Logic

Lecture notes

Phil/Psych 256
Jan. 14, 1997

Logic:

Q: What can propositional logic represent?

	A1: Propositions, i.e.,
		b stands for Babe is peaceful
		m stands for Maa is peaceful
		~b (not b)
		b & m (b and m)
		b V m (b or m)
		b -> m (b implies m)

	A2: Semantics, i.e., truth tables

Q: What about predicate logic?

	A1: Predicates, i.e., 
		peaceful(Babe)   
		peaceful(Maa)
		eats(Babe,vegies)

	A2: Quantifiers, i.e.,
		all, some

		All vegie-eaters are peaceful:
		(x)(eats(x,vegies) -> peaceful(x))

		Some animals eat  vegie-eaters:
		(Ex)(animal(x) & eats(x,vegie-eaters))

		(Ex)(animal(x) & (Ey)(eats(y,vegies) & eats(x,y)))

		(Ex)(Ey)(animal(x) & eats(y,vegies) & eats(x,y))

Q: What about probability?

	A1: P(b) < .9

Q: What can be computed with logic?

	A1: Inferences, e.g.,
		Modus ponens:
			p -> q
			p
			So q.

		Modus tollens
			p -> q
			~q
			So ~p.

	A2: Problem solving:
		1. Planning as deduction
		2. Decision as utility maximizing
		3. Explanation as deduction

	A3: Learning:
		1. Induction
		2. Abduction

	A4: Language

Phil/Psych 256
Jan. 16, 1997

Logic:

Q: What comprises logical deduction?

	1: Database of facts (premises)

	2: Knowledgebase of procedures

	3: Goal expression to be derived

	4: Derivation strategy (search)

e.g.,
	{1}     (1) (x)(Fx -> Gx)       P
	{2}     (2) (x)(Gx -> Hx)       P
	{3}     (3) Fa                  P
	{1}     (4) Fa -> Ga            (1) US(x/a)
	{1,3}   (5) Ga                  (3)(4) MP
	{2}     (6) Ga -> Ha            (2) US (x/a)
	{1,2,3} (7) Ha                  (5)(6) MP
	{1,2}   (8) Fa -> Ha            (3)(7) C
	{1,2}   (9) (x)(Fx -> Hx)       (8) UG(x/a)

Q: What strategies are there?

	A1: Blind search

	A2: Reductio ad absurdum - assert 
	~(Goal) and derive a contradiction
	(Resolution method - see Rips)

	A3: Constrain application of 
	procedures in forward search, promote
	application in backward search (Rips)

Q: What are the disadvantages of logic?

	A1: Less flexible than natural language

	A2: Restricted to verbal information

	A3: Deduction is monotonic (Minsky)

	A4: Deduction is computationally
	explosive

	A5: Deduction not= ab- or in-duction

	A6: Visual reasoning

Q: Is logic psychologically plausible?

	A1: Wason experiment suggests 
	otherwise
	
Readings: Chapter 3 "Rules" and Anderson.

Essay 1: Seven short-answer questions,
	one bonus question.

	Due Feb. 28, 4:00pm, PAS 3289 (box)

	Late policy: No lates accepted!

Further materials

• Bierce, A. (1958), The Devil's dictionary, Dover Publications, New York.
• Mates, B. (1965), Elementary logic, Oxford University Press, Oxford.