Second Order Inductive Logic and Wilmers' Principle M.S.Kliess∗and J.B.Paris† School of Mathematics The University of Manchester Manchester M13 9PL tel. 0161-275-5880 malte.kliess@postgrad.manchester.ac.uk jeff.paris@manchester.ac.uk March 10, 2014 We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship to the first order principles of Regularity and Super Regularity. Key words: Universal Certainty, Second Order Logic, Inductive Logic, Probability Logic, Uncertain Reasoning. 1 Introduction In the framework of Pure Inductive Logic, a rational agent's belief function is usually regarded as a probability function on the set of first order sentences of a certain, fixed language L. This language contains constants representing the objects of the universe ∗Supported by a University of Manchester School of Mathematics Research Studentship. †Corresponding Author. 1 and predicates representing the properties of these objects. This allows an agent to express statements about the universe. So far such statements have, to our knowledge, been limited to first order expressions, allowing the agent to make existential or universal statements about the objects. As the famous Geach-Kaplan statement shows (see e.g. [1]), an agent could increase her expressive power if she were to extend the set of expressions available to include second order statements. As we identify an agent's belief in a statement with the value the agent's belief function assigns to it, allowing the agent to use second order expressions will require her to extend the domain of the her belief function to include second order sentences. Unsurprisingly, this leads to a number of complications. Since Second Order logic is inherently incomplete (see e.g. [10]), we will have to be careful picking a suitable framework for Second Order Inductive Logic. At the same time, we would want to have a suitable interpretation of universal and existential quantification over the predicates in L. In this paper we intend to provide such a framework, allowing an agent to extend her expressive power to second order logic. Once such a framework is given, we can study rational principles involving second order logic. We will give an example of one such principle, called Wilmers' Principle, and provide a representation theorem for second order belief functions that satisfy this principle. We will then consider the consequences of this principle for the thorny question of Universal Truth for first order statements. 2 Second Order probability functions As with traditional Inductive Logic1 for example [2], we will work in a unary language L with predicate symbols Pi and constants ai for i ∈ {1, 2, 3, . . .} = N+ but without function symbols or equality. Let F1L, S1L,QFS1L respectively denote the first order formulae, sentences and quantifier free sentences of L. Let T L denote the set of structures M for L in which the constants ai are interpreted as themselves and |M | = {ai | i ∈ N+}, so every element of the universe of M is denoted by a constant symbol. Similarly we shall use Pj to denote {ai |M |= Pj(ai)}, leaving the M implicit whenever this cannot cause confusion. We say that w : S1L → [0, 1] is a probability function on S1L, if for any θ, φ ∈ S1L, ψ(x) ∈ F1L, 1Actually Inductive Logic is more commonly presented with only finitely many predicate symbols but as we would in any case advocate the rationality of Unary Language Invariance in this context, see for example [8, Chapter 16], this would ultimately lead to the same situation. 