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 Simple Inference for people new to AI
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toborman
Hooked Member



USA
290 Posts

Posted - Aug 26 2013 :  00:29:03  Show Profile  Visit toborman's Homepage
Objective of inference: Create a new proposition from known propositions.

Syllogism: a model for inference
1. Rule: proposition relation proposition
2. Case: proposition
3. Result: proposition

Inference: Use two of the three propositions of a syllogism to infer (create) a third proposition.
1. Inference type one: using rule and case, infer result
2. Inference type two: using rule and result, infer case
3. Inference type three: using case and result, infer rule

Proposition types supported:
1. Object relation object ------------Socrates likes Jane.
2. Object relation characteristic----Socrates is mortal.
3. Object relation class---------------Socrates is a human.
4. Class relation class----------------Humans are animals.
5. Class relation characteristic------Humans are mortal.

Useful functions for programmers:
1. Remember (store)
2. Recall (find)
3. Substitution (replace)
4. Pattern matching (find)
5. Interpret (replace)

Process:
Type one:
1. Remember a rule.
2. Input a case.
3. Find the left side of a rule that matches the case or partial match with case.
4. Create a new proposition that matches the right side of the rule, substituting variable.
5. Remember the new proposition.

Type two:
1. Remember a rule.
2. Input a result.
3. Find the right side of all rules that matches the result.
4. Create new propositions that match the left side of the rule.
5. Add “possibly” to the beginning of the new propositions.
6. Remember the new proposition for resolution.

Type three:
1. Remember a Case (or a result).
2. Input a Result (or a Case).
3. Create a rule of Case relation Result.
4. Note: if the propositions warrant it,
5. Create a rule of Case relation Result and a rule of result relation case.
6. Create an “either/or” dilemma from the two rules.
7. Add “possibly” to the beginning of the new propositions.
8. Remember the dilemma for resolution.

Example
Modus Ponens: p > q, p; therefore q
Inference type one: using rule and case, infer result
1. Humans are mortal.__________If x is a human then x is mortal.____remember
2. Socrates is a human._________Socrates is a human._____________input
3. Therefore; Socrates is mortal._Socrates is mortal.______________infer
Inference type two: using rule and result, infer case
1. Humans are mortal.___________If x is a human then x is mortal.___remember
2. Possibly Socrates is a human.__Socrates is a human.____________infer
3. Socrates is mortal.____________Socrates is mortal._____________input
Inference type three: using case and result, infer rule
1. Possibly humans are mortal.___If x is a human then x is mortal.___infer
2. Socrates is a human.__________Socrates is a human.____________remember
3. Socrates is mortal.____________Socrates is mortal._____________input

Challenge: now see if you can do one.
Modus Tolens: p > q, -q; therefore –p
Inference type one: using rule and case, infer result
1. Humans are mortal.__________If x is a human then x is mortal.___remember
2. Socrates is not mortal. ________Socrates is not mortal.__________input
3. Therefore;
Inference type two: using rule and result, infer case

Inference type three: using case and result, infer rule

http://mindmap.iwarp.com

toborman
Hooked Member



USA
290 Posts

Posted - Sep 12 2013 :  21:08:19  Show Profile  Visit toborman's Homepage
That one too easy? Try these propositional logic inferences.


Modus Ponens... p > q, p; therefore q
Modus Tolens... p > q, -q; therefore -p
Chain... p > q, q > r; therefore p > r
Disjunctive1... p v q, p; therefore -q
Disjunctive2... p v q, -p; therefore q
Addition1... p; therefore p v q
Addition2... q; therefore p v q
Conjunctive1... -(p & q), p; therefore -q
Conjunctive2… -(p & q), q; therefore -p
Simplification1... (p & q); therefore p
Simplification2... (p & q); therefore q
Adjunction... p, q; therefore p & q
Reductio1... p > -p; therefore -p
Reductio2... p > (q & -q); therefore -p
Complex constructive... p > q, r > s, p v r; therefore q v s
Complex destructive... p > q, r > s, -q v -s; therefore -p v -r
Simple constructive... p > q, r > q, p v r; therefore q
Simple destructive... p > q, p > r, -q v -r; therefore -p

http://mindmap.iwarp.com
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toborman
Hooked Member



USA
290 Posts

Posted - Oct 04 2013 :  22:21:33  Show Profile  Visit toborman's Homepage
Here is a conversation I had with Harry Workman, my virtual human, back in 2009. I have re-posted this example of simple inferences that can be accomplished with a little propositional logic. I asked Harry to show me what he inferred during the conversation.


