# Translation Invariance and Miller’s Weather Example

## Abstract

In his 1974 paper “Popper’s qualitative theory of verisimilitude” published in the *British Journal for the Philosophy of Science* David Miller gave his so called ‘Weather Example’ to argue that the Hamming distance between constituents is flawed as a measure of proximity to truth since the former is not, unlike the latter, translation invariant. In this present paper we generalise David Miller’s Weather Example in both the unary and polyadic cases, characterising precisely which permutations of constituents/atoms can be effected by translations. In turn this suggests a meta-principle of the rational assignment of subjective probabilities, that rational principles should be preserved under translations, which we formalise and give a particular characterisation of in the context of Unary Pure Inductive Logic.

## Keywords

Miller’s weather example Verisimilitude for relations Translation invariance Renaming invariance Pure inductive logic Uncertain reasoning## 1 Introduction

^{1}Paraphrased Miller’s example runs as follows.

The primary observation in this present paper is that there is a general result (and conclusion) behind this example. We will first show this in the case of a unary (i.e., monadic) predicate language (where it may well have no great novelty) and then extend our results to a general polyadic (relational) language. On the way we will uncover a ‘meta-principle’ of probability assignment which we will investigate in the unary case.Jones and Smith are closeted away in prison and try to guess the weather outside. Jones thinks it is cool and dry and still whilst Smith also thinks it is cool but otherwise differs in thinking it rainy and windy. They subsequently learn that actually it is hot, rainy and windy. So Smith is right on two scores (rainy and windy) while Jones is wrong on all scores. From this we might conclude that Smith’s guess is closer to the truth than Jones’. But suppose we now replace the propositions ‘hot’, ‘rainy’ and ‘windy’ by the equally expressive ‘hot’, ‘Minnesotan’ (meaning hot and wet or cool and dry) and ‘Arizonan’ (meaning hot and windy or cool and still). In that case Smith’s guess becomes cool, not Minnesotan and not Arizonan, Jones’ becomes cool, Minnesotan and Arizonan. The actual situation is hot and Minnesotan and Arizonan. So now it is Jones who is right on two scores and Smith who is right on none!

To make precise the context we shall work in let \(L_{\vec {P}}\) be a predicate language^{2} with unary relation (i.e., predicate) symbols \(P_1,P_2, \ldots , P_q\) and, for later applications, constant symbols \(a_1,a_2,a_3, \ldots \). Note that by treating \(P_1(a_1), P_2(a_1), \ldots , P_q(a_1)\) as propositional variables our set-up can be considered to extend the propositional calculus in which Miller’s example is formalised.

*atoms*

^{3}of \(L_{\vec {P}}\) are the \(2^q\) formulae \(\alpha _{\epsilon }(x)\) of the form

An atom then is specified by a map \(\epsilon \) and the *Hamming distance between atoms* is defined to be the Hamming distance between these maps, that is the number of arguments on which they give different values.

*translation*of the predicate symbols \(P_1,P_2, \ldots , P_q\) is a family \(\phi _1(x), \phi _2(x),\)\(\ldots , \phi _q(x)\) of quantifier free formulae of \(L^-_{\vec {P}}\) such that as \(\delta \) ranges over \(\Omega _{\vec {P}}\) so the \(2^q\) formulae

^{4}

So if we imagine that the \(P_1(x), P_2(x), \ldots , P_q(x)\) describe certain features of, say, the weather at location *x*, then the \(\phi _1(x), \phi _2(x), \ldots , \phi _q(x)\) would, as in Miller’s example, provide an alternative, and exactly as descriptive, way to describe the weather.

^{5}

*is supported*by the translation \(\vec {\phi }\).

### Theorem 1

Let \(\tau \) be a permutation of \(\Omega _{\vec {P}}\). Then there is a translation \(\vec {\phi } = \langle \phi _1(x), \phi _2(x), \ldots , \phi _q(x)\rangle \) that supports \(\tau \). Conversely, any translation \(\vec {\phi }\) supports a permutation of \(\Omega _{\vec {P}}\) (namely \(\tau _{\vec {\phi }}\)).

*some*tuple of atoms with the desired Hamming distances between them) we can in fact make multiple distance changes simultaneously. In Miller’s example, writing

*C*,

*D*,

*S*and

*M*,

*A*for ‘cool, dry, still’ and ‘Minnesotan, Arizonan’ respectively, (meaning the weather outside Jones and Smith’s prison), the situation may be represented as

*renaming*of atoms which sends (2) to

*constituents of the predicate language*\(L^-_{\vec {P}}\), that is in our notation sentences of \(L^-_{\vec {P}}\) of the form

## 2 The General Polyadic Case

*H*,

*D*, and

*B*stand for, respectively,

*H*and

*D*, with

*l*and

*b*standing for the cities, the respective positions are

^{6}but in terms of

*H*and

*B*they are

*H*,

*B*is an equally expressive pair as

*H*,

*D*.

