# Symmetry and Evidential Support

## Abstract

**:**

## 1. Introduction

## 2. Carnap’s Early Theories of Favoring

## 3. Alternative Approaches to Favoring

## 4. General Conditions on Evidential Favoring

## 5. First Stage of the Proof

For the full details of the proof, I refer the reader to [4]. Here I will simply explain how it works, starting with an overview of the proof strategy: Suppose for reductio that evidential favoring is substantive and antisymmetric and treats predicate permutations identically. By f’s substantivity there exist an ${h}_{1}$, ${h}_{2}$, and e in faithful, adequate language $\mathcal{L}$ such that these three relata are logically independent and $f({h}_{1},{h}_{2},e)$. We will construct another faithful, adequate language ${\mathcal{L}}^{\ast}$ with ${h}_{1}^{\ast}$, ${h}_{2}^{\ast}$, and ${e}^{\ast}$ representing the same propositions as ${h}_{1}$, ${h}_{2}$, and e respectively. Since f concerns a relation among the propositions expressed by sentences, we will have $f({h}_{1}^{\ast},{h}_{2}^{\ast},{e}^{\ast})$. Moreover, ${\mathcal{L}}^{\ast}$ will be constructed so as to make available a predicate permutation $\pi $ such that $\pi \left({h}_{1}^{\ast}\right)={h}_{2}^{\ast}$, $\pi \left({h}_{2}^{\ast}\right)={h}_{1}^{\ast}$, and $\pi \left({e}^{\ast}\right)={e}^{\ast}$. Since f treats predicate permutations identically, $f({h}_{2}^{\ast},{h}_{1}^{\ast},{e}^{\ast})$. But that violates f’s antisymmetry, yielding a contradiction.General Result:If the evidential favoring relation is antisymmetric and substantive, it does not treat predicate permutations identically.

## 6. Second Stage of the Proof

## 7. The Favoring Relation Itself

## Acknowledgements

## References

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1 | This is Carnap’s “firmness” explication of confirmation, as opposed to the “increase in firmness” explication he distinguishes from it in the preface to the second edition of [3]. We will stick with the firmness explication because it is easier to work with, but my criticisms below apply equally well to the increase in firmness explication. |

2 | To prevent logical redundancy in a language’s set of state descriptions, we assume that no literal appears more than once in a state description and the literals appear in alphabetical order. So $Ga\&Gb$ is a state description of ${\mathcal{L}}^{G}$, but $Ga\&Ga\&Gb$ and $Gb\&Ga$ are not. |

3 | On a probabilistic approach like Carnap’s, an evidence set is represented by a conjunction each of whose conjuncts represents one of the set’s members. It makes no difference to the probability calculations if an extra, tautologous conjunct is added to this evidential conjunction. So we can imagine that if we removed empirical facts from an evidence set one at a time, in the end our “evidential conjunction” would be just a tautology. |

4 | Marlos Viana points out to me (in correspondence) that the transition from distributions over state descriptions to distributions over structure descriptions is familiar from Bose–Einstein, Fermi–Derac, and Boltzmann–Maxwell statistics. |

5 | ${\mathfrak{c}}^{\ast}$ can also retrieve the favoring judgment represented in Equation (2)—in fact, any 𝔠-function derived from a regular, probabilistic $\mathfrak{m}$-function will retrieve that judgment. |

6 | This is a metalinguistic statement about how the truth-values of propositions represented by atomic ${\mathcal{L}}^{H}$ sentences relate to the truth-values of propositions represented by atomic ${\mathcal{L}}^{G}$ sentences. None of our object languages contain both G and H as predicates, nor do any of our languages represent the identity relation. |

7 | Since ${s}_{1}$ through ${s}_{4}$ are names we have given to state descriptions of ${\mathcal{L}}^{G}$—and not sentences appearing in ${\mathcal{L}}^{G}$—the disjunctive normal forms written here for ${h}_{1}$, ${h}_{2}$, and e are metalinguistic indications of what those disjunctive normal forms actually look like in ${\mathcal{L}}^{G}$. |

8 | See Maher’s [10] for discussion of arguments by analogy and Carnap’s later views, as well as favoring theories by Hesse and others. Maher freely admits the language-dependence of his own systems. |

9 | For example, [11] develops language dependence problems for Jaynes’s maximum entropy approach. |

