Demonstrative and non-demonstrative reasoning by analogy

Emiliano Ippoliti

Analogy and analogical reasoning have more and more become an important subject of inquiry in logic and philosophy of science, especially in virtue of its fruitfulness and variety: in fact analogy «may occur in many contexts, serve many purposes, and take on many forms»1. Moreover, analogy compels us to consider material aspects and dependent-on-domain kinds of reasoning that are at the basis of the lack of well- established and accepted theory of analogy: a standard theory of analogy, in the sense of a theory as classical logic, is therefore an impossible target. However, a detailed discussion of these aspects is not the aim of this paper and I will focus only on a small set of questions related to analogy and analogical reasoning, namely:

1. The problem of analogy and its duplicity;
2. The role of analogy in demonstrative reasoning;
3. The role of analogy in non-demonstrative reasoning;
4. The limits of analogy;
5. The convergence, particularly in multiple analogical reasoning, of these two apparently distinct aspects and its philosophical and methodological consequences;

§ 1 The problem of analogy and its duplicity: the controversial nature of analogical reasoning

One of the most interesting aspects of analogical reasoning is its intrinsic duplicity: in fact analogy belongs to, and takes part in, both demonstrative and non-demonstrative processes. That is, it can be used respectively as a means to prove and justify knowledge (e.g. in automated theorem proving or in confirmation patterns of plausible inference), and as a means to obtain new knowledge, (i.e. in heuristics and in the generation of conjectures and hypotheses). As it is well known, these two kinds of reasoning are traditionally (e.g. by the logical empiricist philosophy of science) treated as distinct and belonging to two completely independent contexts, namely the context of justification on one hand and the context of discovery on the other: justification is the phase in which hypotheses are confirmed or rejected, discovery is the phase in which the scientific hypotheses are generated. Moreover, analogical reasoning is widely considered not only as one of the main tools in problem-solving activity, but also as an ubiquitous, highly controversial and complex concept: in fact «it can be said that the analogy is, as the tongue of Aesop, at the same time the best and the worse thing»2. Its controversial nature is not accidental and relies on two fundamental properties of analogical reasoning:

1 Shelley 2003, 1 2 Dieudonné 1981, 257

1

a) Ampliativity (i.e. the capability to really extend our knowledge by reaching conclusions which are not included in the premises).

b) Non-monotonicity (i.e. the sensitivity to the entry of new information and premises, which are able to modify the previously obtained conclusions). As a consequence, analogy is an intrinsic time-sensitive kind of inference: it strictly depends on the background knowledge existing at a given time.

Moreover, the very definition of analogy is problematic. The existing orthodox literature agrees in considering analogy as a kind of comparison, which, in short, allows to transfer a known property/information from a sufficiently known source domain S to an at least partially unknown target domain T, by a relation of mapping μ of objects, relations and properties from S into T. In particular, it is possible to distinguish two main conceptions of analogy, namely the inductive and the structural.

Analogy as induction (inductive conception) Analogy is a form of induction (induction on attributes or properties), in virtue of which a single observation is used as a basis for some conclusion. In this sense analogy is a kind of generalization (e.g. Keynes3), which is obtained by a conjunction of material resemblances between domains.

Analogy as shared structure (structural conception) Analogy is a mapping or alignment of «hierarchically structured, causal relationships shared between source and target analogs»4. That is, analogy is ideally an isomorphism of two domains (e.g. Hempel’s nomic isomorphism between Ohm’s law and Poiseulle’s law).

The two conceptions agree on the relevance of overall similarity across domains; nevertheless, the structural conception is based on the mapping between relations (and not on attributes as the inductive conception) and on the systematicity principle, which claims that an analogy is good if it contains mapping between richly structured higher order relations (which are in general the causal ones). Therefore, both conceptions try to specify «a rationale for analogical reasoning»5, that is, to offer an answer to LPA, the Logical Problem of Analogy.
LPA can be formulated as the problem to «find a criterion which, if satisfied by any par

…(Full text truncated)…