*A state formula*of \(L_{\vec {R}}\) for variables \(x_1, \ldots , x_n\) is a formula

*state description (for*\(b_1, \ldots , b_n\)

*)*.

*An atom of*\(L_{\vec {R}}\) is a state formula for

*r*variables. Hence an atom of \(L_{\vec {R}}\) is determined by a map

*q*-tuple of quantifier free formulae \(\psi _i (x_1, \ldots x_{r_i})\) forms a

*translation*of \(L_{\vec {R}}^-\) if the

*is supported by*the translation \(\psi _1, \ldots , \psi _q\) if for each \(\epsilon \in \Omega _{\vec {R}}\),

Unlike the purely unary case however, in the polyadic it is not in general the case that *just any* permutation, or renaming, of atoms is supported by a translation. As we shall prove this will be the case just if the permutation satisfies a certain property (C) from Ronel and Vencovská (2014) which we will define shortly. Interestingly, as we shall subsequently explain, condition (C) is also equivalent to the permutation of atoms generating an automorphism of a certain structure *BL* relevant in Pure Inductive Logic.

- Let \(\Theta (x_1,\ldots , x_n)\) be as in (9) and let \( k_1, \ldots ,k_t \) be distinct numbers from \(\{1,\ldots ,n\}\). Then \(\Theta [x_{k_1},\ldots , x_{k_t}]\) denotes the state formula obtained from (9) by restricting it to \(x_{k_1},\ldots , x_{k_t}\), that is, replacingby$$\begin{aligned} \langle j_1, \ldots , j_{r_i}\rangle \in \{1,2,\ldots , n\}^{r_i} \end{aligned}$$$$\begin{aligned} \langle j_1, \ldots , j_{r_i}\rangle \in \{k_1,\ldots , k_t\}^{r_i}. \end{aligned}$$
- Let \(\Phi (x_{k_1},\ldots , x_{k_t})\) be a state formula, \( m_1, \ldots ,m_s \) distinct numbers anda surjection. Then \((\Phi (x_{k_1}, \ldots ,x_{k_t}))_f\) denotes the state formula \(\Psi (x_{m_1}, \ldots , x_{m_s})\) for which$$\begin{aligned} f :\{m_1, \ldots , m_s\} \rightarrow \{k_1, \ldots , k_t\} \end{aligned}$$$$\begin{aligned} \Psi (x_{f(m_1)}, \ldots , x_{ f(m_s)}) = \Phi (x_{k_1}, \ldots , x_{k_t}). \end{aligned}$$

- (C)
*For*\(\epsilon , \delta \in \Omega _{\vec {R}}\), \(t\le r\)*and distinct*\( j_1, \ldots ,j_t \)*from*\( \{1,\ldots ,r\}\),*if*\(f :\{1, \ldots , r\} \rightarrow \{j_1, \ldots , j_t\}\)*is a surjection then*$$\begin{aligned} \alpha _{\epsilon }(x_1, \ldots , x_r)= & {} (\alpha _{\delta }[x_{j_1}, \ldots ,x_{j_t}])_f \iff \alpha _{\sigma (\epsilon )}(x_1, \ldots , x_r) \\= & {} (\alpha _{\sigma (\delta )}[x_{j_1}, \ldots ,x_{j_t}])_f. \end{aligned}$$

*f*. Let \(\gamma \in \Omega _{\vec {R}}\) be such that

*g*is the identity on \(\{m_1,\ldots ,m_s\}\), it follows that

- (D)
*For distinct*\({m_1},\ldots , {m_s} \in \{1,2,\ldots , r\}\), \(k_1, \ldots , k_t \in \{1,2,\ldots , r\}\),*surjection*\(f :\{m_1, \ldots , m_s\} \rightarrow \{k_1, \ldots , k_t\}\)*and*\(\epsilon , \delta \in \Omega _{\vec {R}}\),$$\begin{aligned} \alpha _\epsilon [x_{m_1}, \ldots , x_{m_s}]= & {} (\alpha _\delta [x_{k_1}, \ldots , x_{k_t}])_f \Longleftrightarrow \alpha _{\sigma (\epsilon )}[x_{m_1}, \ldots , x_{m_s}] \\= & {} (\alpha _{\sigma (\delta )}[x_{k_1}, \ldots , x_{k_t}])_f \end{aligned}$$

*f*to be identity, we have

- (E)
*For distinct*\(m_1, \ldots , m_s \in \{1,2,\ldots , r\}\),$$\begin{aligned} \alpha _\epsilon [x_{m_1}, \ldots , x_{m_s}]= & {} \alpha _\delta [x_{m_1}, \ldots , x_{m_s}] ~ \Longleftrightarrow ~ \alpha _{\sigma (\epsilon )}[x_{m_1}, \ldots , x_{m_s}] \\= & {} \alpha _{\sigma (\delta )}[x_{m_1}, \ldots , x_{m_s}]. \end{aligned}$$

### Theorem 2

Given a permutation \(\sigma \) of \(\Omega _{\vec {R}}\) there is a translation supporting \(\sigma \) just if \(\sigma \) satisfies (C).

