Multiple analogical proportions

Analogical proportions are statements of the form “a is to b as c is to d”, denoted a : b : : c : d, that may apply to any type of items a, b, c, d. Analogical proportions, as a building block for analogical reasoning, is then a tool of interest in artificial intelligence. Viewed as a relation between pairs ( a , b ) and ( c , d ), these proportions are supposed to obey three postulates: reflexivity, symmetry, and central permutation (i.e., b and c can be exchanged). The logical modeling of analogical proportions expresses that a and b differ in the same way as c and d, when the four items are represented by vectors encoding Boolean properties. When items are real numbers, numerical proportions – arithmetic and geometric proportions – can be considered as prototypical examples of analogical proportions. Taking inspiration of an old practice where numerical proportions were handled in a vectorial way and where sequences of numerical proportions of the form x 1 : x 2 : ⋯ : x n : : y 1 : y 2 : ⋯ : y n were in use, we emphasize a vectorial treatment of Boolean analogical proportions and we propose a Boolean logic counterpart to such sequences. This provides a linear algebra calculus of analogical inference and acknowledges the fact that analogical proportions should not be considered in isolation. Moreover, this also leads us to reconsider the postulates underlying analogical proportions (since central permutation makes no sense when n ⩾ 3) and then to formalize a weak form of analogical proportion which no longer obeys the central permutation postulate inherited from numerical proportions. But these weak proportions may still be combined in multiple weak analogical proportions.

Analogical Reasoning as an Inference Scheme

Despite its importance in various fields, analogical reasoning has not yet received a unified formal representation. Our contribution proposes a general scheme of inference that is compatible with different types of logic (deductive, probabilistic, non-monotonic). Firstly, analogical assessment precisely defines the similarity of two objects according to their properties, in a relative rather than absolute way. Secondly, analogical inference transfers a new property from one object to a similar one, thanks to an over-hypothesis linking two sets of properties. The belief strength in the conclusion is then directly related to the belief strength in this meta-hypothesis.

Analogical Reasoning as an Inference Scheme

Analogy plays an important role in science as well as in non-scientific domains such as taxonomy or learning. We make explicit the difference and complementarity between the concept of analogical statement, which merely states that two objects have a relevant similarity, and the concept of analogical inference, which relies on the former in order to draw a conclusion from some premises. For the first, we show that it is not possible to give an absolute definition of what it means for two objects to be analogous; a relative definition of analogy is introduced, only relevant from some point of view. For the second, we argue that it is necessary to introduce a background over-hypothesis relating two sets of properties; the belief strength of the conclusion is then directly related to the belief strength of the over-hypothesis. Moreover, we assert the syntactical identity between analogical inference and single case induction despite important pragmatic differences.

Analogical Inference in Gustav Theodor Fechner’s Inductive Metaphysics

Gustav Theodor Fechner was one of the main proponents of inductive metaphysics in the 19th century. The idea of inductive metaphysics is to use empirical sources and inductive forms of inference in metaphysics. Although this sounds like a research program which might well appeal to scientifically minded philosophers, some of Fechner’s metaphysical conclusions look very suspicious from a scientific viewpoint. For example, Fechner famously argues that the planets and stars are animated by a soul and that the same holds for the whole universe. In the present article it is shown that Fechner’s mystical cosmological views are based on a principle of analogy which is too permissive and that as a result of that his central argument to the conclusion that the earth is animated by a soul is not correct. As the article argues, the principle Fechner bases his inference on has to be supplemented by a causal condition along the lines of Mary Hesse’s account of analogical inferences, a condition Fechner at least implicitly recognizes but fails to meet.

Rereading Jean Lave 30 years on: Analogy and transfer-in-pieces

This paper explores and evaluates some of the criticisms of a cognitive approach to learning leveled by Lave in Cognition in Practice (1988). The paper progresses by initially identifying learning transfer as the focal topic of Lave’s work. Two lines of criticism are identified, one called practice-based and one called concept-based. Through a renewed analysis of one of the empirical cases reported by Lave, I set out to show how this empirical material did not lend itself to Lave’s conclusions, in so far as these conclusions rejected central tenets of cognitive theories of learning. Concerning the concept-based line of criticism, I show how the phenomenon of analogically related uses of words plays a role in defusing one version of the concept-based line of criticism. Concerning the practice-based criticism, I show how an oft-overlooked form of analogical inference, coupled with elements of a cognitive, transfer-in-pieces approach, offers a better account of Lave’s case than the one she offered. The paper concludes with reflections on the lasting significance of Lave’s work.

