A Quick and Dirty Introduction to Logic

This can be considered a followup of my earlier post:
[Link] "A guide to Good debating.."

Any time there is a debate, logic and reasoning(argument)
is used. Debate and logic go hand in hand. A debate cannot
be conceived without logic and reasoning. Since there IS a
lot of debate in this forum and many other ones, it is
important to learn the rules of logic for a valid and meaningful
debate. A debate that does not follow the correct rules of logic
is a total waste of time and energy and is meaningless. It is
also important to realize that the rules of logic are universal
and beyond controversy. There is ONE rule of logic that is
universal. A debate must adhere to that universally recognized
rules of logic. If each stuck to their own rules invented to
suit to their advanatge, a total chaos would result and the
debate would again reduce to meaningless pursuit just as it
would in the absence of any logic. If the participants in a
debate do not agree on the universally agreed set of rules for
logic then they might as well call it quits and not proceed to
the debate, as it would be a total waste of time. The universal
logic is nothing but the one commonly accepted and used almost
intuitivley by all and is used ubiquitously in mathematics and
the sciences, implicitly most of the time. The reason that this
rule of logic is universal and accepted without question is that
besides being intuitive, it also has the best track record of
leading humanity to the truth. It is through correct reasoning
that humanity has been able to advance so much in acquiring
knowledge about the world. This universality and intuitive
correctness of logic will become apparent as I introduce the
rules of logic shortly.

Since it will be quick intro, I will cut down the verbiage
from hereon, and present the material in concise, to the point
manner, assuming readers are intelliegnt enough to fill in any
explanatory details themselves.

Defintions:

Utterance: Anything uttered by humans.

   Utterances can be of two types:

   1. Grammatically incorrect or meaningless utterance (nonsense,gibberish)
      e.g "Obladi Oblada", "Obladi is Obalda", "Hattimatim Tim" etc.

   2. Grammatically correct or meaningful utterance (Sentence)

      e.g, John Doe is an honest man
           Oblitron does not forget

   Sentence can be of two types:

     1. Noncognitive

        When some words in the sentence are not defined,well-defined,
        or when the definition leads to a logical contradiction,
        or when the definition requires a subjective judgement or
        belief to make sense, or when the words are well-defined
        but the sentence as a whole cannot be cognitively
        understood.

       e.g:
        A. God/Soul/Ghost exists, Oblitron does not forget..

           God/Soul/Ghost.. are not clearly defined, and
           some defintions require a prior belief to make
           sense, e.g God = Creator of the universe, assumes
           a creator exists (A belief). Oblitron is not
           defined at all.

        B. "A transcendental thought wave matrix splits
            the holographic projections of mind.

           (Words are well-defined but no cognitive meaning exists
            of the sentence as a whole)

     2. Cognitive

       When the words of the sentence are well-defined and agreed
       on universally, and the sentence as a whole also makes a
       universally agreed on sense.

       e.g  Roses are red.

     Cognitive sentence can be of two types:

       1. Nonpropositional

          No clear "true/false" attribute can be attached,
          e.g : You should do this.
              Do you like to play golf?
              I wish I was in Hawaii!

       2. Propositional (Propositions)

        A clear "true/false" label can be attached to the
        sentence, the true/false label not depending on value
        judgement, but on objective knowledge of facts

        e.g: "X" committed the murder,
             Vacuum fluctuations can cause matter to pop out
             of vacuum etc

        A proposition can be affirmative (Like "IS") or
        negative (Like "IS NOT")

A debate can be meaningful for each participant and end in
a resolution only if propositions are used by all the debators.
Logic and arguments cannot meaningfully apply to noncognitive or
non-propositional sentences. Doing so results in fallacies which
were the subject of my post on fallacies. When fallacies do occur
due to such misuse of logic by any participant, then it also
requires logic by the opponent to point out the fallacies. This
is what happens when a believer tries to use logic to prove God
exists, then a rationalist has to use logic to point out the
fallacy in such argument. Of course one can engage in a
metaphysical discussion or essay using noncogntive sentences,
but it cannot contain any logic or arguments. (Like Therefore..,
because..hence.. etc)

Now on to the the meat, i,e the concepts and rules of logic,
which really apllies to Propositions.