### Proof

*x*,

*y*are in the same \(A_k\) just if \(j_x=j_y\). Let \(h_k= \min \{A_k\}\) for \(k=1,2,\ldots ,t\). A disjunct

*f*maps the members of each \(A_k\) to \(h_k\), the least member of that \(A_k\). Another way of expressing this is that

*g*be a permutation of \(\{1,2, \ldots , r\}\) which, in particular, maps each \(h_k\) to \(j_{h_k}\). Note that this means that for each \(c \in \{1, \ldots , r_i\}\) we have \(g(f(c)) = j_c\). For each \(\epsilon \in \Omega _{\vec {R}}\) define \(\epsilon ' \in \Omega _{\vec {R}}\) by

*g*as above, let \(\xi \in \Omega _{\vec {R}}\) be such that

*f*is an extension of the above defined

*f*, mapping the members of each \(A_k\) to \(h_k\) and \(f(h)=h\) for \(h > r_i\). So

Hence from (21), \(\alpha _{{\gamma }}[x_{j_{h_1}},\ldots , x_{j_{h_t}}]\) with \(\sigma (\gamma )(i, j_1, \ldots , j_{r_i})=1\) equals (20) and (15) with \(\sigma (\epsilon ) (i, 1, \ldots , {r_i})=1\), and thus the disjunction from (14) logically implies the disjunction from (13) and the identity (14) is proved.

*E*), if for some \(\epsilon , \delta \in \Omega _{\vec {R}}\) we have

For the converse suppose that the translation \(\vec {\psi }\) supports a permutation \(\sigma \) of atoms. We need to show that \(\sigma \) satisfies (C).

*i*and \(j_1, \ldots , j_{r_i}\in \{1,\ldots ,r\} \) we have

*i*and \(j_1, \ldots , j_{r_i}\in \{1,\ldots ,r\} \) we have

*The Example continued*For a language

*L*containing one unary predicate \(R_1\) and one binary predicate \(R_2\), let \(\sigma :\Omega _{\vec {R}}\rightarrow \Omega _{\vec {R}}\) be defined by

*f*in condition (C) it is easy to see that the condition is satisfied and hence \(\sigma \) is supported by a translation. From (12) we can see that the translation is \(\psi _1(x)=R_1(x)\) and

*H*,

*B*just as well as in terms of

*H*,

*D*.

^{7}Hence with the \(Q_j\) defined as the \(\psi _j\), \(\Theta (x_1, \ldots , x_n)\) is logically equivalent to

*r*-tuples of variables may well be incompatible if the

*r*-tuples have some variables in common.

Unlike the unary case for the polyadic case we will leave open the question of fully characterising the permutations of constituents (as described in Hintikka (1965)) which are supported by a translation.

## 3 Verisimilitude?

The original motivation for the research in this note came from Miller’s paper (Miller 1974) regarding a measure of closeness of a theory to the truth, where, as explained in e.g., (Miller 1978), the truth is identified with a constituent of a finite propositional language (a complete consistent theory) and the theory with a set of constituents (possibly just one as in the Prisoner Example). First propositional languages were considered (equivalently, unary predicate languages with one constant), then unary predicate languages with no constants, see (Miller 1978).

With predicate languages however, it appears natural to employ languages with constants. In this case we are led to the notion of the quantifier free truth about \(a_1, a_2, \ldots ,a_n\) being a state description for \(a_1,a_2,\ldots ,a_n\) and a quantifier free theory about \(a_1,a_2, \ldots ,a_n\) being a set (disjunction) of such. State descriptions are conjunctions of instantiated atoms and in Sect. 1 we have seen that for unary languages and any permutation of atoms (that is of \(\Omega _{\vec {P}}\)) there is a translation supporting it, which consequently disqualifies Hamming distance between atoms as a measure of verisimilitude. In the polyadic case a permutation of atoms (that is of \(\Omega _{\vec {R}}\)) is supported by a translation just when it satisfies the condition (*C*) and the same conclusion has to be reached regarding the suitability of the Hamming distance between atoms for measuring verisimilitude.