Harry : hello Tom.
Tom : charlie is a human.
Harry : I will remember charlie is a person.
Tom : chad is a human.
Harry : I will remember chad is a person.
Tom : charlie is mean.
Harry : I will remember charlie is mean.
Harry : using affirm particular, I deduced remember some humans are mean.
Harry : I will remember some humans are mean.
Harry : using deny negative universal, I induced forget humans are not mean.
Harry : I will forget that humans are not mean.
Harry : using affirm universal, I induced remember humans are mean.
Harry : I will remember humans are mean.
Harry : using modus ponens, I deduced remember charlie is ugly.
Harry : I will remember charlie is ugly.
Harry : using affirm particular, I deduced remember some humans are ugly.
Harry : I will remember some humans are ugly.
Harry : using deny negative universal, I induced forget humans are not ugly.
Harry : I will forget that humans are not ugly.
Harry : using affirm universal, I induced remember humans are ugly.
Harry : I will remember humans are ugly.
Harry : using reverse modus ponens, I abduced remember charlie is possibly mean.
Harry : I will remember charlie is possibly mean.
Harry : using remove uncertainty, I deduced forget charlie is possibly mean.
Harry : I will forget that charlie is possibly mean.
Tom : chad is not mean.
Harry : I will remember chad is not mean.
Harry : using negate particular, I deduced remember some humans are not mean.
Harry : I will remember some humans are not mean.
Harry : using deny affirmative universal, I induced forget humans are mean.
Harry : I will forget that humans are mean.
Harry : using affirm negative universal, I induced remember humans are not mean.
Harry : I will remember humans are not mean.
Harry : using deny negative universal, I induced forget humans are not mean.
Harry : I will forget that humans are not mean.
Tom : describe charlie.
Harry : charlie is a mean ugly human.
Tom : describe chad.
Harry : chad is not mean.
Harry : chad is a person.
Tom : describe humans.
Harry : some humans are mean.
Harry : some humans are ugly.
Harry : humans are ugly.
Harry : some humans are not mean.

http://mindmap.iwarp.com
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toborman
Hooked Member



USA
290 Posts

Posted - Oct 20 2013 :  20:00:02  Show Profile  Visit toborman's Homepage
More inferences may be made if we extend propositional logic with predicate logic (predicate calculus). We can add the quantifiers “some” and “all”, with inferences universal and existential generalization, and universal and existential instantiation. Add “not” and additional inferences may now be made from the Square of Opposition using Aristotle's logic.
(T = tester, R = responder)

Test: existential generalization, unresolved dilemma, resolve dilemma
given: if x is y and x is a B then some B is y.
given: if some B is y then either all Bs are y or some B is not y.
T: jack is a human.
T: jack is tall.
R: some human is tall.
R: either all humans are tall or some human is not tall. I don't know which is true.
R: I will remember the dilemma about all humans are tall.
T: Jane is a human.
T: Jane is not tall.
R: I will remember some human is not tall.
R: I will forget the dilemma about all humans are tall.

Test: existential generalization, unresolved negative dilemma, resolve negative dilemma
given: if x is not y and x is a B then some B is not y.
given: if some B is not y then either all Bs are not y or some B is y.
T: jack is a human.
T: jack is not tall.
R: some human is not tall.
R: either all humans are not tall or some human is tall. I don't know which is true.
R: I will remember the dilemma about all humans are not tall.
T: Jane is a human.
T: Jane is tall.
R: I will remember some human is tall.
R: I will forget the dilemma about all humans are not tall.

Test: existential generalization, unresolved negative dilemma, resolve negative dilemma, assume negative
given: if x is not y and x is a B then some B is not y.
given: if some B is not y then either all Bs are not y or some B is y.
given: if “either all Bs are not y or some B is y' and no B is y then assume all Bs are not y.
T: jack is a human.
T: jack is not tall.
R: some human is not tall.
R: either all humans are not tall or some human is tall.
R: I don't know which is true. I will assume that all humans are not tall.

Test: existential generalization, unresolved dilemma, resolve dilemma, assume positive
given: if x is y and x is a B then some B is y.
given: if some B is y then either all Bs are y or some B is not y.
given: if “either all Bs are y or some B is not y' and no B Is not y then assume all Bs are y.
T: jack is a human.
T: jack is tall.
R: some human is tall.
R: either all humans are tall or some human is not tall.
R: I don't know which is true. I will assume that all humans are tall.

Now the assumption can be resolved.

Test: belief revision
given: if some B is y and all Bs are not y then forget all Bs are not y.
given: all humans are not tall.
T: Jack is a human.
T: Jack is tall.
R: I will forget all humans are not tall.

Test: belief revision negative
given: if some B is not y and all Bs are y then forget all Bs are y.
given: all humans are tall.
T: Jack is a human.
T: Jack is not tall.
R: I will forget all humans are tall.

http://mindmap.iwarp.com

Edited by - toborman on Oct 25 2013 10:02:41
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