2 (P1) If |= θ, then w(θ) = 1. (P2) If θ |= ¬φ, then w(θ ∨ φ) = w(θ) + w(φ). (P3) w(∃xψ(x)) = limn→∞w ( ∨n i=1 ψ(ai)). To our mind the central problem of (Pure) Inductive Logic can be picturesquely captured as follows: Imagine an agent inhabiting a structure M ∈ T L but having no further knowledge, so in particular the agent has no particular interpretation in mind for the constant and predicate symbols. In that case what probability w(θ) should the agent rationally, or logically, give to θ ∈ S1L? Or more precisely, since we obviously intend for these probability values to be coherent, what probability function w should the agent rationally or logically adopt? In the absence of any clear definition of what is meant here by 'rationally' (which for the purpose of this paper we identify with 'logically') the usual method of tackling this question is by imposing certain ostensibly rational, or at least not irrational, requirements on w and seeing where that leads. For example the symmetry between the constants ai, and between the predicates Pj, from the agent's point of view surely requires that w should satisfy: Constant Exchangeability, Ex w satisfies Ex, if for all θ ∈ S1L and all permutations σ of N+, w(θ(a1, . . . , an)) = w(θ(aσ(1), . . . , aσ(n))). Predicate Exchangeability, Px w satisfies Px, if for all θ ∈ S1L, and all permutations σ of N+, w (θ(P1, . . . , Pm, a1, . . . , an)) = w ( θ(Pσ(1), . . . , Pσ(m), a1, . . . , an) ) whenever θ(Pσ(1), . . . , Pσ(m), a1, . . . , an) is the result of replacing each predicate symbol, Pj, occurring in θ with Pσ(j). Given their standing we shall assume throughout that unless specifically indicated all the probability functions we consider satisfy Ex and Px. Two further principles which we might impose on w are based on the idea that in the presence of no prior knowledge about the universe it would might seem unreasonable to summarily assign zero probability to sentences which could be true. Regularity, Reg w satisfies Reg, if for all θ ∈ QFS1L such that θ is satisfiable, w(θ) > 0. Super Regularity, SReg w satisfies SReg, if for all θ ∈ S1L such that θ is satisfiable, w(θ) > 0. 3 Note that we have separate principles for quantifier-free sentences and sentences that may contain quantifiers. The reason for this is that functions satisfying Regularity in general do not need to also satisfy Super Regularity. The principle SReg2 has been something of a thorn in the side of traditional Inductive Logic. Whilst seemingly quite reasonable it is inconsistent with several other rationally attractive principles such as Johnson's Sufficientness Postulate, see [6], [8], and in consequence is not satisfied by the probability functions comprising Carnap's Continuum of Inductive Methods, though all of them except the pathological c0 do satisfy Reg. Indeed SReg already fails there for consistent but non-tautologous Π1 sentences where the failure is sometimes referred to as the problem of Universal Certainty. In spite of some attempts, see for example [5], to tweak Johnson's Sufficientness Postulate and allow SReg to hold the issue still seems to us problematic. In this paper we shall be investigating a somewhat different approach which under certain circumstance has SReg as a consequence. Returning again to our agent it is clear that for this unary language L there is a certain parallel between the constant symbols and the predicate symbols, as indeed is reflected in the principles Ex and Px. This suggests that from the agent's point of view we could just as well have quantifiers ranging over