*C*)], and the Hamming distance between them (i.e., the Hamming distance between elements of \(\Omega _{\vec {R}}\)) can change translations do preserve Hamming distances between state formulae

*structure*of state descriptions/formulae in the sense which we now explain. In Paris and Vencovská (2015, Chapter 40) the following notion of

*similarity*is introduced:

*State formulae*\(\Theta (x_1, \ldots , x_n)\), \(\Phi (x_1, \ldots , x_n)\)

*are similar, if for distinct*\({m_1},\ldots , {m_s} \in \{1,2,\ldots , n\}\), \(k_1, \ldots , k_t \in \{1,2,\ldots , n\}\)

*and*

*surjection*\(f :\{m_1, \ldots , m_s\} \rightarrow \{k_1, \ldots , k_t\}\),

^{8}of some variables \(x_{m_1}, \ldots , x_{m_s}\) exactly corresponds to the behaviour of some further variables \(x_{k_1}, \ldots , x_{k_t}\) (with

*f*specifying how the \(x_{k_1}, \ldots , x_{k_t}\) are ‘cloned’ by the \(x_{m_1}, \ldots , x_{m_s}\)), the same happens in \(\Phi \), and conversely.

Hence any translated version of the truth carries some information about the truth, namely the shared structure. From this viewpoint then the minimum Hamming distance between translations of state descriptions/formulae might be used to measure how far a theory is from capturing the *structure of the truth*.

## 4 Renaming Invariance and Rationality

Consider the problem Pure Inductive Logic aims to address: How to give a rational assignment of probabilities \(w(\theta )\) to sentences \(\theta \) of an entirely uninterpreted language \(L_{\vec {R}}\)?^{9} The current modus operandi here is to propose principles which we may intuitively feel are somehow ‘rational’ for the probability function *w* to satisfy and investigate their consequences and inter relationships.^{10}

The previous sections suggest a principle of ‘translation invariance’, that *w* should be unaffected by the sort of translation we have considered above. For suppose that within the remit of Pure Inductive Logic we have chosen a ‘rational’ probability function *w* on the set of sentences of the language \(L_{\vec {R}}\) (denoted \(SL_{\vec {R}}\)), that is chosen a probability function satisfying those principles which we judge to demarcate what ‘rational’ means. A caviller now points out that since there is supposed to be no intended interpretation here we could equally well have based our choice on a translation of the original relations - on an equally expressive set of relation symbols of appropriate arities. So, the caviller continues, to be consistent we should still be giving the same probabilities even after making this translation (and regardless of the the names chosen for the relation symbols). In other words, at risk of otherwise being seemingly not consistent we should accept the principle that a rational assignment of probabilities should additionally be invariant under translations.

**Translation Invariance Principle, TIP**

*If*\(\sigma \)

*is a permutation of atoms supported by a translation and*\(w_{\sigma }\)

*is the probability function on*\(SL_{\vec {R}}\)

*determined by*

^{11}

*then*\(w=w_\sigma .\)

It turns out that such a principle of ‘translation invariance’ already exists in equivalent forms in the literature. We shall now discuss this in more detail.

### 4.1 The Unary Case

^{12}Let

*w*be a probability function on the set of sentences \(SL_{\vec {P}}\) of \(L_{\vec {P}}\). Then

*w*is uniquely determined (see for example (Paris and Vencovská 2015, Chapter 7) or [12]) by its values on the state descriptions of \(L_{\vec {P}}\), that is sentences of \(L_{\vec {P}}\) of the form

As in Sect. 1 but using \(\vec {P}\) also in place of \(\vec {Q}\), any permutation \(\tau \) of \(\Omega _{\vec {P}}\) determines a permutation/renaming/translation of atoms and in turn of *state formulae, * that is formulae of \(L^-_{\vec {P}}\) of the form \( \bigwedge _{i=1}^m \alpha _{\epsilon _i}(x_i)\), by sending \( \bigwedge _{i=1}^m \alpha _{\epsilon _i}(x_i)\) to \( \bigwedge _{i=1}^m \alpha _{\tau (\epsilon _i)}(x_i)\).

*w*, each such translation uniquely determines a further probability function \(w_\tau \) on \(SL_{\vec {P}}\) by setting

*w*is invariant under translation in the sense we have discussed above can be seen to be equivalent to satisfying the well know property of

*Atom Exchangeability*:

^{13}

**The Principle of Atom Exchangeability, Ax**

*For*\(\tau \)

*a permutation of atoms and a state description*\(\bigwedge _{i=1}^n \alpha _{\epsilon _i}(b_i)\),

**The Principle of Predicate Exchangeability, Px**

*For*\(\theta \in SL\)*and predicate symbols*\(P_i,P_j\)*of**L*, *if*\(\theta '\)*is the result of transposing*\(P_i,P_j\)*throughout*\(\theta \)*then*\(w(\theta ) = w(\theta ')\).

