Process of Inference

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Process of Inference

• Dr.Shrinivasa Varakhedi
• shrivara_at_gmail.com

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Role of Inference in Life

• Contribution
• Much information we get through Inference
• Little we perceive / listen to and infer a lot
  more.
• Inferential power Faculty of brain
• Every living being uses inference.
• Human mind is considered to be supreme for its
  inferring ability.
• Every action in mind involves inference.
• All tasks require reasoning on knowledge.

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Inference Judgment on evidences

• Standard Examples
• If you see flood in the river at the bottom of
  a hill, you will infer that a big rainfall
  occurred on the hills top.
• After seeing smoke on hills top, you just think
  that Hill has Fire because it has smoke
• These judgments are drawn from the known facts
  Flood and Smoke.
• The flood is sign/mark of rainfall smoke is of
  the fire.

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How to arrive at Inference?

• Causal connection between two cognitive events
• Cognition of Smoke ? Cognition of Fire
• Cognition of Flood ? Cognition of Rainfall
• What made this possible ?
• Not just the perception of smoke on hills and
  flood in river.
• On seeing them, the relation between their
  counterparts triggers in the mind that relation
  leads to inference.

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Instrument of Inference

• Sense organs and Sentences are NOT causes.
• The awareness of relation that smoke and fire
  have is major factor.
• This relation is called Vyapti invariable
  concomitance.
• wherever smoke resides, there resides fire
• ?x smoke (x) ? Fire (x)
• This is NOT causal relation but pervasion /
  coverage. (Of course cause always pervades
  effect)

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VyApti or Invariable concomitance

• VyApti relation is defined as hetu-vyApaka-sAdhya
  -sAmAnAdhikaraNyam by NN school of thought.
• This means that If A is reason (hetu) for B, then
  B is pervasive of A and A B are co-located.
• It can be represented in Predicate Logic as
• ?x Sx ? Hx ? ?x Hx ? Sx
• where S sAdhya, H hetu.
• In FOPL vyApti is expressed as (I should NOT say
  it is vyApti !!!)
• ?x Hx ? Sx ?x Hx ? Sx Hx set ? Sx set

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Generation Process

• Instrument Concomitance
• Cognition
• Mediator Application
• Result Inferential
• Judgement

vyApti-nAnam
parAmarsha
anumiti
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Paramarsha OR Application

• Paramarsha is immediate cause for Anumiti
• vahni-vyApya-dhUmavAn parvataH
• Hill has smoke that is pervaded by Fire
• Paramarsha cognises vyApti relation as well as
  relation with the locus i.e., subject of
  inference.
• VyApti relates smoke with Fire, but it doesnt
  lead to conclude that Fire is located in hills.
• If Smoke is found to be located in hills, then
  fire could be placed on hills.
• Thus vyApti relation and paksha-dharmata
  (residing in subject) are two important concepts
  in inferential process

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Argument Syllogism Nyaya

• NN school of though proposes Five limbed Nyaya or
  syllogism
• hill has fire The thesis / PratijnA
• Because it has smoke Reason / hetu
• Wherever is smoke, there is fire as in kitchen
  udAharaNa (with vyApti) / Eg.
• hills has smoke that is pervaded by Fire
  Application / upanaya
• The hills has fire Conclusion / nigamanam

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Syllogism and Nyaya

• Aristotelian system of logic admits three limbed
  argument
• Major premise All men are mortal.
• Minor premise Socrates is a man.
• Conclusion Socrates is mortal.
• On the contrast, five limbed syllogism is
  admitted in NN school of thought
• It is necessary to invoke akAnkshA expectancy
  in hearers mind
• This is called parArtha meant for others.
  svArtha is inference for self.

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Technical Terms (Beware of them!)

• Paksha Subject of Inference
• sAdhya predicated property to be proved
• Probandum
• Hetu Reason / evidence (prover property)
• vyApti pervasion - Invariable concomitance
• Paksha-dharmatA Being related with Paksha
• Sapaksha Loci where probandum is determined
• Vipaskha Loci where probandum is known to be
  absent

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Major points in NN theory of Inference

• NN system deals with mental / psychological
  process
• This process involves mental events / states.
• Inference and its causal factors are cognitive
  episodes
• Never NN system talked of form content
  separation
• Propositions returns truth value and cognitions
  return content
• No deductive mechanism as in FOPL

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Deduction Method and P-Logic

• A complete deductive mechanism based on form or
  syntax
• The semantics of AND, OR, NOT, IF-THEN is
  captured by truth-table
• With the flavour of Boolean Algebra (, , ) you
  may find logic more mathematical and easier
• This mechanism is thru the power of Form
• Form and Content are separated
• Form is nothing but shape that helps to
  manipulate
• Content is information required

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P- logic ? B - logic

• In general a logic is defined by
• syntax what expressions are allowed in the
  language.
• Semantics what they mean, in terms of a mapping
  to real world
• proof theory how we can draw new conclusions
  from existing statements in the logic.
• Propositional logic is the simplest..
• Predicate logic is an extension of Pro.Logic
• Boolean Logic is new version of P-logics.