the predicates as over the constants, in other words second order quantifiers ∀X and ∃X where the second order variable X is intended to range over the predicates Pi. To this end let F2L, S2L respectively denote the second order formulae and sentences of L and say that w : S2L → [0, 1] is a probability function on S2L, if for any θ, φ ∈ S2L, ψ(x), η(X) ∈ F2L, (P1) If |= θ, then w(θ) = 1. (P2) If θ |= ¬φ, then w(θ ∨ φ) = w(θ) + w(φ). (P3) w(∃xψ(x)) = limn→∞w ( ∨n i=1 ψ(ai)). (P4) w(∃X η(X)) = limn→∞w ( ∨n i=1 η(Pi)). So here we have added (P4) to the original requirements, reflecting the idea that the Pi exhaust the second order universe just as (P3) reflected the idea that the ai exhaust the first order universe. This extension from first to second order in this way is in fact hardly more than notational (though it will enable us to state and investigate new principles). For example if M ∈ T L then M automatically becomes a second order structure by taking the second order 2Also referred to as Cournot's Principle, see [3]. 4 universe of M to simply be the set of interpretations of the relation symbols in M , i.e. {{ai |M |= Pj(ai)} | j ∈ N+}. For this reason we shall not distinguish between M as a first or second order structure and similarly we shall use T L for 'both' versions. Notice then that the condition θ |= φ in (P1-2) is actually equivalent to ∀M ∈ T L, if M |= θ then M |= φ since if a sentence ψ ∈ S2L has a model then it has a model with denumerably many elements and subsets in its universe and hence by suitable naming a model in T L. With this extended notion of probability function all the standard properties can be proved just as before (e.g. see [7, Proposition 2.1], [4] or [8, Lemma 3.8]): Lemma 1. Let w be a probability function on S2L. Then for θ, φ ∈ S2L, (a) w(¬θ) = 1− w(θ). (b) If |= ¬θ, then w(θ) = 0. (c) If θ |= φ, then w(θ) ≤ w(φ). (d) If θ ≡ φ, then w(θ) = w(φ). (e) w(θ ∨ φ) = w(θ) + w(φ)− w(θ ∧ φ). Lemma 2. Let w be a probability function on S2L and ∃x1 . . . ∃xk θ(x1, . . . , xk, ~P ,~a), ∃X1 . . . ∃Xk ψ(X1, . . . , Xk, ~P ,~a) ∈ S2L. Then w(∃x1 . . . ∃xk θ(x1, . . . , xk, ~P ,~a)) = lim n→∞ w ( ∨ i1,...,ik≤n θ(ai1 , . . . , aik , ~P ,~a) ) w(∀x1 . . . ∀xk θ(x1, . . . , xk, ~P ,~a)) = lim n→∞ w ( ∧ i1,...,ik≤n θ(ai1 , . . . , aik , ~P ,~a) ) w(∃X1 . . . ∃Xk ψ(X1, . . . , Xk, ~P ,~a)) = lim n→∞ w ( ∨ i1,...,ik≤n ψ(Pi1 , . . . , Pik , ~P ,~a) ) w(∀X1 . . . ∀Xk ψ(X1, . . . , Xk, ~P ,~a)) = lim n→∞ w ( ∧ i1,...,ik≤n ψ(Pi1 , . . . , Pik , ~P ,~a) ) (1) 5 Just as structures in T L extend naturally from S1L to S2L so do probability functions on S1L extend uniquely to probability functions on S2L. Precisely: Theorem 3. Let w be a probability function on S1L. Then w extends uniquely to a probability function on S2L. Furthermore if w satisfies Ex and Px on S1L then they are preserved in this extension to S2L. Proof Let w be a probability function on S1L, let A be the subsets of T L of the form {M ∈ T L |M |= θ} for θ ∈ QFS1L. Define a finitely additive measure μ on A by μ{M ∈ T L |M |= θ} = w(θ). By a compactness argument if φ and θn are in QFS1L for n ∈ N and {M ∈ T L |M |= φ} = ⋂ n∈N {M ∈ T L |M |= θn} then for some k {M ∈ T L |M |= φ} = ⋂ n≤k {M ∈ T L |M |= θn} and from this it follows that μ preserves all infs in A. By Carathéodory's Extension Theorem then μ extends uniquely to a countably additive measure on the σ-algebra generated by A. Since {M ∈ T L |M |= ∃xθ(x)} = ⋃ n∈N+ {M ∈ T L |M |= θ(an)}, {M ∈ T L |M |= ∃X θ(X)} = ⋃ n∈N+ {M ∈ T L |M |= θ(Pn)}, all the sets {M ∈ T L |M |= θ} for θ ∈ S2L will be in this σ-algebra and if we define w+ on S2L by w+(θ) = μ{M ∈ T L |M |= θ} then w+ will satisfy (P1-4). Since w+ agrees with w on QFS1L, by a result of Gaifman, [4], (essentially by induction on the length of formulae using (1)) w+ agrees with w on S1L and hence provides the required extension of w to S2L. Furthermore it is now easy to see by induction on quantifier complexity that this is the only possible extension of w to S2L and if w satisfies Ex and Px on QFS1L then it also does so on S2L  Given this result we shall in future not particularly distinguish between a probability function defined on S1L and its extension to S2L. 6 3 Wilmers' Principle So far we have established some initial technical results, allowing us to work with second order sentences in the framework of Pure Inductive Logic. In this section we suggest and discuss a rational principle for Second Order expressions that rational agents may want to accept as defining their beliefs. The motivation for the principle is the following idea: Suppose we have a First Order formula θ(x) with just one free variable. Then in the agent's ambient structure M θ(x) defines the subset of the universe {ai |M |= θ(ai)}. A rational agent then might feel that there should be a name for this set in the language, and thus θ(x) defines not only a subset of the universe, but also a predicate of the language, i.e. {ai |M |= θ(ai)} = {ai |M |= Pj(ai)} for some (unary) Pj in the agent's language. The formal definition of this principle in terms of probability functions is given by: Wilmers' Principle, WP Let w be a probability function on S2L. Then w satisfies Wilmers' Principle, WP, if w(∃X ∀x (θ(x)↔ X(x))) = 1 whenever θ(x) ∈ F1L.3 This principle is based on the original suggestion by George Wilmers that it would be rational that w gave probability 1 to all the tautologies of second order monadic logic. The current version then is but a fragment of what was originally intended. The main aim of this section is to provide a certain representation theorem for the probability functions satisfying Wilmers' Principle. This will turn out to be useful in the next section when we consider also Reg and SReg. We first need some notation and apparatus. For each n ∈ N+, define ℘(n)∞ × n∞ and structures Mf,g as follows: • Let ℘(n) denote the power set of {1, 2, . . . , n}, let ℘(n)∞ denote the set of functions f : N+ → ℘(n) and let n∞ denote the set of function g : N+ → {1, 2, . . . , n}. So ℘(n)∞ × n∞ denotes the set of all pairs 〈f, g〉 with f ∈ ℘(n)∞ and g ∈ n∞. 3As far as the actual statements of results in this paper are concerned it would make no difference if we took instead the version of Wilmers' Principle with θ(x) ∈ F2L. 7 • For each pair 〈f, g〉 ∈ ℘(n)∞ × n∞, define the structure Mf,g for L with finite universe {e1, e2, . . . , en} by a Mf,g i = eg(i) and P Mf,g j = {ek | k ∈ f(j)}. So Mf,g |= Pj(ai) ⇐⇒ Mf,g |= Pj(eg(i)) ⇐⇒ g(i) ∈ f(j). [Here a Mf,g i and P Mf,g j are the interpretations of ai and Pj in Mf,g.] Let n ∈ N+ and let μn be a normalized, σ-additive measure on ℘(n)∞ × n∞. We say that • μn is invariant under Ex if for any θ ∈ S2L and any permutation σ : N+ → N+, μn{〈f, g〉 |Mf,g |= θ}) = μn({〈f, gσ〉 |Mf,g |= θ}, • μn is invariant under Px if for any θ ∈ S2L and any permutation σ : N+ → N+, μn{〈f, g〉 |Mf,g |= θ)}) = μn({〈fσ, g〉 |Mf,g |= θ}. We will use the measures μn to construct probability functions that only give weight to those structures Mf,g that instantiate precisely n distinguishable constants. For