**The Strong Negation Principle, SN**

*For*\(\theta \in SL\), \(w(\theta )=w(\theta ')\)*where*\(\theta '\)*is the result of replacing each occurrence of the predicate symbol*\(P_i\)*in*\(\theta \)*by*\(\lnot P_i\).

Each of the principles Px and SN have an evident claim to rationality on the grounds of *symmetry*. Namely in the completely uninterpreted situation envisaged here it would be irrational to assign probability values which broke the existing symmetries in the language. However, the reasons for the rationality of Atom Exchangeability are less easy to appreciate, as indeed the surprise inherent in Miller’s example shows.

For this reason one might require the constraints imposed on one’s choice of probability function to at least include Px+SN. That being the case one might argue not that we should always have \(w= w_\tau \) for \(\tau \) a permutation of \(\{0,1\}^q\) but simply that \(w_\tau \) was, as far as these constraints were concerned, an equally good choice, i.e., that the \(w_\tau \) should also satisfy at least Px+SN.

Thus we would be advocating here a sort of *meta-principle*, namely that if one initially proposed that adherence to principles *X*, *Y*, *Z* etc. determined what constituted a choice of probability function *w* being rational, then on secondary consideration *w* should additionally be such that all the \(w_\tau \) also satisfy *X*, *Y*, *Z* etc..^{14}

In the particular case of Px+SN this meta-principle yields a new principle which, for \(q>2\), lies strictly between Px+SN and Ax:

**The Unary Principle of Inculcated Px+SN, I(Px+SN)**

*For every permutation*\(\tau \)*of*\(\Omega _{\vec {P}}\), \(w_\tau \)*satisfies Px+SN.*

### Theorem 3

Let *w* be a probability function on \(SL_{\vec {P}}\). Then the probability function \(w_\tau \) satisfies Px+SN for each permutation \(\tau \) of \(\Omega _{\vec {P}}\) just if \(q >2\) and \(w=w_\sigma \) for every even permutation \(\sigma \) of \(\Omega _{\vec {P}}\) or \(q \le 2\) and *w* satisfies Ax.

### Proof

We first show this in the forward direction. Let \(\mathcal {S}_{\vec {P}}\) be the group of permutations of \(\Omega _{\vec {P}}\). Let *H* be the subgroup of \(\mathcal {S}_{\vec {P}}\) of permutations \(\tau \) such that \(v_\tau =v\) for all probability functions *v* on \(SL_{\vec {P}}\) which satisfy Px+SN. Equivalently, *H* is generated by the permutations of atoms which just transposes \(P_i,P_j\) and those which just transpose \(P_i, \lnot P_i\). Notice that for \(q>2\) all such permutations are even whilst for \(q \le 2\)*H* also contains odd permutations. Note too that a probability function *v* satisfies Px+SN just when \(v_\tau =v\) for all \(\tau \in H\).

*H*(and so is non-trivial). In fact it is a normal subgroup since if

Now suppose that \(q >2\). Then since the only non-trivial normal subgroups of \(\mathcal {S}_{\vec {P}}\) are itself and the alternating group *A* of even permutations it must be the case that \(\bigcap _{\sigma \in \mathcal {S}_{\vec {P}}} K_{w_\sigma }\) is one of these and hence, by taking \(\sigma \) to be the identity permutation, it follows that \(A \subseteq K_w\). Hence \(w=w_\sigma \) for every \(\sigma \in A\). (As we shall see later in this case of \(q>2\), *H* only contains even permutations and we cannot obtain a stronger result here.)

We now turn to the case \(q \le 2\). When \(q =1\), \(\mathcal {S}_{\vec {P}}\) is the only non-trivial normal subgroup of \(\mathcal {S}_{\vec {P}}\) so in this case we must have \(K_w=\mathcal {S}_{\vec {P}}\), in other words *w* satisfies Ax. When \(q=2\), \(\mathcal {S}_{\vec {P}}\) has two non-trivial proper normal subgroups. However both of these only contain even permutations while *H* contains some odd permutations so again we must have \(K_w=\mathcal {S}_{\vec {P}}\) and Ax follows.

Turning to the converse direction this is clear in the case of \(q \le 2\). For \(q>2\), suppose that \(w=w_\sigma \) for all \(\sigma \in A\). Then certainly since \(H \subseteq A\) for \(q>2\), *w* is invariant under permutations of predicate symbols and permutations replacing \(P_i\) by \(\lnot P_i\), so *w* satisfies Px+SN, and so does \(w_\sigma \) for \(\sigma \) an even permutation. Also since \(w_\sigma =w_\tau \) for all even permutations \(\sigma , \tau \), it is easy to see that this must also hold for all odd permutations \(\sigma , \tau \) and hence by the same argument as in the even case, for \(\sigma \) an odd permutation \(w_\sigma \) must satisfy Px+SN too. \(\square \)

At this point one might question whether even for \(q>2\) the new principle I(Px+SN), amounting to \(A \subseteq K_w\), really is strictly between Px+SN and Ax. (As Theorem 3 shows it is equivalent to Ax for \(q\le 2\).) We now construct examples of probability functions which show that this is the case.

