Hasty Generalizers and Hybrid AbducersHasty Generalizers and Hybrid Abducers

External Semiotic Anchors and Multimodal RepresentationsExternal Semiotic Anchors and Multimodal Representations

Department of Philosophy and Computational Philosophy Laboratory, University of Pavia, Italy

Department of Philosophy, Sun Yat-sen University, Canton, China

Workshop on Abduction and Induction in AI and Scientific Modeling (AIAI06), Workshop on Abduction and Induction in AI and Scientific Modeling (AIAI06),

ECAI2006, Riva del Garda, Italy, August 29, 2006ECAI2006, Riva del Garda, Italy, August 29, 2006

Integrating Induction and Abduction 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

Van Benthem (2000) on Van Benthem (2000) on AbductionAbduction and and Induction Induction

• 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 “DeductionDeduction” either)

Induction in Induction in Organic Agents 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

Organic InductionOrganic InductionHuman beings mess thing up Human beings mess thing up above the simplest levels of above the simplest levels of complexity. This is particularly complexity. This is particularly true of true of inductive inferencesinductive inferences: it : it seems there is a tendency for seems there is a tendency for hasty and unfounded hasty and unfounded generalizations.generalizations.

But not every generalization But not every generalization from a single case is bad from a single case is bad (that is a fallacy). Hasty (that is a fallacy). Hasty generalization is a generalization is a prudent prudent strategystrategy, especially when , especially when risks are high: survival skills risks are high: survival skills are sometimes exercised are sometimes exercised successfully but not successfully but not rationallyrationally. . We have a cognitive error but We have a cognitive error but not a strategic error. This fact not a strategic error. This fact always stimulated the always stimulated the theorists to say something theorists to say something helpful about the problem of helpful about the problem of induction – MILL - (and on induction – MILL - (and on abduction - PEIRCE) both abduction - PEIRCE) both fallacious but strongfallacious but strong..

The Human agent is The Human agent is genetically and culturally genetically and culturally endowed with a kind of endowed with a kind of rational survival kit rational survival kit (Woods, 2004) also (Woods, 2004) also containing some containing some strategicstrategic uses of fallacies.uses of fallacies.

For example:For example:

Hasty generalizationHasty generalization

1.1. Cynthia is a bad driver. Cynthia is a bad driver.

2.2. Women are bad drivers.Women are bad drivers.

It is sometimes worse not to It is sometimes worse not to generalize in this way.generalize in this way.

• 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.

Mimetic Induction – Mimetic Mimetic Induction – Mimetic Abduction Abduction

Ideal Agents 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.

Agent-Based reasoning and Agent-based Logic

• We will exploit the framework of agent-based reasoning as illustrated by Gabbay and Woods (Woods 2004; Gabbay, Woods 2005), so adopting the perspective of a cognitive agent.

• In the agent-based reasoning above (Gabbay and Woods, 2001) logic can be considered a formalization of what is done by a cognitive agent: logic is agent-based.

Agent-Based reasoningAgent-Based reasoning

Agent Based Reasoning consist in describing and analyzing the Agent Based Reasoning consist in describing and analyzing the

reasoning occurring in problem solving situations where the agent reasoning occurring in problem solving situations where the agent

access to cognitive resources encounters limitations such asaccess to cognitive resources encounters limitations such as

1.1. Bounded Information Bounded Information

2.2. Lack of Time Lack of Time

3.3. Limited Computational Capacity. Limited Computational Capacity.

Actually Happens Rule: to see what agent should do we should Actually Happens Rule: to see what agent should do we should

have to look first to what they actually do. Then, if there is have to look first to what they actually do. Then, if there is

particular reason to do so, we would have to repair the accountparticular reason to do so, we would have to repair the account

(Woods, 2005).(Woods, 2005).

Agent-Based logic and the framework of Non-Monotonic Logic

• Classical logic as a complete system• Deduction and modus ponens (the “truth

preserving feature”)• Non Monotonic Logic: new information can

compel us to revise previous generated hypotheses (Decision-Making Process and the “casual truth preserving feature”)

• Not-only-deductive reasoning

Agent-based reasoning and Agent-based reasoning and Actually happens ruleActually happens rule

This rule is a particular attractive assumption about human cognitive behaviour mainly for two reasons:

• beings like us make a lot of errors

• cognition is something that we are actually very good at (strategic rationality and cognitive economies)

Fallacies IFallacies I• It is in this framework that fallacious ways of

reasoning are seen as widespread in human beings’ cognitive performances, and nevertheless they can in some cases be redefined and considered as good ways of reasoning.