this, it will be convenient to have a single second order sentence expressing this. Let ξn be the sentence ∃x1 . . . ∃xn [ ∧ 1≤i<j≤n ∃X ¬(X(xi)↔ X(xj)) ∧ ∀z n∨ i=1 ∀X (X(xi)↔ X(z)) ] (2) for n ∈ N+. Theorem 4. Let μ be a normalized, σ-additive measure on ℘(n)∞×n∞ invariant under Ex and Px and suppose that μ{〈f, g〉 | ran(f) = ℘(n) and ran(g) = n} = 1. (3) Let wμ be the function on S2L defined by wμ(θ) = μ{〈f, g〉 |Mf,g |= θ} for θ ∈ S2L Then (i) wμ is a probability function on S2L satisfying Px + Ex. (ii) wμ satisfies Wilmers' Principle, WP, and wμ (ξn) = 1. Conversely let v be a probability function on S2L satisfying Px + Ex + Wilmers' Principle and suppose that v (ξn) = 1. Then v = wμ for some normalized, σ-additive measure μ on ℘(n)∞ × n∞ invariant under Ex + Px and satisfying (3). 8 Proof For (i) it is straightforward to check that wμ satisfies (P1-2). For (P3) let ψ(x) ∈ F2L with x the only free variable. Then wμ(∃xψ(x)) = μ{〈f, g〉 |Mf,g |= ∃xψ(x)} = μ { n⋃ k=1 {〈f, g〉 |Mf,g |= ψ(ek)} } = μ { ∞⋃ i=1 {〈f, g〉 |Mf,g |= ψ(ai)} } , since g is onto for almost all 〈f, g〉, = lim m→∞ μ { 〈f, g〉 ∣∣∣Mf,g |= m∨ i=1 ψ(ai) } , by σ-additivity of μ, = lim m→∞ wμ ( m∨ i=1 ψ(ai) ) , and thus (P3) holds for wμ. (P4) now follows using an analogous argument, but without the requirement of f being onto, for a sentence ∃X η(X) ∈ S2L. Since μn is invariant under Ex and Px, it follows that wμn satisfies Ex + Px. For (ii), we will first show that wμ satisfies Wilmers' Principle. Let θ(x) ∈ F2L with x the only free variable and let 〈f, g〉 ∈ ℘∞ × n∞ with f onto. Then there is j ∈ N+ such that {ek |Mf,g |= θ(ek)} = {ek | k ∈ f(j)} = {ek |Mf,g |= Pj(ek)}. Hence, Mf,g |= ∀x (Pj(x)↔ θ(x)), and in turn, Mf,g |= ∃X ∀x (X(x)↔ θ(x)). By (3) then, μ{〈f, g〉 |Mf,g |= ∃X ∀x (X(x)↔ θ(x))} = 1 and hence wμ(∃X ∀x (X(x)↔ θ(x))) = 1. An analogous proof shows that wμ(ξn) = 1. For the converse let v be a probability function on S2L with the properties as given. For φ ∈ QFS1L set μ{〈f, g〉 ∈ ℘(n)∞ × n∞ |Mf,g |= φ} = v(φ). (4) Then μ is a finitely additive normalized measure on the algebra A of these subsets of ℘(n)∞ × n∞. Furthermore μ preserves infs, i.e. is a pre-measure. To see this suppose 9 that φ, θn ∈ QFS1L for n ∈ N and⋂ n∈N {〈f, g〉 |Mf,g |= θn} = {〈f, g〉 |Mf,g |= φ}. (5) Suppose that there is no k ∈ N such that⋂ n≤k {〈f, g〉 |Mf,g |= θn} = {〈f, g〉 |Mf,g |= φ}. (6) Then for each k we can find a structure Mfk,gk to satisfy {θn |n ≤ k} ∪ {¬φ}. Taking an ultraproduct of these structures with respect to a non-principle ultrafilter yields a structure M satisfying the set of sentences {θn |n ∈ N} ∪ {¬φ} (7) whose universe is still the set of ek, k = 1, 2, . . . , n and, when restricted to the original language, is of the form Mf,g for some 〈f, g〉 ∈ ℘(n)∞ × n∞. Furthermore since the θn, φ ∈ QFS1L this Mf,g also satisfies (7). But that contradicts (5), so such a k must exist and the preservation of this inf follows. By Carathéodory's Extension Theorem we can now uniquely extend μ to the σ-algebra generated by A. Since v satisfies Ex and Px then μ as defined by (4) will be invariant under Ex and Px and this property will be retained in this extension of μ. We now need to show that μ{〈f, g〉 | g is onto} = 1. (8) Since v(ξn) = 1 for some ai1 , ai2 , . . . , ain v ( ∧ 1≤k<j≤n ∃X ¬(X(aik)↔ X(aij)) ) > 0. and for some Pqk,j , 1 ≤ k < j ≤ n, v ( ∧ 1≤k<j≤n ¬(Pqk,j(aik)↔ Pqk,j(aij)) ) > 0. By definition this left hand side is equal to μ { 〈f, g〉 ∣∣∣M〈f,g〉 |= ∧ 1≤k<j≤n ¬(Pqk,j(aik)↔ Pqk,j(aij))) } , 10 and similarly for disjunctions of such formulae ∧ 1≤k<j≤n ¬(Pqk,j(aik)↔ Pqk,j(aij)). But since v(ξn) = 1, by (1), the limit probability of these disjunctions is 1, giving (8). Hence by the first part wμ is a probability function and by definition it agrees with v on QFS1L. So by induction on the quantifier complexity of φ ∈ SL2, v(φ) = wμ(φ) = μ{〈f, g〉 |Mf,g |= φ} It remains to show that μ satisfies (3). Notice that from (8) we are already half way there. Since v(WP ∧ ξn) = 1, μ{〈f, g〉 |Mf,g |= WP ∧ ξn} = 1. Let 〈f, g〉 be in this set with g onto. Then since M〈f,g〉 |= WP we have that for any Pi, Pj there are Pk, Pr, Pm such that M〈f,g〉 |= ∀x (Pk(x)↔ >) M〈f,g〉 |= ∀x (Pr(x)↔ ¬Pi(x)) M〈f,g〉 |= ∀x (Pm(x)↔ (Pi(x) ∧ Pj(x))) Hence the P M〈f,g〉 j form a Boolean Algebra. Also since Mf,g |= ξn every pair em 6= er are separated by some P M〈f,g〉 j so this Boolean Algebra must be all subsets of {e1, e2, . . . , en}. I.e. f must be onto and (3) follows.  In the case n = 1 there is only one choice for 〈f, g〉 and similarly μ and in this case wμ is cL0 from Carnap's Continuum of Inductive Methods for L. For n > 1 there are infinitely many choices for μ invariant under Ex and Px. For example for n = 2 and 0 < c < 1 we can define μc on the standard basis subsets of ℘(2) ∞ × 2∞ by μc { 〈f, g〉 ∣∣∣ n∧ k=1 f(jk) = sk ∧ m∧ r=1 g(ir) = qr } = 2−m m∏ k=1 c|sk|(1− c)2−|sk|. If we now extend μc to a normalized σ-additive, Ex and Px invariant measure measure on ℘(2)∞ × 2∞ then all the wμc will be different. We now give a version of a 'Ladder Theorem' which provides a classification of the probability functions satisfying Wilmers' Principle and will be useful in the next section. In this theorem the ξn are as given in (2). Theorem 5 (Ladder Theorem). Let w be a probability function on S2L satisfying Wilmers' Principle. Then w can be represented as w = λ0w0 + ∑ n∈N+ λnwn (9) 11 with the wn satisfying Wilmers' Principle, wn(ξn) = 1 and w0(ξn) = 0 for n > 0, λn ≥ 0 and ∑ n λn = 1. Proof Let w be a function on S2L satisfying Wilmers' Principle. For n > 0, if w(ξn) = 0 let λn = 0 and take wn to be any probability function satisfying Wilmers' Principle and wn(ξn) = 1. On the other hand if w(ξn) = λn > 0 set wn = w(* | ξn). Since ξn does not contain any constant or predicate symbols wn satisfies Ex and Px and similarly wn inherits Wilmers' Principle from w since w gives value 1 to any instance of this principle. Clearly wn(ξn) = 1. Now consider ŵ = w − ∑ n∈N+ λn * wμn and let λ0 = ŵ(>). If λ0 = 0, we are done. Otherwise w0 = λ−10 ŵ, is a probability function satisfying Ex, Px and Wilmers' Principle. Also for n > 0, w0(ξn) = 0 since wm(ξn) = 0 for 0 < m 6= n and w(ξn) = λn = λnwn(ξn).  We have already seen that there are probability functions w satisfying Wilmers' Principle in whose Ladder Representation λn > 0. The same is true for λ0: We are now in a position to show that: Proposition 6. There are probability functions satisfying Wilmers' Principle with λ0 > 0 in the Ladder Representation. Proof Let ξn be as above. Since the set {¬ξn |n ∈ N+} together with all instances of Wilmers' Principle is consistent it has a model M ∈ T L. Define the probability function VM : S2L→ {0, 1} by VM(θ) = { 1 if M |= θ, 0 if M |= ¬θ. (10) As given VM does not satisfy Ex or Px. However by a method introduced into this field by Gaifman, see [4] or [7, Theorem 12.3], we can use VM to construct a probability function w which does satisfy Ex and Px and which gives non-zero probability to the same sentences. Indeed in this case it will continue to give all the instances of Wilmers' Principle and the ¬ξn probability 1 so for this w we must have that λ0 = 1.  