Let \(\epsilon _1,\epsilon _2, \ldots , \epsilon _{2^q}\) list the elements of \(\Omega _{\vec {P}}\). For \(\sigma \in \mathcal {S}_{\vec {P}}\) it will be convenient to also treat \(\sigma \) as a permutation of these subscripts, that is \(\sigma (\epsilon _i)=\epsilon _{\sigma (i)}\).

^{15}\(w_{\vec {c}}\) by

^{16}

*i*,

*j*th entry \(c_i^{j-1}\). Since we can find suitable \(\vec {c}\) yielding positive Vandermonde determinant this means that for any odd permutation \(\tau \)

*H*does (and conversely in fact, see (Hill and Paris 2013)). Then since \( \nu \notin H\) the polynomials

*u*satisfies Px+SN whilst \(u \ne u_\nu \) since they give different values on

*Principle of Constant Exchangeability*, Ex, that is that for any sentence \(\theta (a_1, a_2, \ldots , a_n)\) and distinct \(j_1,j_2, \ldots , j_n\),

### 4.2 The Polyadic Case

We now turn to the case of the polyadic language \(L_{\vec {R}}\) and *w* a probability function on \(SL_{\vec {R}}\).

In this case too, equivalent forms of the Invariance Under Translation Principle already exist in the literature, see (Paris and Vencovská 2015, Chapters 39, 40; Paris and Vencovská 2011; Ronel and Vencovská 2014). Most directly, the principle is equivalent to the the Permutation Invariance Principle, PIP. This is a special case of the ‘ultimate’ symmetry principle of Pure Inductive Logic, INV, which we will briefly explain to start with.

In Paris and Vencovská (2015, Chapters 23, 39) we have argued that assigning probabilities to (classes of logically equivalent) sentences of an entirely uninterpreted language \(L_{\vec {R}}\) could be imagined as a task to be performed by an agent who knows that s/he is in a structure *M* for \(L_{\vec {R}}\) with universe \(\{a_1, a_2, \ldots \}\), in which each constant symbol \(a_i\) is interpreted as \(a_i\), but having no information as to what sentences of \(SL_{\vec {R}}\) hold in their ambient structure *M*. Since a rational agent in such a situation would presumably wish to respect symmetry, this picture clearly helps to motivate the symmetry principles which we have mentioned already. Our attempt to capture that which underlies *all* symmetry principles in Pure Inductive Logic, see the foregoing, was based on the observation that any symmetry of the (classes of logically equivalent) sentences of \(SL_{\vec {R}}\) corresponds to an automorphism of the set of all possible structures as above, along with the set of its definable subsets, in the following sense:

*M*for \(L_{\vec {R}}\) with universe \(\{a_1,a_2, a_3, \ldots \}\) where each constant symbol \(a_i\) of the language is interpreted in

*M*by the element \(a_i \). Let \(BL_{\vec {R}}\) be the two-sorted structure with universe \(\mathcal{T}L_{\vec {R}}\) together with the sets

**The Invariance Principle, INV**

*If*\(\eta \)*is an automorphism of*\(BL_{\vec {R}}\)*then*\(w(\theta )=w(\eta ( \theta ))\) for \(\theta \in SL_{\vec {R}}\).

As discussed in Paris and Vencovská (2015), INV in its full generality may be too strong, possibly denying rationality to almost all probability functions. This has indeed been proved for languages \(L_{\vec {P}}\) with only unary predicates: there is only one (from various points of view a not-entirely-suitable) probability function on \(SL_{\vec {P}}\) satisfying INV, see (Paris and Vencovská 2015, Chapter 23). The situation in the polyadic remains intriguingly open.

Whilst INV in the purely unary context, after corroborating the intuition for previously known symmetry principles, has been shown to just go too far, in the polyadic context INV has yielded a further interesting symmetry principle which obtains from INV by imposing an additional requirement on the \(\eta \), namely that they map state descriptions to state descriptions. In Paris and Vencovská (2015, Chapter 39) this has been proved to be equivalent to the principle PIP which we will state precisely after introducing some further definitions from Paris and Vencovská (2015).

