Inductive Generalizations and Manipulative Abduction

1. Integrating Abduction and Induction in AI, Edinburgh, UK, July 29, 2005 Inductive Generalizations and Manipulative Abduction Lorenzo Magnani Department of Philosophy and Computational Philosophy Laboratory, University of Pavia, Italy Department of Philosophy, Sun Yat-sen University, Canton, China

2. Integrating Induction and Abduction • Induction in Organic Agents • Mimetic Inductions • Ideal and Computational Inductive Agents • Mimetic Abductions • Ideal and Computational Abductive Agents • Sentential, Model-Based and Manipulative Abduction • A Cognitive Integration: Samples, Induction, and Abduction

3. Organic Induction Human beings mess thing up above the simplest levels of complexity. This is particularly true of inductive inferences: it seems there is a tendency for hasty and unfounded generalizations. But not every generalization from a single case is bad (that is a fallacy). Hasty generalization is a prudent strategy, especially when risks are high: survival skills are sometimes exercised successfully but not rationally. We have a cognitive error but not a strategic error. This fact always stimulated the theorists to say something helpful about the problem of induction – MILL - (and on abduction - PEIRCE) both fallacious but strong. • The Human agent is genetically and culturally endowed with a kind of rational survival kit (Woods, 2004) also containing some strategic uses of fallacies. • For example: • Hasty generalization • Cynthia is a bad driver. • Women are bad drivers. • It is sometimes worse not to generalize in this way. Van Benthem (2000) on Abduction and Induction • The kid on touching the element on his mother’s kitchen stove learns in one case never to do that again (primitive induction) • This is not an offense to inductive reasoning. • MILL provides “Methods” for Induction • PEIRCE integrates Abduction and Induction through the syllogistic framework where the two non-deductive inferences can be clearly distinguished. • Indeed, it is not easy to give a crystal-clear definition of them, either independently or in their inter-relationship. (Of course, this is not easy for “Deduction” either) Induction in Organic Agents • Hasty Generalization, Secundum Quid, Biased Statistics, Other Fallacies • Strategic versus Rational thinking (conscious but often tacit) • Mill says that institutions rather than individuals are the embodiment of inductive logics

4. Mimetic Induction – Mimetic Abduction Ideal Agents • Kid’s performance is a strategic success and a cognitive failure. • Human beings are hardwired for survival and for truth alike so best strategies can be built and made explicit, through self-correction and re-consideration (for example Mill’s methods). • Mill’s methods for induction, Peirce’s syllogistic and inferential models for abduction Inductive and Abductive Agents • Ideal Logical Inductive and Abductive Agents • Ideal Computational Inductive, Abductive, and Hybrid Agents • Merely successful strategies are replaced with successful strategies that also tell the more precise truth about things.

5. creative, selective • what is abduction? • theoretical abduction (sentential, model-based) • manipulative abduction (mathematical diagrams, construals) scientific discovery diagnosis

6. creative, selective • what is abduction? • theoretical abduction (sentential, model-based) • manipulative abduction (mathematical diagrams, construals) scientific discovery diagnosis

7. SENTENTIAL Theoretical Abduction MODEL-BASED

8. Model-based cognition • Simulative reasoning • Analogy • Visual-iconic reasoning • Spatial thinking • Thought experiment • Perception, sense activities • Visual imagery • Deductive reasoning(Beth’s • method of semantic tableaux, • Girard’s “geometry” of proofs, etc.) • Emotion SENTENTIAL Peirce stated that all thinking is in signs, and signs can be icons, indices, or symbols. Moreover, allinferenceis a form of sign activity, where the word sign includes “feeling, image, conception, and other representation” (CP 5.283), and, in Kantian words, all synthetic forms of cognition. That is, a considerable part of the thinking activity ismodel-based. Of course model-based reasoning acquires its peculiar creative relevance when embedded in abductive processes Theoretical Abduction MODEL-BASED

9. Mathematical Diagrams (also Model-Based) manipulative abductionnicely introduces to hypothesis generationin active, distributed, and embodied cognition The activity of “thinking through doing” is made possible not simply by mediating cognitive artifacts and tools, but by active process of testing and manipulation. Thinking through doing Construals Manipulative Abduction

