Logic and Errors

Methodlogy and Concepts:

Statement: statement is generally meant group of words that have meaning but in the logic statement group of words that have two terms, first term is used under subject and the second term under predicate. the statements in logic are called prepositions or premis

Premises: the premises that has the major term is callled major premises and the premises that has the minor term is called minor premise.

Term: The predcate of major premises is called major term ans is denoted by S and the terms which are common in both the premises are called middle term which is denoted by M. for example:
             All men are mortal (Major Premises)
             Ram is man (Minor premise)
             Ram is mortal (Conclusion)
Here 'mortal is a major tern and its subject is ot minor term hence because it is in minor premise also so it is middle term. Minor term here is "ram".

In logic unit of argument is called prepostion or premise and unit of though is called term. A preposition is different from a sentecnce that its predicate either affirm or deny the subject,i.e., if we say that "I have taken food". in this sentence predicate neither affirns nor denies the subject but if we say that "a man is mortal" then the word mortal asffirms the  man so this is a prepositions and "I have taken food" is justa sentence. now the question is how will we identify the major and minor premises in a syllogism (group of Logical statements). It is possible by identyfying the major and minor terms. For this purpose study the 1st and 2nd premises and identify the preducate which had vital and wide concept. It will be tha major term and the terms which come in the both premises will be the middle terms. a terms may have one word or more than one word. moreover, minor term will be used as subject of the conclusion and its predicate will be the major term.

Type of Syllogism:
1. Categorical: Here all the prepositions are categorical in positive or negative form and no doubtfullness is seen at all, e.g.,
             All graduates are eligible (major)
             All eligible are men (Minor)
             Some Eligible are graduates  (Con...)

2. Hypothetical: The syllogism of this type uses prmise of conditions "If he works had, he will succeed." Here the first part is called antecedent and the second part is called consequent. in this type of syllogism the major premise is hypothetical losing some thing and the minor premise will be categorical, e.g., If he comes, I shall meet Him. In this syllogism
              If he come   (Major)
              I shall meet him  (Minor)

3. Disjunctive: In this type of syllogism the major premise is disjunctive and the other two premises are categorical, e.g.,
            Either he is an hionest man or a thief.  (Major)
            He is an honest man (Minor)
            He is not a thief (concl....)

4. Relational:  Here relations between various terms are shown orderl, e.g.,
           A>B, B>C, C>D
           so A>D  (Concl....)

5. Deilmma: In such type of syllogism the major premise is double hypthetical and minor premise is a double hypthotical and minor  premise is of disjunctive type and the conclusion will either disjunctive or categorical,.eg.
   (i) If these books confirm to Quran they are reliable if they do noot they ary are superfluous.
   (ii) Either they confirm to Quaran or they do not
   (iii) Either they are reliable or superfluous
In the above example the seconf and third premises are disjunctive.

6. Copula: The verb used in the premises in logic is called copula. for this purpose "is" and "are" generally used and logicians always talks in the term of the present

Types pf Prepositions;
 1. From the quantative point of view:-from the quality of point of view it is of two types: 

          a) affirmative i.e., All  graduates are men. 

          b) Negative:- No man is harworking.

 2. From the quantative Point of View: This view also has two types: 

           a) Universal:- in which the subject is universal and refers to all without exception,i.e., 

               Human beings are mortal. 

           b) Particular-in which the subject does not refer to all,i.e., 

               some people are not hardworking

The prepositions which begins with proper noun will also be treated as universal. Similarly, the propositions which uses phrase like "all but one", 90% about 50% etc will be treated as particular prepositions. the words like some times, often generally, etc, also make the prepositions particular. The prepositions which use words like few, little, hardly, scarely, etc, are considered to be negative used in prepositions then the preposition will become affirmative. On the basis of above divisions we can say that there are four types of categorical prepositions in logic;

1. Universal Affirmative:- It is called "A" prepositions, ie,
    All labourers are hard working
    all children are true speaking

2. Universal negative:-they are callled "E" prepositions, ie,
    No minister is dedicated
    No human being is happy etc.

3. particular Affirmative: They are callef "I" prepositions, ie,
    some men are hard-working
    some people are lobourious

4. particular Negative:-they are called "O" prepositions,ie,
    Some men are not hardworking
    some ministers are not honest

Disturb of terms in propositions:
    In case of A propositions only subject is disturbed
    In case of E propositions both subject and predicate are disturbed
    In case of I propositionsneither subject nor predicate is disturbed
    In case of O propositions only predicate is disturbed

Now, question is how will the terms be disturbed. Take the example of A propositions "Man is Mortal". Here the term man has two meaning. the first man means human being and the secong, is man means animality. if this term 'man has used as denotation, ie, man means human being, the term man will be disturbed. on the other hand, if man means animality, ie, connotation of the term, it will not be disturbed because there are other animals besides man and the propositions will not remain universal. Similarly, in the case of E propositions both subject and predicate are disturbed, ie, no  man is happy. here no man refers to all human being and happy also refers to all without exception. so both can be distributed ie, for logical conclusions. in the third case of I proposition neither subject nor predicate can be disturbed, ie, some men are not  hardworking. here neither the term men refers to 'all' neither the term hardworking refers to all so both of them cannot be disturbed. In case of O propositions only predicate is di9sturbed, ie, some men are not honest. here the term men does not refer to all but the term honest may refer to all so it can be distributed. Distribution of the middle or other terms helps us to arrive at logical conclusion.