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Propositional Logic Syntax

• Symbols (e.g., letters, words) are used to
  represent facts about the world, e.g.,
• P represents the fact Andrew likes chocolate
• Q represents the fact Andrew has chocolate
• These are called atomic propositions
• Logical connectives are used to represent and ?,
  or ? , if-then ?, not ?.
• Statements or sentences in the language are
  constructed from atomic propositions and logical
  connectives.
• P ? ?Q Andrew likes chocolate and he doesnt
  have any.
• P ?Q If Andrew likes chocolate then Andrew has
  chocolate

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Propositional Logic Semantics

• What does it all mean?
• Sentences in propositional logic tell you about
  what is true or false.
• P ? Q means that both P and Q are true.
• P ? Q means that either P or Q is true (or both)
• P ? Q means that if P is true, so is Q.
• This is all formally defined using truth tables.

We now know exactly what is meant in terms of the
truth of the elementary propositions when we get
a sentence in the language (e.g., P gt Q v R).
X Y X v Y T T TT F T F T
TF F F
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Proof Theory

• How do we draw new conclusions from existing
  supplied facts?
• We can define inference rules, which are
  guaranteed to give true conclusions given true
  premises.
• For propositional logic useful one is modus
  ponens
• If A is true and Agt B is true, then conclude B
  is true.

A, A ?B B
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Rules of Inference
?Q P ? Q _____ ? ?P

• P
• ______
• ? P?Q

Modus tollens
Addition
P ? Q Q ? R _______ ? P ? R
P?Q _____ ? P
Hypothetical syllogism
Simplification
P Q ______ ? P?Q
P?Q ?P _____ ? Q
Conjunction
Disjunctive syllogism
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Proof Theory and Inference

• So, let P mean It is raining, Q mean I carry
  my umbrella.
• If we know that P is true, and P gt Q is true..
• We can conclude that Q is true.
• Note that certain expressions are equivalent
• think about P gt Q and ? P v Q.

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More complex rules of inference

• Other rules of inference can be used, e.g.,
• This is essentially the resolution rule of
  inference, used in Prolog.
• Consider
• What can we conclude?

A v B, ? B v C A v C
sunny v raining ? raining v umbrella
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Proof Theory (Logic Vs Nyaya)

• NN Theory has developed a complete system of
  proof checking the validity of an inference thru
  hetvAbhAsa fallacies that are based on
  content.
• P-Logic gives proof for valid conclusion thru its
  axiom-based testing methods, which is purely
  mechanical.
• NOTE Validity properness or Being according
  to rules Truth Correspondent to the reality.

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Truth and Validity

• Truth correspondance to the reality
• Test is based on Content Not on the form
• Logic excludes this test from its scope
• Validity coherence among the Ps
• Test is based on Form Not on the content
• Logic explains such formal tests
• (Yo can say inference is valid if it passes thru
  the formal tests)

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Truth and Validity

• According to P-Logic, any conclusion of a valid
  argument must be true if all the premises are
  true
• This shows that Validity of an argument
  guarantees you about the truth of the conclusion
• Whereas NN system decides the truth of a
  conclusion basing on its content (Even true
  inference may deduce from untrue cognition!)

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Is NN system deemed to be Logic ?

• According to me NN is Not LOGIC in the sense that
  P-Logic is called so.
• However it can anytime take inputs from different
  systems and can be improved to do logic
• More over, it must be noted that Never Indian
  systems tried to separate the Form and Content
  or Syntax and Semantics
• They knew that there is a small line between
  them. Panini has used this and achieved mechanism
  in his system.

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Beyond Mechanism

• The hardcore logicians believe that everything
  could be reduced to the level of Formal structure
  and be processed mechanically
• It is NOT so. Ultimately you must stop somewhere
  in basic level of semantics, for semantics is the
  supreme
• NN system holds that vyApti relation is a basic
  relation of the whole reality, which you must
  have somewhere at the end. Accept it nowYet to
  be established!

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Summary

• I conclude my presentation with the following
  remarks..
• NN system should be restudied in this different
  context. It should open for new borrowings (Like
  power of deductive mechanism of B-logic).
• The inferential techniques developed by NN
  systems may be useful for Relational Logic (The
  logic based on relations of concepts)
• NN system is Not a LOGIC that has limited scope.

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THANKS