10. Thinking through doing Construals Manipulative Abduction

11. Samples, Induction, Abduction “If we think that a sampling method is fair and unbiased, then straight generalization gives the best explanation of the sample frequencies. But if the size is small, alternative explanations, where the frequencies differ, may still be plausible. These alternative explanations become less and less plausible as the sample size grows, because the sample being unrepresentative due to chance becomes more and more improbable. Thus viewing inductive generalization as abductions show why sample size is important. Again, we see that analyzing inductive generalizations as abductions shows us how to evaluate the strengths of these inferences (Josephson, p. 42).” “If we do not think of inductive generalizations as abductions we are at a loss to explain why such inference is made stronger and more warranted, if in connecting data we make a systematic search for counter-instances and cannot find any, than it would be just take the observation passively. Why is the generalization made stronger by making an effort to examine a wide variety of types of A’s? The answer is that it is made stronger because the failure of the active search of counter-instances tend to rule out various hypotheses about ways in which the sample might be biased, that is, is strengthens the abductive conclusion by ruling out alternative explanations for the observed frequency (Josephson 2000)” Manipulative abduction can be considered a kind of basis for further meaningful inductive generalizations. For example different construals can give rise to different inductive generalizations. If “an inductive generalization is an inference that goes from the characteristics of some observed samples of individuals to a conclusion about the distribution of those characteristics in some larger populations” (Josephson) what characterizes the sample as “representative” is its effect (sample frequency) by reference to part of its cause (populations frequency): this should be considered a conclusion about its cause. • Samples and Manipulative Abduction • ConstrualsManipulative abduction is the correct way for describing the features of what are called ``smart inductive generalizations'', as contrasted to the trivial ones. For example, in science construals can shed light on this process of sample ``production'' and ``appraisal'': through construals, manipulative creative abduction generates abstract hypotheses but in the meantime can originate possible bases for further meaningful inductive generalizations through the identification of new samples (or of new features of already available sample, for instance in terms of the detection of relevant circumstances). Different generated construals can give rise to different plausible inductive generalizations.

12. LOGICAL IDEAL ABDUCTIVE and INDUCTIVE SYSTEMS • - symbolic: they activate and “anchor” meanings in material communicative andintersubjective mediators in the framework of the phylogenetic, ontogenetic, and cultural reality of the human being and its language. They originated in embodied cognition and gestures we share with some mammals but also non mammals animals (cf. monkey knots and pigeon categorization, Grialou, Longo, and Okada, 2005); • - abstract: they are based on a maximal independence regarding sensory modality; strongly stabilize experience and common categorization. The maximality is especially important: it refers to their practical and historical invariance and stability; • rigorous: the rigor of proof is reached through a difficult practical experience. For instance, in the case of mathematics, as the maximal place for convincing reasoning. Rigor lies in the stability of proofs and in the fact they can be iterated. • Mathematics is the best example of maximal stability and conceptual invariance. • Flach and Kakas (2000). A useful perspective on integration of abduction and induction: • explanation (hypothesis does not refer to observables – selective abduction [but abduction creates new hypotheses too]) • generalization – genuinely new (hypothesis can entail additional observable information on unobserved individual, extending the theory T) • Imagine we have a new abductive theory T’ = T H constructed by induction: an inductive extension of a theory can be viewed as set of abductive estensions of the original theory T. • controversies on IAI are of course open and alive • cf. the cognitive analysis of the origin of the mathematical continuous line as a pre-conceptual invariant of three cognitive practices (Theissier, 2005), and of the numeric line (Châtelet, 1993; Dehaene, 1997; Butterworth, 1999). • logical systems are in turn sets of proof invariants, sets of structures that are preserved from one proof to another or which are preserved by proof transformations. They are the result of a distilled praxis, the praxis of proof: it is made of maximally stable regularities. • MAXIMIZATION OF MEMORYLESSNESS characterizes demonstrative reasoning. Its properties do not yield information about the past, contrarily for instance to the narrative and not logical descriptions of non-demonstrative processes, which often involve “historical”, “contextual”, and “heuristic” memories.

13. Thanks lorenzo.magnani@unipv.it