10 | The full definition of faithfulness also ensures that if we think of propositions as sets of worlds, a faithful language will have $\sim x$ represent the complement of the proposition represented by x, $x\&y$ represent the intersection of the propositions represented by x and y, and $x\vee y$ represent the union of those propositions. |

11 | This idea is familiar from the hypothetico-deductivist theory of confirmation; see [12] for discussion. |

12 | Evidential favoring will not be substantive if evidence can only discriminate among hypotheses by logically ruling some out, as suggested by the falsificationism of [14]. Miller [15] is moved to falsificationism in part by language dependence issues related to those discussed in this article. Nevertheless, falsificationism remains a minority view of evidential favoring. |

13 | Again, these are metalinguistic indications of what the disjunctive normal forms look like, not actual object-language expressions of the evidence and hypotheses. |

14 | Because $\mathcal{L}$ is faithful, its state descriptions express a set of mutually exclusive, exhaustive propositions. Since the state descriptions of ${\mathcal{L}}^{\ast}$ express these same propositions, ${\mathcal{L}}^{\ast}$ will wind up faithful as well. |

15 | Technically, the order of some conjuncts in the disjunctive normal form of e may be changed. But here and elsewhere in the argument that follows, I treat a sentence of a faithful language as interchangeable with other sentences in that language logically equivalent to it. If we think of propositions as sets of possible worlds, then any two logically equivalent sentences in a faithful language express the same proposition. Since f relates the propositions expressed by sentences, any two equivalent sentences in a faithful language will enter into f-relations in exactly the same ways. |

16 | One might wonder what the predicates of ${\mathcal{L}}^{\ast}$ mean—for instance, what property of the a-tuple is represented by the predicate ${G}^{\ast}$? We can construct the meaning of ${G}^{\ast}$ from the meanings of the (presumably well-understood) predicates of $\mathcal{L}$. Each state description of $\mathcal{L}$ says something about a, and there are tuples in the world that make that state description true when referred to by a. Each state description of ${\mathcal{L}}^{\ast}$ is a synonym of an $\mathcal{L}$ state description; an ${\mathcal{L}}^{\ast}$ state description says the same thing as its $\mathcal{L}$ counterpart. ${G}^{\ast}a$ is equivalent to a disjunction of state descriptions of ${\mathcal{L}}^{\ast}$, so we can determine which tuples make ${G}^{\ast}a$ true when referred to by a. ${G}^{\ast}$ expresses the property of belonging to the set containing just those tuples. |

17 | Depending on the size of $\mathcal{L}$ and the particular constitution of ${h}_{1}$, ${h}_{2}$, and e, this assignment strategy may require ${\mathcal{L}}^{\ast}$ to have more predicates, and thus more state descriptions, than $\mathcal{L}$. The relevant calculations and a general recipe for making this work are described in [4, Appendix A]. |

18 | Quick proof: If we let $\#{h}_{1}$ represent the number of ${h}_{1}$-sds and so forth, the proportion of its state descriptions that ${h}_{1}$ shares with e is $(\#{h}_{1}e+\#{h}_{1}{h}_{2}e)/(\#{h}_{1}+\#{h}_{1}{h}_{2}+\#{h}_{1}e+\#{h}_{1}{h}_{2}e)$ and the proportion of its state descriptions that ${h}_{2}$ shares with e is $(\#{h}_{2}e+\#{h}_{1}{h}_{2}e)/(\#{h}_{2}+\#{h}_{1}{h}_{2}+\#{h}_{2}e+\#{h}_{1}{h}_{2}e)$. If the unstarred versions of the two equalities are met, these proportions are equal. So if the unstarred equalities are both met, the Proportional Theory indicates no favoring. Contraposing, if the Proportional Theory indicates a favoring relation at least one of the unstarred equalities is violated. |

19 | In fact, ${D}^{\u2020}$ and ${E}^{\u2020}$ must be chosen so that it is logically possible for the tuple represented by a to have any combination of the properties represented by ${D}^{\u2020}$, ${E}^{\u2020}$, and the predicates of $\mathcal{L}$. This is required so that the state descriptions of ${\mathcal{L}}^{\u2020}$ will express a set of mutually exclusive, exhaustive propositions and ${\mathcal{L}}^{\u2020}$ can be faithful. The possibility of finding properties of the tuple expressible by ${D}^{\u2020}$ and ${E}^{\u2020}$ that are independent in the relevant sense of each other and of the properties represented in $\mathcal{L}$ is guaranteed by an assumption I call the Availability of Independent Properties. For a defense of that assumption see [4, p. 502]. |