Modern Analogue Approach Applied to High-Resolution Varved Sediments—A Synthesis for Lake Montcortès (Central Pyrenees)

In Quaternary paleosciences, the rationale behind analogical inference presupposes that former processes can be explained by causes operating now, although their intensity and rates can vary through time. In this paper we synthesised the results of different modern analogue studies performed in a varved lake. We discuss their potential value to obtain best results from high resolution past records. Different biogeochemical contemporary processes revealed seasonality and year-to-year variability, e.g., calcite precipitation, lake oxygenation, production and deposition of pollen and phytoplankton growth. Fingerprints of the first two of these processes were clearly evidenced in the varve-sublayers and allow understanding related to past events. Pollen studies suggested the possibility of identifying and characterizing seasonal layers even in the absence of varves. Marker pigments in the water column were tightly associated with phytoplankton groups living today; most of them were identified in the sediment record as well. We observed that 50% of these marker pigments were destroyed between deposition and permanent burying. In another study, seasonality in the production/distribution of branched glycerol dialkyl glycerol tetraethers (brGDGTs) and derived temperature estimates were investigated in catchment soils and particles settling in the lake. The signatures of brGDGTs in depositional environments mainly were representative of stable conditions of soils in the catchment that last over decades; no brGDGTs seemed to be produced within the lake. The main contribution of this review is to show the advantages and limitations of a multiproxy modern-analogue approach in Lake Montcortès as a case study and proposing new working hypotheses for future research.

Confirmation Based on Analogical Inference: Bayes Meets Jeffrey

Certain hypotheses cannot be directly confirmed for theoretical, practical, or moral reasons. For some of these hypotheses, however, there might be a workaround: confirmation based on analogical reasoning. In this paper we take up Dardashti, Hartmann, Thébault, and Winsberg’s (2019) idea of analyzing confirmation based on analogical inference Bayesian style. We identify three types of confirmation by analogy and show that Dardashti et al.’s approach can cover two of them. We then highlight possible problems with their model as a general approach to analogical inference and argue that these problems can be avoided by supplementing Bayesian update with Jeffrey conditionalization.

On the Usefulness of the Modern Analog Approach to Interpret High Resolution Records: The Case of Varved Sediments from Lake Montcortès (Central Pyrenees)

In the Quaternary paleosciences, the rationale behind analogical inference presupposes that former natural changes can be explained by causes operating now, although their intensity and rates can vary through time. In this paper we synthesise synthetize the results of different modern analog studies and discuss their value to obtain the best inferences from high resolution past records. This synthesis is based on the following: 1) The monthly monitoring of calcite precipitation reveals a strong connection with primary producers and between-years variability; this precipitation produces a seasonal signal with imprint on varve formation. 2) Clear pollen sedimentation peaks occur in spring/summer and fall/winter that coincide with temperature, precipitation, relative humidity and winds; this pattern converges with the two-layer coupled varves representing the same seasonality. 3) We assess the lake’s contemporary oxygenation dynamics over a three- year period; a combination of sedimentary REDOX proxies revealed different scenarios of oxic/anoxic shifts since 1500 CE. 4) We investigate presence of seasonality in the production/distribution of glycerol dialkyl glycerol tetraethers and derived temperature estimates in soils and particulate matter. Branched glycerol dialkyl glycerol tetraethers signatures and some derived temperature estimates proxies appear to mainly depend on the non-seasonal shifts in soil properties. 5) Currently we examine relationships and similarities between extant phytoplankton and derived pigments in water and traps, and their correspondence with subfossil pigments; some preliminary results are presented here.Keywords: high resolution, endogenic varves, calcite precipitation, pollen sedimentation, meromixis, freshwater glycerol dialkyl tetraether, subfossil pigments, long-term ecology.