Propositional sentences will often be denoted by the
shorthand p, q etc. (e.g  p = "It is raining")

A proposition usually has a subject (About which/whom the proposionj
is making the true/false declaration) and predicate (The attribute
of the subject to which the true/false characterization applies)

e.g in All ravens are black (Here Raven = Subject, Black = Predicate)

Propositions are of three types:
   (1) Categorical (p)
   (2) Hypothetical(If p, then q), (Also equivalent to : p, unless q)
   (3) Disjunctive (p or q)

Examples of each above:

    1. All ravens are black (Ravens: Subject, Black: Predicate).
       States an unconditional truth between subject and predicate

    2. If it rains, the streets get wet (Here It rains = antecedent,
       streets get wet = consequent). A Hypothetical proposition
       states a conditional link between antecedent and consequent
       propositions.

    3. Either A or B committed the murder.
       A disjucntive proposition says that one of two propositgions are
       true, but does not specify which one. The above really means that
       one of the two propositions (p,q) below is true:

       A committed the murder (p)
       B committed the murder (q)

       If p and q can both be true then it is called mutually
       non-exclusive proposition(alternative) proposition. If
       p and q cannot both be true then it is a mutually
       exclusive disjunctive proposition.

Categorical propositions are of 4 types : A, E, I and O

   Ex: A - All ravens are black (All S are P)
       E - No raven is white (No S is P)
       I - Some cats are black (Some S are P)
       O - Some cats are not black (Some S are not P)

Disrtibuted term : A term that refers to all members of a set.
  (e.g All professors, ALL mammals etc).

From definition of distributed term we conclude that:

* In A type propostions the subject is distributed (e.g Raven)
   but not the predicate (black)

* In E type proposition both subject and predicate are distributed.

* In I type proposition both subject and predicate are undistributed

* In O type proposition predicate(black) is distributed but subject
  is undistributed


Premise: A proposition which is assumed to be true to logically
infer a conclusion from them using valid rules of inference.

Note: It is very important that in a debate, all the premises used
by all the participants are accepted by all, or else the debate will
end up in a chaos just as it does if noncognitive sentences are used
in the debate. One must state his/her premises clearly for a
meaningful debate, if not already obvious from the context.

Syllogism : An example of three propositions, where the third one
is a valid conclusion inferred from the other two as premises.

It is of the form P1  (premise 1)
                  P2  (Premise 2)
       Therefore  C   (Conclusion)

  Major term: Predicate of the conclusion of the syllogism (P)
  Minor term: Subject ,, ,,     ,, ,,   ,,     ,,   ,,,    (S)

  Major Premise: Premise containing the major term
  Minor premise: Premise containing the minor term
  Middle term: Term common in the Major and Minor Premises. (M)

 example:

      P1: All professors are educated  (major premise)
      P2: Some Bengalis are professors (minor premise)
      C : Therefore, some Bengalis are educated (Middle term: professors)

  In the above: Major term = educated, minor term = Bengali
                Middle term = professor

 Valid Syllogisms:

  Since there are 4 different way of arranging the subject,
  predicate and middle term in the premises thus we have four
  different "figures" of syllogism viz:


  Figure 1       Figure 2         Figure 3        Figure 4
   M - S          S - M            M - S           S - M
   P - M          P - M            M - P           M - P

Hence  P - S     Hence  P - S     Hence P - S      Hence P - S


In each figure the premises and the conclusion can be of any of the 4
different types of categorical propositions viz A,E,I and O so we cab
have categorical syllogisms of the form AAA-1, AAA-2,.., AAE-1, AAE-2,..
EAO-1,... (256 total)

(Where -1,-2, etc refers to the figure type above)

But not all syllogisms will be valid. The rules of valid syllogism
make many of those syllogism invalid. The rules of valid
syllogism are based on the very basic rules of thought, which
are intuitively obvious as true, but are nevertheless important
in building up complex forms true statements, or by contradicting
these obviously true statements it is possible to build up false
complex statements which may not appear as obviously false, the
falseness may be hidden by the complexity the proposition.
The falsity of the complex proposition can only be seen as false
by breaking into simpler statements and seeing that one or more
of the simpler propositions violate one of these axioms.
These rules thus form the axioms of thought. These axioms are:

(1) Identity Law, i.e A=A, where A is the attribute of any
    object in an abstract sense. e.g black = black

(2) Law of non contradiction: "A" cannot be = "not-A", e.g black
    cannot be "not black". A Black ball cannot be a not black
    (e.g white, green etc).

