( A ∧ B ∧ A ) + p z ( A )= p z ( A ) .Noticing in addition that p z ( A ∧ B ∧ A ∧ B ) = p z ( A ∧ B ) , p z ( A ∧ B ∧ C ∧ A ∧ B ) = p z ( A ∧ B ∧ C ) , 0 ≤ p z ( A ∧ B ∧ C ∧ A ∧ B ) ≤ p ( B ∧ B ) =
0, and 0 ≤ p z ( A ∧ C ∧ A ∧ B ) ≤ p z ( A ∧ A ) =
0, wehave p ∗ x ( B | A ) = p ∗ x ( A ∧ B ) p ∗ x ( A )= p ∗ y ( B | A ) p y ( B | A ) × p z ( A ∧ B ) p z ( A )= p ∗ y ( B | A ) · p z ( B | A ) p y ( B | A )= p ∗ y ( B | A ) .And, likewise, we may show that p ∗ x ( B | A ) = p ∗ y ( B | A ) . p ∗ x ( C | A ∧ B ) = p ∗ x ( C ∧ A ∧ B ) p ∗ x ( A ∧ B )= [ p ∗ y ( B | A ) / p y ( B | A )] · p z ( C ∧ A ∧ B )[ p ∗ y ( B | A ) / p y ( B | A )] · p z ( A ∧ B )= p z ( C | A ∧ B ) .45imilarly, p ∗ x ( C | A ∧ B ) = p z ( C | A ∧ B ) and p ∗ x ( C | A ) = p z ( C | A ) .It is an easy consequence of Definition 5.1 iv) and Theorem 5.4 that { A ∧ B , A ∧ B , A } behaves as a partition under p z when { A , A } and { B , B } both behave as partitions under p z . Consequently, p ∗ x (( A ∧ B ) ∧ ( A ∧ B )) = p ∗ x (( A ∧ B ) ∧ A ) = p ∗ x (( A ∧ B ) ∧ A ) = p ∗ x (( A ∧ B ) ∨ ( A ∧ B ) ∨ A )= p z ( A ∧ B ∧ (( A ∧ B ) ∨ ( A ∧ B ) ∨ A )) · p ∗ y ( B | A ) p y ( B | A )+ p z ( A ∧ B ∧ (( A ∧ B ) ∨ ( A ∧ B ) ∨ A )) · p ∗ y ( B | A ) p y ( B | A )+ p z ( A ∧ (( A ∧ B ) ∨ ( A ∧ B ) ∨ A ))= p z ( A ∧ B ) · p ∗ y ( B | A ) p z ( B | A ) + p z ( A ∧ B ) · p ∗ y ( B | A ) p z ( B | A ) + p z ( A )= p z ( A ) · p ∗ y ( B | A ) + p z ( A ) · p ∗ y ( B | A ) + p z ( A )= p z ( A )[ p ∗ y ( B | A ) + p ∗ y ( B | A )] + p z ( A )= p z ( A ) + p z ( A ) = x - z Dependence, x - y Rigidity, x - z Rigidity and Partition conditionsall obtain.Conversely, suppose now that the x - z Rigidity, x - y Rigidity, x - z Rigidityand Partition conditions hold. Then, starting from the Theorem of Total46robability, p ∗ x ( C ) = p ∗ x ( C | A ∧ B ) · p ∗ x ( A ∧ B )+ p ∗ x ( C | A ∧ B ) · p ∗ x ( A ∧ B ) + p ∗ x ( C | A ) · p ∗ x ( A )= p z ( C | A ∧ B ) · p ∗ x ( A ∧ B )+ p z ( C | A ∧ B ) · p ∗ x ( A ∧ B ) + p z ( C | A ) · p z ( A )= p z ( C ∧ A ∧ B ) p z ( A ∧ B ) · p ∗ x ( A ∧ B )+ p z ( C ∧ A ∧ B ) p z ( A ∧ B ) · p ∗ x ( A ∧ B ) + p z ( C ∧ A )= p ∗ x ( B | A ) · p ∗ x ( A ) p z ( B | A ) · p z ( A ) · p z ( C ∧ A ∧ B )+ p ∗ x ( B | A ) · p ∗ x ( A ) p z ( B | A ) · p z ( A ) · p z ( C ∧ A ∧ B ) + p z ( C ∧ A )= p ∗ y ( B | A ) · p z ( A ) p y ( B | A ) · p z ( A ) · p z ( C ∧ A ∧ B )+ p ∗ y ( B | A ) · p z ( A ) p y ( B | A ) · p z ( A ) · p z ( C ∧ A ∧ B ) + p z ( C ∧ A )= p ∗ y ( B | A ) p y ( B | A ) · p z ( C ∧ A ∧ B )+ p ∗ y ( B | A ) p y ( B | A ) · p z ( C ∧ A ∧ B ) + p z ( C ∧ A ) .We have generalized three forms of updating, Bayesian conditionaliza-tion, Jeffrey conditionalization and Adams conditioning to the co ¨ordinatedsetting. While less well known than the other two, Adams conditioning isparticularly apt in the present setting as likelihoods may often be the busi-ness of another lab. Our conditional is meant to model regularities—and yet discovery of reg-ularities is rare indeed. It behooves us, therefore, to look for a less strictaccount of discovery. One option would be to use a finite frequency inter-pretation: 47 efinition 6.5 (Relative Frequency of Conditionals and Negated Condi-tionals) . The frequency of a regularity A → B for a laboratory x is |{h y , z i| R xyz and if T ∈ v ( y , A ) then T ∈ v ( z , B ) }| .The frequency of ¬ ( A → B ) for a laboratory x is |{h y , z i| R xyz and if T ∈ v ( y i , A ) then F ∈ v ( z , B ) }| . The relative frequency, of course, is the frequency divided by |{h y , z i| R xyz }| and |{h y , z i| R xyz }| as appropriate.Another option would be to elaborate a scheme for assigning betting quo-tients.Definition 6.5 yields a notion of graded regularity. While in certainrare cases a regularity always holds, there are also regularities which holdonly for a certain percentage of cases. (For example, a 90%-regularity.)Finally, it is obvious that the probability of conditionals and conditionalprobabilities are different, as p ( A → B ) is defined when p ( A ) =
0, while p ( B | A ) is not. References [1] Adams, Ernest ,
The Logic of Conditionals: an Application of Probabilityto Deductive Logic , Dordrecht: Reidel, 1975.[2] Armendt, Brad, ‘Is there a Dutch Book Argument for ProbabilityKinematics?’,
Philosophy of Science , (1980): 583–588[3] Belnap, Nuel D., Jr, ‘A useful four-valued logic’, in J. Michael Dunn& George Epstein (eds.), Modern Uses of Multiple-Valued Logic: InvitedPapers from the Fifth International Symposium on Multiple-Valued Logicheld at Indiana University, Bloomington, Indiana, May 13–16, 1975 , Epis-teme , volume 2, Dordrecht: Reidel, 1977, pp. 8–37.[4] Belnap, Nuel D., Jr, ‘How a computer should think’, in Gilbert Ryle(ed.)
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