Types of Logical Inferences:
    A. Immediate Inference
    B. Mediate Inference

A. Immediate Inference:- Sometimes premises are arranged in different forms. out of them some are true and some are false, ie, "some students are industrious", is a premise. If it is considered to be true then "some students are not industrious" is also true but if we say that "no students is industrious" it is false. All these arrangements are based on certain rules. This is called immediate inference. We shall discuss here only two methods of Immediate inference:

1. Conversion:- Here we drive conclusions by taking the following rules into account:
a) Predicate becomes the subject and the subject becomes the predicate.

b) Quality of proposition does not undergo and change, ie, Affirmative propositions will lead to affirmative conclusion and negative proposition well lead to negative conclusion.
c) A proposition is converted into I proposition. I proposition is converted into I and E into E itself.
d) Proposition O can not be converted at all. Examples will illustrates these point

    * All ministers are politicians (A propositions). its converted from is
                 Some politicians are ministers (I pro-positions)
    * Some men are har-working ((I pro-position) It converted from is:
                 Some hard working are men (I pro-Position)
    *  No man is immortal (E Proposition). is converted as
                  No Immortal is man (E proposition)
     *  Some students are not intelligent(O Proposition). it can be converted as
                  Some intelligent are not students.

2. Observations: Observed from of inference is based on the following rules:

a) Subject does not undergo any change but preicate is changed into contradictory. Here contradictory does not mean the antonym of the term. It is not used in the logic. for making contradictory of the term 'not' not be added before it, ie, contradictory of "kind" is not "cruel" but "not Kind" is the real contradictory of 'kind".

b) Negative propostions are changed into affirmative and vice-versa, ie, A is observed into E and E is observted into A proposition.

c) The quality of the proposition will remain the same. If the proposition is universal it will remain universal and if it is particular it will remain particular in the observed from the conclusion.

Following example will illustrated these points clearly
     1. all religious people are contended people (Proposition A)
        No religious is not contended people (E)
    
     2. No man is fully impartial (E)
        All men are not fully impartial (A)

     3. Some men are hard of hearing. (O)
        Some men are not hard of Hearing (I)

     4. Some doctors are not good (O)
        Some doctors are good (I)

Validity table: In order to test the validity of the conclusions we may also use the following table Here
    
              T & t = Truth
              F & f =  False
              d       = Doubtful

How to use the Table: "All men are honest" is a proposition 'A'. If it is considered to be true then the conclusion proposition, E, ie, "no man is honest" will be false, 'Some honest are men' will be true conclusion of O Proposition will also be false. No if this proposition is considered to be false then proposition E, ie, "no man is honest is doubtfuk and proposition I, ie, "Some men are honest" is also doubtful but the proposition O, ie, "Some men are not honest" is true.

"Some students are active " iis an I Proposition if it is considered to be true then conclusion of E proposition if false, and conclusion of A and O porpositiona are doubtful which is clear from the third column of the table. If the same proposition is considered to be false then conclusions of E and O proposition will be true and A propositions willbe false which is evident from the second column of the table.

B. Immediate Inference: Arriving conclusion on the basis of the middle term is called mediate inference. There are two premises-one is major and the other is minor with one conclusion. This group of arguments is technically known as syllogism. An example will illustrate the point:

       All mammals are vertebrates (Major)
       The whale is a mammal (Minor)
       The whale is a certebrate (Concl...)

Here the word Mammal is common in the both premises. it is middle term. This does not occur at all in the conclusion even if on the basis of its conclusion has been drawn. The categorical syllogismis thus essentially a process of comparision. Each of the terms in the conclusion is compared is compared with the middle terms in premises and conclusion is arrived at on the basis of the middle term. For arriving at right and valid conclusions following rules or canons must be kept in view:

1. In every syllogism there must be three terms and not more than three, ie, the major term, the minor term and the mroddle term. "good characters are like by the people. GB Shah uses good characters in his works. There are four terms in the above ptwo premises becasue goog characters has two different meaning in the two premises. so no valid conclusion can be drawn here. The middle term must have the identical meaning in both the premises and it must lie able to be distributed.

2. Every categorical syllogism must containn only three premises-major, minor and the conclusion,eg,
          Man is Mortal (1)
          Cow is mortal (2)
          Ram is a man (3)
There are three major premises excluding conclusion so no conclusion can be drawn from them.

3. The middle term myst be distributed in at least one of the premises. It is a must because it is the standars of comparision. It must be used in at least one  premise to its universal extent, eg, sedimentary rocks are stratified substances. Some metamorphic rocks are stratified substances. Here the term strafied substances is not distributed at all because sedimentary rocks agree with one part of the stratified substances and metamorphic rocks agree with another part. Thus there are four terms.