12 4 Regularity and Super Regularity Concerning the regularity and super regularity4 of probability functions satisfying Wilmers' Principle the first point to notice is that for n > 0 the wn in (9) do not satisfy Reg since, for example, for m > n m∧ j=1 Pj(aj) ∧ ∧ 1≤i≤m i 6=j ¬Pj(ai)  is inconsistent with ξn and hence gets probability 0 according to wn. Hence if λ0 = 0 and only finitely many of the λn are non-zero in the representation (9) then w will not satisfy Reg. However we do have: Theorem 7. If w satisfies Wilmers' Principle and either λ0 > 0 or infinitely many λn are non-zero in the Ladder Representation of w then w satisfies Reg. Proof We start by considering the case when infinitely many of the λn are non-zero. By the Disjunctive Normal Form Theorem it is enough to show that w gives non-zero probability to quantifier free sentences of the form m∧ i=1 k∧ j=1 P εi,j j (ai) where the εi,j ∈ {0, 1} and P 1 = P, P 0 = ¬P . Pick n large (i.e. compared with m, k) with λn > 0 and let 〈f, g〉 ∈ ℘(n)∞ × n∞ be such that ξn and Wilmers' Principle hold in Mf,g. For j = 1, . . . , n pick distinct ~δj = 〈δ1,j, δ2,j, . . . , δn,j〉 agreeing on the first m coordinates with 〈ε1,j, ε2,j, . . . , εm,j〉 for j ≤ k and such that the vectors 〈δi,1, δi,2, . . . , δi,n〉 are also all distinct (possible since n is large). From the proof of the converse in Theorem 4 ran(f) = ℘(n) so for each ~δ = δ1, δ2, . . . , δn ∈ {0, 1}n there is a κ(~δ) ∈ N+ such that Mf,g |= n∧ i=1 P δi κ(~δ) (ei). So Mf,g |= n∧ j=1 n∧ i=1 P δi,j κ(~δj) (ei), 4Which continue to apply only to first order sentences. 13 and Mf,g |= ∃X1, . . . , Xn ∃x1, . . . , xn n∧ j=1 n∧ i=1 X δi,j j (xi). Hence from Theorem 4, wn ( ∃X1, . . . , Xn ∃x1, . . . , xn n∧ j=1 n∧ i=1 X δi,j j (xi) ) = 1. Noticing that by the choice of δi,j these Xj and xi must be distinct we now obtain by Lemma 2, Ex and Px that wn ( n∧ j=1 n∧ i=1 P δi,j j (ai) ) > 0, (11) and in turn, as required, wn ( k∧ j=1 m∧ i=1 P εi,j j (ai) ) > 0. Turning now to the case when λ0 > 0. Let μ0 be the measure for w0 as given in the proof of Theorem 3, so μ0 { M ∈ T L ∣∣∣M |= WP ∧ ∃x1 . . . ∃xn ∧ 1≤i<j≤n ∃X ¬(X(xi)↔ X(xj)) } = 1. Let M ∈ T L be in this measure 1 set, without loss of generality say M |= ∧ 1≤i<j≤n ∃X ¬(X(ai)↔ X(aj)). Then since M |= WP , just as before the sets {ai | 1 ≤ i ≤ n, M |= Pj(ai)} form a Boolean Algebra which must be all subsets of {a1, a2, . . . , an}. The demonstration that Reg holds now proceeds just as in the case above.  We now improve this Theorem to conclude SReg. Theorem 8. If w satisfies Wilmers' Principle and either λ0 > 0 or infinitely many λn are non-zero in the Ladder Representation of w then w satisfies SReg. 14 Proof We first give the proof for the case when infinitely many λn are non-zero in the Ladder Representation of w. Since for this unary language L any consistent sentence in S1L is logically equivalent to a disjunction of sentences of the form 5 m∧ i=1 P ~δi(ai) ∧ s∧ i=1 ∃xP ~τi(x) ∧ q∧ i=1 ∀x¬P ~γi(x), where P ~δi(ai) is short for ∧r j=1 P δi,j j (ai) etc., to show that w satisfies SReg it is enough, by Px, to show that for n large compared m and r and λn > 0 there is some choice of distinct predicate symbols, R1, . . . , Rr say, such that wn  m∧ i=1 R ~δi(ai) ∧ ∧ ~δ 6=~δ1,...,~δm ∀x¬R~δ(x)  > 0. Assume that 〈0, 0, . . . , 0〉 is amongst these ~δk. [Changing 〈0, 0, . . . , 0〉 to some other vector ~δk only requires a minor modification of the proof which follows. Notice that there must be some ~δk otherwise the sentence is actually a contradiction!] By Theorem 7 wn ( m∧ i=1 P ~δi(ai) ) > 0. (12) Let ψ = m∧ i=1 P ~δi(ai). From Wilmer's Principle for wn wn ( ∃X ∀x ( X(x)↔ m∨ i=1 P ~δi(x) )) = 1 so with (12) there is some predicate symbol Rr+1 such that wn ( ψ ∧ ∀x ( Rr+1(x)↔ m∨ i=1 P ~δi(x) )) > 0. (13) We now claim that for t = 1, . . . , r there are distinct predicate symbols R1, . . . , Rt such that wn(ζt(R1, R2, . . . , Rt)) > 0 (14) 5See for example [8, Chapter 10]. 