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					Logic
Logic is the science of reasoning.
Reasoning is a formal activity
The notion of form has wider implication in logic. It pertains to the form of
proposition as well as the form of argument. The notion of form refers to the
norm/ rule/ laws that constitute expression. All expression follows a grammar.
Our thoughts are formal – they are structured. The structurality of thoughts
presupposes law. They are laws of thoughts. There are three laws of thoughts,
they are, 1. Law of identity; 2. Law of contradiction; 3. Law of excluded-
middle

Law of identity: p is identical with itself. It asserts that if any statement is true
then it is true. (If p stands for a true proposition then p is true only.)
Law of contradiction asserts that „no statement can be both true and false‟. (If p
is a true statement then p cannot be false at the same time.)
Law of excluded-middle asserts that „any statement is either true or false‟. (If p
is a true statement (¬p) its negation is false, both cannot be true together and both
cannot be false together)

Proposition is a logical sentence.
The form of proposition is constituted of terms.
A simple proposition is constituted of at least two terms; they are, the subject
term and the predicate term.
The subject term and predicate term refer to two different classes. They are
related by a copula. Copula is a ‘to be’ verb.
Example, All men are mortal.
Here, in the above, proposition „men‟ refers to the subject class term and the term
„mortality‟ represents predicate class. The copula „is‟ relates the subject and the
predicate terms.
There are four type of categorical propositions used in Aristotelean logic. Their
types are made with reference to the quality and quantity of the propositions.
The categorical Propositions are:

                                             Logical form of Propositions
   1.   A: All men are mortal (Universal Affirmative)    All s is p
   2.   E: No men are mortal (Universal Negative)        No s is p
   3.   I: Some men are mortal (Particular Affirmative) Some s is p
   4.   O: Some men are not-mortal (Particular Negative) Some s is not-p



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    Two Inferential Process of Deduction:
      1. Immediate Deductive Inference
      2. Mediate Deductive Inference

    Immediate Deductive Inference: Conclusion is deduced from one of the given
    propositions. Conversion and Obversion are deductive inferences.




    Conversion:
    The Rules of Conversion:
      1. The conversion proceeds with interchanging the subject term and the
         predicate term, i.e. the subject term of the premises becomes the predicate
         term of the conclusion and the predicate term of the premise becomes the
         subject of the conclusion.

                       For Example, No Hungarians are Cricketers (Convertend)
                                  No Cricketers are Hungarians (Converse)

    The given proposition is a premise is otherwise called as Convertend, where as
    the conclusion drawn from the premise is called Converse.

       2. The quality of the premise (convertend) remains same. The quantity of the
          proposition may change.



                               Table of Valid Conversion
    Convertend                                    Converse

A: All S is P (All students are smart)          I: Some P is S (Some smart persons
 are students)
    E: No S is P (No student is tall)     E: No P is S (No tall persons are students)
    I: Some S is P (Some students are poets)    I: Some P is S (Some poets are
    students
    O: Some S is not-p                          (Conversion is not valid)




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Obversion:
Rules of Obversion
1. Obversion is one of the immediate inferences.
2. To obvert a proposition, we change its quality and replace the predicate term
by its complement.

Instances:
A: All poets are emotional (Obvertend)
E: No poets are non-emotional. (Obverse)

E: No singers are barbarians
A: All singers are non-barbarians.

I: Some politicians are statesmen
O: Some politicians are not non-statesmen.

O: Some teachers are not-cricketers
I: Some teachers are non-cricketers.


Syllogism

A syllogism is a deductive argument in which conclusion is inferred from two
premises.

The standard syllogistic argument will have 3terms and 3 propositions.

The term that occurs as the predicate term of the conclusion is called the „major
term’.

The term that occurs as subject term of the conclusion is ‘minor term’.

The term, which does not appear in the conclusion but appears only in the
premises, is called ‘middle term’.

Major premises
Minor premises
Conclusion.




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Mood & Figure of Syllogism

The standard form of categorical propositions determines the mood of the
syllogism


For example, in an argument like;

A: All men are sincere (Major Premise)
I: Some men are hard working (Minor Premise)
I: Therefore, Some hard working persons are sincere (Conclusion)

The mood of the above argument is: A I I


The different positions of the middle term determine the figure of the syllogism.

Ist                                    2nd




      3rd                              4th




For Example, A: All scholars are IITians
           I: Some scholars are scientists
           Therefore, Some scientists are IITians.

In the above argument „scholar‟ is the middle term. It appears in the subject place
of major premise and also the subject place of the minor premise. Hence, it
constitutes the 3rd figure.




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Rules and fallacies of Syllogism:
An argument in syllogism becomes fallacious if and only iff it violates the rules
of syllogism. Here forth we are stating about some of the rules of syllogism and
some of the fallacies
    1. In a syllogism there must be at least three terms. If an argument involves
       four terms then we cannot draw a valid conclusion.