4. the term is not distributed in the premises cannot be distributed in the conclusion. an example will illustrate the point:
     All rational beings are responsible for their action (1)
     Brutes are not rational being (2)
     Brutes are not responsible for their action (3)

here the major term responsible for their action is distributed in the conclusions but it was not distributed in the major premises becasue it means that only those who are responsible for there action who are not rational being.

5&6: Two negative premises yeild no conclusion and if one of the premises is negative, the conclusion muse be negative,ie,
       No student dislikes games (1)
       No game is fully satisfactory (2)
No conclusion can be drawn from these two premises.
       No human being dislikes justice (1)
       Students are human being (2)
       Students do not dislike justice (concl...)      

Here one of the two premises is negatie that makes the conclusion negative, but double negatives used in a premise makes it affirmative and conclusion based in it will be correct, eg,

       No one who is not throughtly upright is to be trusted
       This man is not throughtly upright
       This man is not to be trusted.

7. No conclusion can be drawn from two particular premises
8. If one of the premises is particular the conclusion must be partcular.

Four Possible arguments in a LOGIC;

      S= Subject or Minor Term
      P= Predicate of Major Term
      M= Middle Term

On the basis of the four figures shown above and four types of propositions there are 4x4x4x4=256 types of syllogism possible. but there are only 19 valid syllogism which are given below and another are invalid. these four figures and their numbers must kept in view.

   Figure 1      Figure II      Fugure III    Figure IV
   1. AAA      5. EAE        09. AAI        15. AAI
   2. EAE      6. AEE        10. IAI         16. AEE
   3. AII        7. AIO         11. AII         17. IAI
   4. EIO       8. AOO       12. EAO       18. EAO
                                     13. OAO       19. EIO
                                     14. EIO

Some examples will illustrate these points

1. All spirituals love humanity  (A)
   All indians are spirituals    (A)
   All indians love spirituality (A)
AAA sylogism is here. thus conclusion is correct according to Figure no 1

2. All students are hardworking  (A)
   Anil is a students            (I)
   Anil is hard working          (I)
Syllogism no 3 (ALL) is Here according to Figure no 1

3. No human being is animal  (E)
   All animals have four legs(A)
   No man has four legs      (E)
this conclusion is not valid becasue EAE is not in Figure 4

4. Some studetns are faithful (I)
   All students are Indian    (A)
   Some Indians are faithful  (I)
Syllogism no 10 (IAI) is here. Thus it is valid according to figure no 3

5. All animals like grass   (A)
   Lion does not like grass (O)
   Lion is not an animal    (O)
This conclusion is valid because AOO is their in the figure no 2 so this conclusion is correct.

Rules of Hyoithetical Syllogism: It si based on the conclusion between a supposition or condition and its consequences. It starts by the word, if, supposing, granted that, as etc. This part of the syllogism which expresses the condition is called antecedent ans the clause stating the result is called comsequent. In such type of syllogism the hypothetical premise is the major premise and the categorical proposition is the major premise. Following rules must be kept in view whole arriving at valid conclusions in such cases.

Either affirm the antecedent or deny the consequent, eg,
 1. If he were well, he would leave the place
 2. he has not left the place
 3. Conclusion (He is not well)
Here the consequent is denied. in the following example antecedent is affirmed
 1. If it rains toady, he shall not go to school
 2. It is raining
 3. Conclusion (He shall not go to school)
So always affirm the antecedent or deny the consequent as the case may be for valid conclusions. If by mistake, consequent is affirmed the following invalid conclusion will be drawn
 1. If prefect justice prevailed, the rich would not be permitted to exploit the poor.
 2. The rich are not premitted to explot the poor
 3. Conclusion 9Perfect justice is prevailing)
The above rules is also followed in disjunctive syllogism.

Common Errors in Logic

1. Fallacy of ambiguity:
If middle term or phrase are ambigious wrong conclusion may be drawn,ie,
  1. Partisans are not faithful
  2. republicans are partisans
  3. Conclusion (Republicans are not faithful)
Here middle term Partisans has two meaning in the two premises. In premises No 1, it means those who are prejudicely interested in some cause but in the secong premise it denoted the members of a particulars party

2. Fallacy of composition:
When we accept a thing or term to be true of a whole which just holds true of its parts only, eg,
  1. All the angles of a triangle are less than 180*
  2. ABC are the angles of this triangle
  3. Conclusion (ABC are less than 180*)
Here all the angles in the first premise means each and every angle taken separately but in the second premise it means A, B, and C taken collectively. it means what is true of part can not be necessarily true of the whole

3.Fallacy of Divison
It is opposite composition. If we consider that what is true of the whole will be true of parts when taken separately it will also give wrong conclusion, eg,
  1. All the birds fly in the sky
  2. Ostrich is a bird
  3. Conclusion (Ostrich fly in the sky)
Ostrich is an exception which does not fly in the sky

4. Fallacy of Irrelevant conclusion
Fro some time assume as wrong conclusion to be right if it si near to the tight conclusion. This makes our efforts, futile, eg,
  1. Those who takes shelter in slums are sometimes butchered by the rich
  2. Rich people are always antagonistic to those who lives in slums
  3. Conclusions (Rich butcher the pooor because they hate them