(3) Law of Excluded Medium: If "X" is one of the possible/
    admissible attributes of "A", then A must be either "X" or
    "not X". here is no third possibility. A ball is either
    black or not black.

In debates we more often encounter complex statements, not the
simpler axiomatic statements. Since the complex statements
are not easily judged for its validity using these basic rules
of thought we have to utilize some higher level rules that
are derivable form these basic axioms but are more easily suited
to ascertain the validity of the complex statements. Therein
lies the importance of knowing the rules of logic(syllogism).
So let us state the valid rules of syllogism that are in turn
derivable from these basic axioms of thought:

  Rule-1: There can be only three terms in a syllogism (S,P,M) and
          the meaning of the major or minor terms cannot be changed
          between the premise and conclusion. A violation of the
          latter rule is called fallacy of equivocation.

  Rule-2: A valid syllogism must have the middle term distributed
          at least in one of the two premises.

         Fallacy of undistributed middle (When Rule-2 is violated):

         P1: All dogs are mammals
         P2: All cats are mammals
         C : Therefore ALL cats are dogs
         (Middle term mammal not distributed in either pr6emise.
          All mammals were not implied in either P1 or P2).

  Rule-3: A major term cannot be more distributed in the conclusion
         than in the major premise.

         Fallacy of Illicit major(When Rule-3 is violated):

         P1: All dogs are mammals
         P2: No cat is a dog
         C : therefore, No cat is a mammal

        (Mammal distributed in conclusion which says that ALL mammals
         are not cats, but not distributed in major premise P1)

  Rule-4: A minor term cannot be more distributed in the conclusion than
          in the minor premise.

        Fallacy of Illicit minor(When Rule-4 is violated):

        P1: All Caucasians are white
        P2: Some Amercans are Caucasians
        C : Therefore, All Americans are white

       (Minor term Americans distributed in conclusion, but not in
        minor premise P2).

  Rule-5: Premises of the syllogism cannot both be negative

  Fallacy of negative Premises(When Rule-5 is violated):

  P1: No PhD is a clerk
  P2: No wrestler is a PhD
  C : No/All wrestler is/is not a PhD (All 4 combinations)

The next six rules are actually derivable from the previous five
rules:

 Rule-6: In a valid syllogism, if any premise is negative, then so
         will be the conclusion. Violation of this rule gives rise
         to Fallacy of drawing Affirmative conclusion from a
         negative premise.

 Rule-7: I a valid syllogism, if both the premises are affirmative,
         then so will be the conclusion.

 Rule-8: In a valid syllogism at least one premise must be universal,
         i.e of A or E type.

 Rule-9: In a valid syllogism if one of the premise is particular
         (I or O type) then so will be the conclusion.

 Rule-10: In a valid syllogism it can never be the case that the
          major premise particular and the minor premise is negative.

 Rule-11: In a valid syllogism the conclusion of two universal
          premises cannot be particular.

Based on the rules above the valid forms of syllogisms reduce to the
following 15:

AAA-1, EAE-1, AII-1, EAO-1, EAE-2, AEE-2, EIO-2, AOO-2, IAI-3, AII-3,
OAO-3, EIO-3, AEE-4, IAI-4, EIO-4

Example of using the above to judge the validity of a syllogism

A: All snakes are reptiles (major Premise)
O: Some snakes are not dangerous animals (Minor Premise)
O: Therefore some dangerous animals are not reptiles (Conclusion)

The above syllogism is invalid (since it is of type AOO-3, not
among the 15 valid forms) It is inavlid beacuse it violates
Rule-3 (Fallacy of illicit major). Major term reptile is
undistributed in major premise than in the conclusion. (This is
also an example of arrving at a valid conclusion from an
invalid inference).