• A fallacy is a pattern of poor reasoning which appear to be a pattern of good reasoning ( Hansen, 2002).

Fallacies IIFallacies II

Formal fallacy Informal fallacy

Deductive argument which has an invalid

form (not Truth Preserving Reasoning)

(expl. Affirming the Consequent)

Any other invalid mode of reasoning

whose failing is not in the shape of the

argument

(expl. Ad hominem, Hasty Generalization,…)

The Toddler and the Stove

• A sample of Hasty Generalization•X% of all observed A's are B''s: (The stove touched burns)

•Therefore X% of all A's are Bs: (All the stoves burn)

THE STOVE THE STOVE TOUCHED BURNSTOUCHED BURNS

HASTY HASTY GENERALIZATIONGENERALIZATION

ALL THE STOVES ALL THE STOVES

BURNBURN

FALLACIES IFALLACIES I(LOGICAL PERSPECTIVE)(LOGICAL PERSPECTIVE)

FORMALFORMAL

INFORMALINFORMAL

BAD REASONIGSBAD REASONIGS

DEDUCTIVE INVALID DEDUCTIVE INVALID ARGUMENTS (ARGUMENTS (NOT TRUTH NOT TRUTH PRESERVINGPRESERVING FEATURES)FEATURES)

INDUCTIVE INVALID INDUCTIVE INVALID ARGUMENTSARGUMENTS

FALLACIES IIFALLACIES II(AGENT-BASED (AGENT-BASED PERSPECTIVE)PERSPECTIVE)

LIMITED COGNITIVE LIMITED COGNITIVE SETTINGSETTING

ACTUALLY HAPPENS ACTUALLY HAPPENS RULE RULE

FALLACIES ARE FALLACIES ARE “BETTER THAN “BETTER THAN NOTHING” NOTHING” (RATIONAL (RATIONAL SURVIVAL KIT) SURVIVAL KIT)

COGNITIVE ECONOMIES COGNITIVE ECONOMIES CASUAL TRUTH PRESERVING CASUAL TRUTH PRESERVING FEATURE OF FALLACIESFEATURE OF FALLACIES

GOOD EPISTEMIC ACTIONS GOOD EPISTEMIC ACTIONS IN PRESENCE OF “BAD” IN PRESENCE OF “BAD” REASONINGSREASONINGS

ABDUCTION AS A FALLACIOUS ABDUCTION AS A FALLACIOUS ARGUMENTARGUMENT

BEING-LIKE-US AS HASTY BEING-LIKE-US AS HASTY GENERALIZERSGENERALIZERS

Abduction as an example of fallacy considered in Agent-Based

Reasoning

AbductionAffirming the Consequent

Abduction that only generate plausible

hypotheses (selective or creative)

Abduction considered as “Inference to

the Best Explanation.”

• what is abduction?• theoretical abduction (sentential, model-based)

• manipulative abduction (mathematical diagrams, construals)

creative, selectivecreative, selective

scientific scientific discoverydiscovery

diagnosisdiagnosis

• what is abduction?• theoretical abduction (sentential, model-based)

• manipulative abduction (mathematical diagrams, construals)

creative, selectivecreative, selective

scientific scientific discoverydiscovery

diagnosisdiagnosis

Theoretical AbductionTheoretical Abduction

SENTENTIALSENTENTIAL

MODEL-BASEDMODEL-BASED

Theoretical AbductionTheoretical Abduction

SENTENTIALSENTENTIAL

MODEL-BASEDMODEL-BASED

Peirce stated that all thinking is in signs, and signs can be icons, indices, or symbols. Moreover, all inference is 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 is model-based. Of course model-based reasoning acquires its peculiar creative relevance when embedded in abductive processes

•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

Model-based Model-based cognitioncognition

Manipulative AbductionManipulative Abduction

Mathematical Diagrams Mathematical Diagrams (also Model-Based)(also Model-Based)

ConstrualsConstruals

Thinking through doingThinking through doing

manipulative abduction nicely introduces to

hypothesis generation in 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.

