Inductive Reasoning • How observations and beliefs support other beliefs Induction, Analogy, Metaphor • In some ways, opposite of deductive reasoning & Blending P Æ Q Q Therefore: P is more likely • Inherently Uncertain • Adds New Knowledge (unlike deduction) – Everyday life – Scientific reasoning Abduction or Specific Induction Abduction Schema • Because conditionals typically express causal •Q reasoning, we often explain q via p (modus ponens) • If P then Q – Blown fuses prevent electrical appliances from • Therefore: P working – Hair dryer has blown fuse. • Not logically valid, but useful! • But, often need to explain q – Hair dryer not working. – Blown fuse? • Abduction – explanation for an event via a causal relationship General Induction or Explanation-Based Learning Inductive Generalization • Explain new experiences based on • Observe instances have generalizations property F and infer all other members of the • Use generalizations in later reasoning class have the same property F(a) F(b) F(c) believe P likely QPexplained by ÆQ Therefore: (Ax)F(x) • Induction can be descriptive or explanatory 1 Constraints on Induction Factors that limit Induction • Induction is inherently fallible • Expectations • For any given set of observations, an infinite – Robins have red number of inductive breasts until the year generalizations follow 7693 and then green • All zebras will have stripes until the year 3000 but breasts thereafter. thereafter become spotted. – Robins have either red • Cognitive processes must breasts or green constrain the induction process – Similarity, Availability, Framing breasts Constraints on Induction Constraints on Induction • Temporal Contiguity • Similarity – Pavlov’s Dogs – Similarity depends on • Availability of perception, entrenched Observations knowledge, contextual relevance, frames, etc. Constraints on Induction Constraints on Induction • Frequency • Framing – Golden retrievers are – Constrains domains too friendly to be good hypotheses come from watchdogs – Prevents consideration of implausible hypotheses – Can be biologically based 2 Kids Use Categories to Category-Based Induction Constrain Induction • Categories used to constrain induction • My Cadillac Seville has a 6-cylinder engine – Sevilles? – Cadillacs? – Sedans? –Cars? – Vehicles? Factors Influencing Coverage v. Typicality Category-Based Induction • Coverage more important than typicality • Typicality for strength of induction Dogs have a LAA. Whales have a LAA. Dogs have a LAA. Dogs have a LAA. Mammals have a LAA. Mammals have a LAA. Cats have a LAA. Whales have a LAA. • Coverage Therefore: Mammals have a LAA. – Average similarity btw exemplar in premise (dog/whale) and category in conclusion (mammal) Inductive Strength, Similarity, Coverage and Analogy • People less willing to generalize from an • Object 1 has properties A and B exemplar to a more abstract category than • Object 2 has properties A, B, and also C. to a less abstract category. • Therefore, it is likely that object 1 also has Chimps have a LAA. Chimps have a LAA. property C. Primates have a LAA. Mammals have a LAA. • Analogical Reasoning – Understanding new situations by projecting knowledge from previous situations of a similar sort 3 Analogical Reasoning Analogical Reasoning Source Target Source Target person bird person bird chair tree chair ? house nest backyard tree • Source (Base) Analog – domain you have a lot of • Heart of Analogy is Establishment of Mappings knowledge about – Mappings – correspondences between domains • Target Analog – domain you are trying to draw inferences about • Neal’s 2nd set of mappings more complete & • Personification – using concepts relevant to people to coherent reason about other domains Constraints on Analogical Structure Mapping Thinking • Similarity • Overall Similarity – Similarity of both attributes and relations • Identification of Consistent Structural • Relational/Structural Similarity Parallels – Similarity of relations • Purpose • Attributes X is red X is large • Relations X collides with Y X is larger than Y Steps in Structure Mapping Analogy & Problem Solving (1) Set up mappings • Gick & Holyoak between domains • Duncker’s Tumor (2) Discard attributes Problem -- hot, massive • Impenetrable Fortress (3) Map relations from source to target • 10% solve problem w/no --more-massive-than hints --revolves-around • 75% solve problem when (4) Observe systematicity given Impenetrable -- discard isolated relations Fortress problem and hint -- keep relations governed by to apply it higher-order relations 4 Analogical Problem Solving Correspondences btw Problems • Construct Representation of Source & • Military Problem • Radiation Problem • Initial State Goal –use army • Initial State Goal – use rays to Target to capture fortress destroy tumor • Resources – Sufficiently large • Resources – sufficiently • Select Source as Potential Analog army powerful rays • Operators – Divide army, • Operators – reduce ray • Construct Mapping move army, attack w/army intensity, move ray source, • Constraints – Unable to send administer rays • Extend Mappping entire army along one road • Constraints – unable to safely administer high-intensity rays • Solution – Send small groups from one direction safely along multiple roads • Solution –administer low- simultaneously intensity rays from multiple • Outcome – Fortress captured directions simultaneously by army • Outcome – tumor destroyed by rays Correspondences btw Problems • Convergence Schema • Initial State Goal –use force to overcome a central target • Resources – sufficiently great force • Operators – reduce force intensity, move source of force, apply force • Constraints – unable to apply full force along one path safely • Solution – apply weak forces along multiple paths simultaneously • Outcome – central target overcome by force 5.