20 | What about the sentences in ${\mathcal{L}}^{\u2020}$ that are synonyms of $\varphi $-sds? For our particular ${h}_{1}$, ${h}_{2}$, and e, $\mathcal{L}$ has 32 state descriptions 24 of which are $\varphi $-sds. Each $\varphi $-sd has a synonym that is a disjunction of 4 ${\mathcal{L}}^{\u2020}$ state descriptions. ${\mathcal{L}}^{\prime}$ has one more predicate than $\mathcal{L}$, so it has 64 state descriptions. We have already assigned synonyms to 9 of them, leaving 55. Of the 24 $\varphi $-sds in $\mathcal{L}$, use ${\mathcal{L}}^{\u2020}$ to split 17 of them into disjunctions of 2 state descriptions in ${\mathcal{L}}^{\prime}$, and split the other 7 into disjunctions of 3 state descriptions in ${\mathcal{L}}^{\prime}$. This will assign meanings to the remaining 55 state descriptions of ${\mathcal{L}}^{\prime}$ and ensure that each $\varphi $-sd has a synonym in ${\mathcal{L}}^{\prime}$. (For a generalization of the math involved here, including how big ${\mathcal{L}}^{\u2020}$ and ${\mathcal{L}}^{\prime}$ will typically have to be, see [4, Appendix A], especially notes 68 through 70.) |

21 | A bit of thought will also reveal that because the state descriptions of ${\mathcal{L}}^{\u2020}$ express a mutually exclusive, exhaustive set of propositions the state descriptions of ${\mathcal{L}}^{\prime}$ do so as well. This makes ${\mathcal{L}}^{\prime}$ faithful. |

22 | For one such position, see [16]. |

23 | Notice how much stronger this result is than standard “underdetermination of theory by evidence” arguments. Underdetermination of theory by evidence typically argues that while some e may favor ${h}_{1}$ over ${h}_{2}$, we can always manufacture an ${h}_{3}$ that is just as well supported by e as ${h}_{1}$. The present result shows that even if e, ${h}_{1}$, and ${h}_{2}$ are selected for us, there will be no favoring of ${h}_{1}$ over ${h}_{2}$ by e unless some logical entailment holds among them. It is not just that an outlandish theory can be made up that accounts for the data; it is that no theory under serious consideration accounts for the data better than any of the others. |

24 | In correspondence Paul Bartha asks me to consider whether the set of languages to which a favoring theory should apply can be restricted further than I have suggested on an a priori basis. His example is a language whose single constant represents a tuple of 10 emeralds and whose three predicates represent the properties “tuple elements 1 through 6 are green,” “tuple elements 7 through 9 are green,” and “tuple element 10 is green.” Bartha suggests that this language can be ruled out (and in particular that a formal favoring theory need not treat permutations of its predicates identically) on the grounds that some predicates talk about more objects than others, or alternatively that some atomic sentences convey more information than others. My response is that the proper individuation of objects for purposes of projection should be determined by our empirical evidence. For example if we are making predictions about emeralds, is evidence more significant if it is about more emeralds or if the emeralds it talks about have a greater total mass? (What if emeralds 7 through 9 in Bartha’s example together weigh more than emeralds 1 through 6?) Put another way, should a scientific language for predicting features of emeralds have constants that pick out individual emeralds or individual 1-gram chunks of emerald mass? I do not see how this kind of question could be answered a priori, but if we try to answer it using empirical evidence and then develop our evidential favoring relation from languages that individuate objects properly, we will wind up in the same kind of circle as we saw with $np$. Once more something beyond our evidence will have to play a decisive role. |

**Table 1.**${\mathfrak{m}}^{\u2020}$- and ${\mathfrak{m}}^{\ast}$-values for language ${\mathcal{L}}^{G}$.

Name | State description | ${\mathfrak{m}}^{\u2020}$ | ${\mathfrak{m}}^{\ast}$ |
---|---|---|---|

