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The Reality of Logic. David Davenport Computer Eng. Dept., Bilkent University, Ankara 06533 - Turkey. email: [email protected]. Outline. Background Logic & its Problems Fuzzy & other deviant logics Truth A Cognitive “solution” The new Reality. Don't Trust Me. - PowerPoint PPT Presentation
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The Reality of LogicThe Reality of Logic
David DavenportComputer Eng. Dept., Bilkent University, Ankara 06533 - Turkey.email: [email protected]
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.