Manipulative AbductionManipulative Abduction

ConstrualsConstruals

Thinking through doingThinking through doing

Samples, Induction, AbductionSamples, Induction, Abduction

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.

““If we do not think of inductive If we do not think of inductive generalizations as abductions generalizations as abductions we are at a loss to explain why we are at a loss to explain why such inference is made such inference is made stronger and more warranted, stronger and more warranted, if in connecting data we make a if in connecting data we make a systematic search for counter-systematic search for counter-instances and cannot find any, instances and cannot find any, than it would be just take the than it would be just take the observation passively. Why is observation passively. Why is the generalization made the generalization made stronger by making an effort to stronger by making an effort to examine a wide variety of types examine a wide variety of types of A’s? The answer is that it is of A’s? The answer is that it is made stronger because the made stronger because the failure of the active search of failure of the active search of counter-instances tend to rule counter-instances tend to rule out various hypotheses about out various hypotheses about ways in which the sample might ways in which the sample might be biased, that is, is be biased, that is, is strengthens the abductive strengthens the abductive conclusion by ruling out conclusion by ruling out alternative explanations for the alternative explanations for the observed frequency (Josephson observed frequency (Josephson 2000)”2000)”

• Samples and Manipulative Abduction

• Construals Manipulative 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.

““If we think that a sampling If we think that a sampling method is fair and unbiased, method is fair and unbiased, then straight generalization then straight generalization gives the best explanation of gives the best explanation of the sample frequencies. But if the sample frequencies. But if the size is small, alternative the size is small, alternative explanations, where the explanations, where the frequencies differ, may still be frequencies differ, may still be plausible. These alternative plausible. These alternative explanations become less and explanations become less and less plausible as the sample size less plausible as the sample size grows, because the sample grows, because the sample being unrepresentative due to being unrepresentative due to chance becomes more and more chance becomes more and more improbable. Thus viewing improbable. Thus viewing inductive generalization as inductive generalization as abductions show why sample abductions show why sample size is important. Again, we see size is important. Again, we see that analyzing inductive that analyzing inductive generalizations as abductions generalizations as abductions shows us how to evaluate the shows us how to evaluate the strengths of these inferences strengths of these inferences (Josephson, p. 42).”(Josephson, p. 42).”

LOGICAL IDEAL ABDUCTIVE and INDUCTIVE SYSTEMS LOGICAL IDEAL ABDUCTIVE and INDUCTIVE SYSTEMS

- - symbolicsymbolic: they activate and “anchor” : they activate and “anchor” meaningsmeanings in in material material communicative andcommunicative and intersubjective intersubjective mediatorsmediators in in the framework of the phylogenetic, ontogenetic, and the framework of the phylogenetic, ontogenetic, and cultural reality of the human being and its language. They cultural reality of the human being and its language. They originated in embodied cognition and gestures we share originated in embodied cognition and gestures we share with some mammals but also non mammals animals (cf. with some mammals but also non mammals animals (cf. monkey knots and pigeon categorization, Grialou, Longo, monkey knots and pigeon categorization, Grialou, Longo, and Okada, 2005);and Okada, 2005);

-- abstract abstract: they are based on a : they are based on a maximalmaximal independence independence regarding sensory modality; strongly stabilize experience regarding sensory modality; strongly stabilize experience and common categorization. The maximality is especially and common categorization. The maximality is especially important: it refers to their practical and historical important: it refers to their practical and historical invariance and stability;invariance and stability;

-rigorousrigorous: the rigor of proof is reached through a difficult : the rigor of proof is reached through a difficult practical experience. For instance, in the case of practical experience. For instance, in the case of mathematics, as themathematics, as the maximal maximal place for convincing place for convincing reasoning. Rigor lies in the stability of proofs and in the reasoning. Rigor lies in the stability of proofs and in the fact they can be iterated.fact they can be iterated.

Mathematics is the best example of maximal stability and Mathematics is the best example of maximal stability and conceptual invariance. conceptual invariance.

• 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.

• 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).

• 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.

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 extensions of the original theory T.

controversies on IAI are of course open and alive

Thanks

lorenzo.magnani@unipv.it