${s}_{1}$ | $Ga\&Gb$ | $1/4$ | $1/3$ |

${s}_{2}$ | $Ga\&\sim Gb$ | $1/4$ | $1/6$ |

${s}_{3}$ | $\sim Ga\&Gb$ | $1/4$ | $1/6$ |

${s}_{4}$ | $\sim Ga\&\sim Gb$ | $1/4$ | $1/3$ |

Expresses same | ||
---|---|---|

State description | ${\mathfrak{m}}^{\ast}$ | proposition as |

$Ha\&Hb$ | $1/3$ | ${s}_{2}$ |

$Ha\&\sim Hb$ | $1/6$ | ${s}_{1}$ |

$\sim Ha\&Hb$ | $1/6$ | ${s}_{4}$ |

$\sim Ha\&\sim Hb$ | $1/3$ | ${s}_{3}$ |

Expresses same | ||
---|---|---|

State description | ${\mathfrak{m}}^{\ast}$ | proposition as |

${G}_{1}o\&So$ | $1/4$ | ${s}_{1}$ |

${G}_{1}o\&\sim So$ | $1/4$ | ${s}_{2}$ |

$\sim {G}_{1}o\&So$ | $1/4$ | ${s}_{4}$ |

$\sim {G}_{1}o\&\sim So$ | $1/4$ | ${s}_{3}$ |

(i) | each ${h}_{1}^{\ast}$-sd to an ${h}_{2}^{\ast}$-sd and vice versa, | |

(ii) | each ${e}^{\ast}$-sd to itself, | |

(iii) | each ${h}_{1}^{\ast}{h}_{2}^{\ast}$-sd to itself, | |

(iv) | each ${e}^{\ast}{h}_{1}^{\ast}$-sd to a ${e}^{\ast}{h}_{2}^{\ast}$-sd and vice versa, | |

(v) | each ${e}^{\ast}{h}_{1}^{\ast}{h}_{2}^{\ast}$-sd to itself, and | |

(vi) | each ${\varphi}^{\ast}$-sd to a ${\varphi}^{\ast}$-sd. |

(i) | ${s}_{1}:{F}^{\ast}\overline{{G}^{\ast}}\phantom{\rule{4.pt}{0ex}}{B}_{1}^{\ast}{B}_{2}^{\ast}{B}_{3}^{\ast}$ | ${s}_{2}:\overline{{F}^{\ast}}{G}^{\ast}\phantom{\rule{4.pt}{0ex}}{B}_{1}^{\ast}{B}_{2}^{\ast}{B}_{3}^{\ast}$ | |

(ii) | ${s}_{3}:{F}^{\ast}{G}^{\ast}\phantom{\rule{4.pt}{0ex}}{B}_{1}^{\ast}{B}_{2}^{\ast}\overline{{B}_{3}^{\ast}}$ | ||

(iii) | ${s}_{4}:{F}^{\ast}{G}^{\ast}\phantom{\rule{4.pt}{0ex}}{B}_{1}^{\ast}\overline{{B}_{2}^{\ast}}{B}_{3}^{\ast}$ | ||

(iv) | ${s}_{5}:{F}^{\ast}\overline{{G}^{\ast}}\phantom{\rule{4.pt}{0ex}}{B}_{1}^{\ast}\overline{{B}_{2}^{\ast}}\overline{{B}_{3}^{\ast}}$ | ${s}_{6}:\overline{{F}^{\ast}}{G}^{\ast}\phantom{\rule{4.pt}{0ex}}{B}_{1}^{\ast}\overline{{B}_{2}^{\ast}}\overline{{B}_{3}^{\ast}}$ | |

(v) | ${s}_{7}:{F}^{\ast}{G}^{\ast}\phantom{\rule{4.pt}{0ex}}\overline{{B}_{1}^{\ast}}{B}_{2}^{\ast}{B}_{3}^{\ast}$ |

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Titelbaum, M.G.
Symmetry and Evidential Support. *Symmetry* **2011**, *3*, 680-698.
https://doi.org/10.3390/sym3030680

**AMA Style**

Titelbaum MG.
Symmetry and Evidential Support. *Symmetry*. 2011; 3(3):680-698.
https://doi.org/10.3390/sym3030680

**Chicago/Turabian Style**

Titelbaum, Michael G.
2011. "Symmetry and Evidential Support" *Symmetry* 3, no. 3: 680-698.
https://doi.org/10.3390/sym3030680