15 where ζt(R1, R2, . . . , Rt) is ψ ∧ ∀x ( Rr+1(x)↔ r∨ i=1 P ~δi(x) ) ∧ t∧ j=1 ∀x (Rj(x)↔ (Rr+1(x) ∧ Pj(x))). To see this suppose that we have such a ζt(R1, R2, . . . , Rt) for some t < r and distinct R1, . . . , Rt. We wish to show it for t+ 1. Since by Wilmers' Principle wn(∃X∀x(X(x)↔ (Rr+1(x) ∧ Pt+1(x)))) = 1 there is a predicate symbol Rt+1 such that wn(ζt+1(R1, R2, . . . , Rt, Rt+1)) > 0 (15) where ζt+1(R1, R2, . . . , Rt, Rt+1) is ζt(R1, R2, . . . , Rt) ∧ ∀x (Rt+1(x)↔ (Rr+1(x) ∧ Pt+1(x))). Unfortunately we seem to have no immediate guarantee that Rt+1 differs from R1, . . . , Rt. So suppose, without loss of generality, that in fact Rt+1 = R1 6= R2, R3, R4, . . . , Rt. By Px for distinct relation symbols S1, S2, . . . , St different from P1, P2, . . . , Pr, R1, R2, . . . , Rt, Rr+1 we have that wn(ζt+1(R1, R2, . . . , Rt, Rt+1)) = wn(ζt+1(S1, S2, . . . , St, S1)) > 0. We can take infinitely many such S1, S2, . . . , St so a pair of these must overlap in the sense that their conjunction has non-zero wn probability. Without loss of generality we may suppose then that wn(ρ) > 0 where ρ is ζt+1(R1, R2, . . . , Rt, Rt+1) ∧ ζt+1(S1, S2, . . . , St, St+1) and St+1 = S1 (and Rt+1 = R1). But ρ |= ∀x (Rt+1(x)↔ (Rr+1(x) ∧ Pt+1(x))) ∧ ∀x (St+1(x)↔ (Rr+1(x) ∧ Pt+1(x))) so ρ |= ∀x (Rt+1(x)↔ St+1(x)) and in turn ρ |= ζt+1(R1, R2, . . . , Rt, St+1). Hence wn(ζt+1(R1, R2, . . . , Rt, St+1)) ≥ wn(ρ) > 0 and the R1, R2, . . . , Rt, St+1 now are all different. 16 Having now found such distinct R1, R2, . . . , Rr let ζ = ζr(R1, R2, . . . , Rr), so wn(ζ) > 0. It remains to show that ζ |= m∧ i=1 R ~δi(ai) ∧ ∧ ~δ 6=~δ1,...,~δm ∀x¬R~δ(x). Since for i = 1, . . . ,m ζ |= ζ ∧ ψ |= Rr+1(ai) ∧ P ~δi(ai), it follows that ζ |= m∧ i=1 R ~δi(ai), which gets us halfway there. Now suppose that ~δ is not amongst the ~δ1, . . . , ~δm. Notice that by our initial assumption at least one of the coordinates of ~δ is 1. In this case ζ ∧R~δ(x) |= Rr+1(x) and since ζ ∧Rr+1(x) |= Pi(x)↔ Ri(x), ζ ∧R~δ(x) |= ζ ∧Rr+1(x) ∧R ~δ(x) |= P ~δ(x) ∧ m∨ i=1 P ~δi(x) |= ⊥, which gives that ζ |= ∀x¬R~δ(x). We now have, as required, that wn  m∧ i=1 R ~δi(ai) ∧ ∧ ~δ 6=~δ1,...,~δm ∀x¬R~δ(x)  ≥ wn(ζ) > 0 with the R1, R2, . . . , Rr are distinct. The corresponding result to Theorem 8 when λ0 > 0 follows similarly.  A straightforward corollary here then is: Corollary 9. If the probability function w satisfies Wilmers' Principle and Reg then it satisfies SReg. 17 It is worth noting that strengthening Wilmers' Principle to allow the formula θ(x) to be second order does not provide any correspondingly stronger results than we already have. Firstly Reg is perforce first order and the second order version of SReg trivially cannot be proved because it would require the satisfiable negations of instances of Wilmers' Principle to have non-zero probability in the very presence of Wilmers' Principle! In this note we have considered one possible second order principle in relation to just one first order consequence i.e. regularity. 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