The Reality of LogicThe Reality of Logic

David DavenportComputer Eng. Dept., Bilkent University, Ankara 06533 - Turkey.email: david@bilkent.edu.tr

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 argumentsor The 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 inRussell’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

Fuzzy Logic - examples

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 defn.

• Sentences• abstract grammar

• utter word at time.

• Purpose• communication

• manipulation!

Situation in which word “CAT” is heard & cat is seen

audiosenses

visualsenses

“CAT”

Meaningful Utterances

W Umind

W U

mind

U

mind

?

• Making sense of utterances

• Selecting therelevant model

• True, False & Unknown

Some Other Possible Some Other Possible Relations between Relations between

Utterance, Mind & World Utterance, Mind & World

W U1

mind

U2

U3

W1

U

mind1

mind2W2

W

U1

mind1

mind2U2

W U

mind1

mind2

UW1

mind

W2

W3

(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 MentalModel

Utterance

Truth

Truth

Map or ModelTruth

Utterances about truth

W U

mind

W U

mind

(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

Uliar

mind

U1

mind

U2

Paradox Lost

Philosophical whirlpool - stay clear!

U

mind

U1

mind

U2

U1

mind

U2

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.