      The fallacy is called fallacy of four terms

      Example: All men are mortal
          Some scholars are sincere.

      There is no term common in the above argument, which makes it
      impossible to draw a valid conclusion.


   2. The middle term must be distributed at least once in the premises. If the
      middle term is not distributed in any of the premises then the arguments
      commits the fallacy of undistributed middle.

            Example:
                         All students are scholars
                         Some scholars are technocrats
                         Therefore, Some technocrats are students

            As you know that A proposition which is universal affirmative
            prop., does not distribute its predicate term. So the predicate term
            „scholar‟ is not distributed in the major premise and it is also not
            distributed in the minor premise. The minor premise is „I‟ prop.,
            which does not distribute any term. Therefore, the argument
            commits the fallacy of undistributed middle.




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   3. If both the premises are negative then no conclusion follows. It commits
      the fallacy of exclusive terms.


             Ex.          No judges are sentimental
                          No judges are singers
                          Therefore, No singers are sentimental

       The argument is fallacious because in negative proposition whether it
universal negative or particular negative proposition the terms (the subject term
and the predicate term) exclude each other. Their exclusion implies exclusion of
the relationship of middle terms.

   4. If a term is distributed in conclusion it must be distributed in the respective
      premises. If this condition is not fulfilled them it leads to the fallacy of
      either Illicit Major or Illicit Minor.



   Illicit Major:         All students are regular
                          No hardworking persons are students.
                          Therefore, No hardworking persons are regular

   As the E proposition distributes its predicate term, the term „regular‟ in the
   conclusion is distributed. It is the major term and as major term it has
   appeared in the predicate place of the major premise, which is undistributed. It
   is because A‟ proposition distributes only its subject term not the predicate
   term. Hence, the argument commits the fallacy of Illicit Major


   Illicit Minor: All students are singers
             All students are poets
       Therefore, All poets are singers

   The predicate term poet in the conclusion is distributed which is the minor
   term. As a minor term it must be distributed in the minor premise. The minor
   premise is A type of proposition which distributes only the subject term. The
   poet occurs as the predicate term in the minor premise and remains
   undistributed. Thus the argument commits the fallacy of illicit minor.

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   In a syllogism if one of the premise is particular then the conclusion must be
   particular proposition


   In a syllogism if one of the premise is negative then conclusion would be
   negative proposition.


Induction:
Inductive Generalization

Causality

Casual Relations:
        1. Necessary Condition
        2. Sufficient Condition

NC: the presence of oxygen is a necessary condition for combustion to occur.

            Et1 → Et2

We can legitimately infer cause from effect only in the sense necessary condition.

And we can legitimately infer effect from cause only in the sense of sufficient
condition.

Postulates of Induction

   1.       Law of Causality
   2.       Law of Uniformity of Nature
   3.       Law of Conservation of Energy

Induction by Simple Enumeration:

The method of arriving at general or universal propositions from particular facts
of experience is called „inductive generalization‟.


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Mills‟ Method for Understanding Causal Relation:

    Method of Agreement
    Method of Difference
    Joint Method

    Method of Concomitant Variation
    Method of Residues

Science
Science replaced truth by authority.

Simple view of Scientific Method

Induction used in Scientific Prediction:
                 uniformity of nature
                 conservation of energy
                 causality

             Limits of Observation

Is observation theory laden?

             The Problem of Induction
              Problem of certainty (Hume & Russell)
              Different generalization can be made looking at the past
                instances. (Goodman)
Ex. „GRUE‟ [Bule / Green]
Ex. 1. All emeralds are blue (t/f) before   2. All emeralds are green (t) after
Inductive Generalization based on large number of observation


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           Context of Discovery
           Context of Justification

Certainty of the conclusion is replaced by Probability.

Probability is based on the consistency of the available data.

Ex. Laws in science are not absolutely proven to be true, rather generalization
which is high probability of being true.

Justification for Induction:

    „Invariable and unconditional‟ causal connection
Law of causation is established by empirical grounds – confronts a paradox.

*causal relation is proved by experience
*conclusion presupposes law of causation
*problem of circularity of definition.

Karl Popper‟s Falsifiability thesis:

“Empirical method is continuously to expose a theory to the possibility of being
falsified”

Formulation of Conjectures/ hypothesis
Increasing the degree of Falsifiability

    Any theory, which is shown to be false, is discarded or at the very least,
     modified. Science thus progresses by means of conjectures and refutation.


Verisimilitude (Approximation of truth) : Truth content Vs Falsity content

Hypothesis – corroborates with reality

Corroboration – belief in the approximate truth of theory.




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