The Reality of Logic David Davenport Computer Eng. Dept., Bilkent University, Ankara Turkey.

Outline Background –Logic & its Problems –Fuzzy & other deviant logics –Truth A Cognitive “solution” The new Reality (non) standard disclaimer

Logic Deductive vs. Inductive Classical Logic The study of valid arguments orThe study of consistent beliefs “ … is a paragon of clarity, elegance, and efficiency.” - Quine Founded on three “laws,” –Identity A=A –Excluded Middle A or ~A{ only two possible truth values } –Contradiction ~(A and ~A){ nothing can be true & false}

Logic Declarative sentences“Is it true that x” –Contrast with questions, commands, etc. –Grammatical substitution!{ feels right to native speaker! } Arguments –Statement & supporting reasons –Conclusion given premises{ deduce or infer } Valid Argument –no possible situation where premises are all true but the conclusion is not.{ entailment } Dependent on meaning & truth!

Logic - problems? Reference “The King of France is bald” Borderline cases “Ted Bartlett is fat” Others... Monotonicity, semantics, temporal, modal, etc.

Fuzzy Logic Roots in Russell’s Vague Logic & Jan Lukasiewicz’s multivalued logic Denies Law of Excluded Middle (A or ~A) t(s) = 0 or t(S) = 1 vs. 0 <= t(S) <= 1 “Snow is White” “Grass is green” “grass is 85% green”

Fuzzy Logic - examples

Truth The common notion –reality, historical, mathematical, logical Existing Theories of truth –Correspondence Theory –Coherence Theory –Pragmatic Theory –Others (Deflationary, Semantic, Appraisal, etc.)

Computational Systems Modeling the world Purpose is “prediction” States of model map to states of the world Rely on causality Multiple models But no mind, no model! world model mind world model

Computation and Cognition Cognitive agents –satisfy needs in complex world –are computational systems Mental Models, “connect” to the world –(causal links, accurate reflection, corresponding states, etc.) Linguistic utterances mind world utterances

Mind/representation must be logical! Traditionally, “if a & b & c then z” –but, not very realistic. Alternative, Inscriptors “if z then a & b & c” –naturally fuzzy, predictive, but require “not” –models scientific & invalid reasoning Representation a b c z

Language Learn words ostensibly by verbal def n. Sentences abstract grammar utter word at time. Purpose communication manipulation! Situation in which word “CAT” is heard & cat is seen audio senses visual senses “CAT”

Meaningful Utterances W U mind W U U ? Making sense of utterances Selecting the relevant model True, False & Unknown

Some Other Possible Relations between Utterance, Mind & World W U1U1 mind U2U2 U3U3 W1W1 U mind 1 mind 2 W2W2 W U1U1 mind 1 mind 2 U2U2 W U mind 1 mind 2 U W1W1 mind W2W2 W3W3 (a) (e) (c) (b) (d)

And Truth… Matching is coherence Correspondence –but of utterance & mental model Mind-dependent notion of truth –shared language, environment & senses World Mental Model Utterance Truth Map or Model Truth

Utterances about truth W U mind W U (a) (b) “Snow is white is false” “It is true that snow is white” “Snow is white”

The Liar Paradox “This sentence is false” Paradox found U liar mind U1U1 U2U2

Paradox Lost Philosophical whirlpool - stay clear! U mind U1U1 U2U2 U1U1 U2U2

Summary Mind as computational system making predictions to guide actions to satisfy needs. must (of necessity) be inherently logical Need to store/represent info. inscriptor formulation, defused borderlines! Utterances meaning truth & utterances about truth defused liar paradox!

Some “Conclusions” Reality does not come pre-cut and labeled. Truth is a relation btw utterance & mind. Representation is predictability.

Thank you.