*permutes state formulae*if for each

*n*and (distinct) variables \(x_{j_1},\ldots , x_{j_n}\), \(\digamma \) permutes the state formulae \(\Phi (x_{j_1},\ldots , x_{j_n})\) in these variables (up to logical equivalence). Properties (

*A*) and (

*B*) are defined as follows:

- (
*A*) -
*For each state formula*\(\Theta (x_{k_1},\ldots , x_{k_t})\)*and surjective mapping*\(\tau :\{m_1, \ldots , m_s\} \rightarrow \{k_1, \ldots , k_t\}\),$$\begin{aligned} (\digamma (\Theta (x_{k_1},\ldots , x_{k_t}))_\tau = \digamma (\Theta (x_{k_1},\ldots , x_{k_t})_\tau ). \end{aligned}$$ - (
*B*) -
*For each state formula*\(\Phi (x_{j_1},\ldots , x_{j_n})\)*and (distinct)*\(i_1,i_2,\ldots ,i_k \in \{j_1, \ldots , j_n\}\)$$\begin{aligned} \digamma (\Phi ) [x_{i_1},\ldots ,x_{i_k}]~=~ \digamma (\Phi [x_{i_1},\ldots ,x_{i_k}]). \end{aligned}$$

**The Permutation Invariance Principle, PIP**

*If*\(\digamma \)

*is a permutation of state formulae of*\(L_{\vec {R}}\)

*satisfying*(

*A*)

*and*(

*B*)

*then for a state description*\(\Phi (b_1, \ldots , b_n)\),

*A*) and (

*B*), by mapping \(\Theta (x_1, \ldots ,x_n)\) to \(\sigma (\Theta (x_1, \ldots ,x_n))\) as in (34) (and analogously for any other

*n*-tuple of distinct variables), just when \(\sigma \) satisfies the condition (

*C*). We remark that PIP, and hence TIP, is also equivalent to the principle which states that similar state descriptions get the same probability (

*Nathanial’s Invariance Principle*, NIP), cf. (Paris and Vencovská 2015, Chapter 41).

In the purely unary context, PIP is equivalent to Ax. As Ax does in the unary case, in the polyadic PIP implies Px and SN (the general formulation of these principles are as in the unary case except that in Px we need to say that we exchange predicate symbols *of the same arities*). How PIP relates to the Inculcated Px+SN, that is, to the requirement that Px and SN hold not only for *w* but also for any \(w_\sigma \) where \(\sigma \) is a permutation of \(\Omega _{\vec {R}}\) satisfying (C) and \(w_\sigma \) is defined as in (35), remains an open question. Although much of the reasoning used in the proof of Theorem 3 could be used with \(\mathcal {S}_{\vec {P}}\) replaced by the group of all permutations of \(\Omega _{\vec {R}}\) satisfying (*C*), we lack sufficient insight into the structure of this group to allow us to draw interesting conclusions.

## 5 Conclusion

Inspired by Miller’s Weather Example and its underlying notion of a *translation* we have considered the extent to which a simple permutation of atoms can be formulated, or explained, in terms of a translation. It turns out that this is always the case for purely unary languages whilst for general polyadic languages it requires the permutation to also satisfy a certain property (*C*). This also establishes a precise connection between translations and those permutations that can be extended to automorphisms of the overlying structure since (*C*) is again exactly the additional ingredient needed in that case too.

A salient feature of Miller’s Weather Example is that it reveals an underlying rational commitment to adopting beliefs that are *translation-proof*. Whilst the most rigid meaning one might give to that expression is that beliefs should be *translation-invariant* we argue that within the context of Pure Inductive Logic a more catholic interpretation might be that the translation preserves the *rationality* of the beliefs, rather than the actual quantitative beliefs themselves. Formally this leads to a meta-principle which we have characterised for certain rationality criteria in unary languages, showing it (for languages with at least three predicate symbols) to lie strictly between simply observance of these criteria and full preservation of belief values under translation.

## Footnotes

- 1.
- 2.
Miller’s example is set within a propositional language. Working within a predicate language will however allows us to generalize it, in particular to polyadic languages.

- 3.
Note that atoms are not the same thing as what is commonly called atomic formulae of the language. Carnap refers to them as

*Q-predicates*and Hintikka and Niiniluoto as*attributive constituents*. - 4.
Up to logical equivalence. Throughout we will usually, for convenience, identify formulae which are logically equivalent rather than actually syntactically identical.

- 5.
Recall our convention of identifying formulae even if they are formally only logically equivalent.

- 6.
We do not introduce a precise notion of distance here wishing just to convey the intuition. It could be the Hamming distance between atoms, see below, in which case we should also incorporate the prisoners’ opinion about, and the truth of,

*D*(*b*,*b*),*D*(*l*,*l*),*B*(*b*,*b*) and*B*(*l*,*l*). However since*B*(*x*,*x*) and*D*(*x*,*x*) coincide, this would make no difference to comparisons of descriptions using*H*,*D*and*H*,*B*. - 7.
Note that this means that the notation, if the need ever arose to write this out, would require us to talk about \(\epsilon _{ \langle \Theta , \langle k_1,\ldots ,k_r\rangle \rangle }(i,j_1,\ldots , j_{r_i})\) which are values from \(\{0,1\} \) such that \(\Theta (x_1, \ldots ,x_n) \models R_i(x_{k_{j_1}}, \ldots ,x_{k_{j_{r_i}}})\) just when \(\epsilon _{ \langle \Theta , \langle k_1,\ldots ,k_r\rangle \rangle }(i,j_1,\ldots , j_{r_i})=1\); the \(\langle k_1,\ldots ,k_r\rangle \) are from \( \{1,\ldots ,n\}^r\),

*i*from \(\{1, \ldots , q\}\) and \(\langle j_1,\ldots ,j_{r_i}\rangle \) from \(\{1,\ldots ,r\}^{r_i}\). Note also in particular that tuples \(\langle k_1,\ldots ,k_r\rangle \) and \(\langle j_1,\ldots ,j_{r_i}\rangle \)*with repeats*are included. - 8.
By the

*behaviour of the variables*\(x_{m_1}, \ldots , x_{m_s}\)*in*\( \Theta \) we mean all the information contained in \(\Theta \) that involves just these variables and no others. - 9.
See for example (Paris and Vencovská 2015, Chapter 1) for further details.

- 10.
For the definition of a probability function in this context and general background see for example (Paris and Vencovská 2015) or [12].

- 11.
Using the condition (E) and (Paris and Vencovská 2015, p 42), or [12], we can see that this definition does yield a probability function.

- 12.
So in particular \(\Omega _{\vec {P}}\) is the set of maps from \(\{1,2, \ldots , q\} \) to \(\{0,1\}\).

- 13.
For more details on Atom Exchangeability see (Paris and Vencovská 2015, Chapter 14).

- 14.
Another such ‘meta-principle’ in this area is

*(Unary) Language Invariance*, see for example (Paris and Vencovská 2015), which again applies not simply to a single probability function*w*but to a family of related probability functions to which*w*belongs. - 15.
See (Paris and Vencovská 2015, Page 51) for more details.

- 16.
The \(w_{\vec {c}}\), and in turn the probability functions

*v*,*u*to be introduced shortly, immediately satisfy the ubiquitous*Constant Exchangeability Principle,*Ex, see for example (Paris and Vencovská 2015, Page 33).

## Notes

## References

- Hill, A. J., & Paris, J. B. (2013). An analogy principle in inductive logic.
*Annals of Pure and Applied Logic*,*164*, 1293–1321.CrossRefGoogle Scholar - Hintikka, J. (1965). Distributive normal forms in first-order logic.
*Studies in Logic and the Foundations of Mathematics*,*40*, 48–91.CrossRefGoogle Scholar - Miller, D. (1974). Popper’s qualitative theory of verisimilitude.
*British Journal for the Philosophy of Science*,*25*(2), 166–177.CrossRefGoogle Scholar - Miller, D. (1978). The distance between constituents.
*Synthese*,*38*(2), 197–212.CrossRefGoogle Scholar - Miller, D. (2006).
*Out of error*. Farnham: Ashgate Publishing Company.Google Scholar - Niiniluoto, I. (1998). Verisimilitude: The third period.
*British Journal for the Philosophy of Science*,*49*(1), 1–29.CrossRefGoogle Scholar - Paris, J. B. & Vencovská, A. (April 2015). Pure inductive logic. In
*The Association of Symbolic Logic Perspectives in Mathematical Logic Series*. Cambridge University Press.Google Scholar - Paris, J. B. & Vencovská, A. (2011). A note on Nathanial’s invariance principle in polyadic inductive logic. In Banerjee, M., & Seth, A. (Eds.),
*Logic and its applications, ICLA, Proceedings of the 4th Indian Logic Conference*, Dehli, India (pp. 137–146).Google Scholar - Popper, K. R. (1972).
*Objective knowledge*. Oxford: Clarendon Press.Google Scholar - Ronel, T., & Vencovská, A. (2014). Invariance principles in polyadic inductive logic.
*Logique et Analyse*,*228*, 541–561.Google Scholar - Tichý, P. (1974). On Popper’s definitions of Verisimilitude.
*British Journal for the Philosophy of Science*,*25*(2), 155–160.CrossRefGoogle Scholar - Vencovská, A. Pure inductive logic—Nesin Maths Village Summer School Course Notes. Available at http://www.maths.manchester.ac.uk/~jeff/lecture-notes/NesinNotes.pdf

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.