When one or more premise(s) of a syllogism are not categorical
propositions we can have the following 4 types of syllogisms:

( in the following -> = implies, v = "OR", ~ = "NOT"):

 (1) Modus Ponens

   A rule of inference of the form:

   p -> q  (If p then q, or p implies q)
   p
   -------
   therfore q

  Ex: "If Tuesday is the 14th, then Friday must be the 17th.
      Tuesday is the 14th. Therefore, Friday is the 17th."

  Fallacy due to mis-stating Modus Ponens is known
  as Fallacy of Affirming the Consequent(q):

   p -> q
   q
   -------
   therefore p

  A concrete example of Fallacy of Affirming the Consequent:

   If it is raining then there is cloud in the sky
   There is cloud in the sky
   Therefore it is raining

 (2) Modus Tollens

    A rule of inference of the form:

     p -> q
        ~ q
    -------
    therefore ~ p

  Ex: "If it had rained this morning, then the grass would
       still be wet But the grass is not wet. Therefore, it did
       not rain this morning."

    Fallacy of denying the antecedent in Modus Tollens is known
    as Fallacy of Denying the Antecedent:

     p -> q
         ~p
    --------
    therefore ~q

 (3) Hypothetical Syllogism

     A rule of inference of the form:

     p -> q
     q -> r
     -------
     therefore p -> r

   Ex: "If Debbie is promoted, then Gene will be, too. But if Gene
      is promoted, then Kim will be angry. Therefore, if Debbie is
      promoted, then Kim will be angry."

 (4) Disjunctive Syllogism)

    A rule of inference of the form:

     p v q
       ~ p (or q)
    ------
    therefore q (or p)

  Ex: "Either Ellen brought him to the party or Keith did. But
      Ellen didn't. So, Keith brought him to the party."

    Fallacious form of Disjunctive Syllogism:

     p v q
       p (or q)
    ------
    therefore ~q (or p)


 Readers are invited to come up with examples for the fallacious
 forms of 2 and 4 above (Home Work!). I already provided example
 of 1. There is no fallacious form of 3.

 It is important to realize that a valid reasoning can be based
 on false premises. Or conversely an invalid reasoning can be
 based on true premises. It must be borne in mind that with
 true premises, a valid inference will necessarily lead to a
 true conclusion, never a false conclusion. But with false
 premises, a valid inference can lead to either true or false
 conclusion. An invalid inference can lead to any combination of
 true and false premises and conclusions. In many debates it is
 seen thatreligious apologists state true premises from Science
 but come to false conclusions (Which they claim as true)
 through invalid inferences (By resorting to fallcies like red
 herring, false dichotomy, ignoratio elenchi etc). Another
 common fallacy is to state a conclusion but not based on any
 premise or through valid inferences, like non sequitar.

 On the other hand the fact that an argument is valid doesn't
 necessarily mean that its conclusion is true, as it may be
 based on false premises. This is also seen in the many
 arguments of theists where they state as premise such
 falsehoods as "Order in nature violates 2nd Law of
 Thermodynamics" when trying to refute evolution and defend
 creationism. That premise is a false statement of science.

A valid(Sound) argument is one that does not commit any fallacy
mentioned here or any of the other fallacies listed in my post
earlier and based on true premises. (Those fallacies are
ultimately reducible to the fallacies and violation of rules
covered in this discussion). So if any debator purports to be a
honest debator he/she has to be willing to subject his/her
arguments to the rules of logic and convince oneself and others
that no such rule was violated. Without this sincerity of the
debators, the debate really becomes meaningless for the
debators, but may still be informative for those spectators who
appreciate and understand rules of logic, who can figure out
which debator is correct or wrong. It is not expected that one's
logic will be consistently free from fallacies. One can
unwittingly commit fallacies. To err is human. A sincere
debator is expected to admit to having committed a fallacy
unwittingly. But it is not expected that one will commit
fallacies wittingly (To mislead the opponent and gain unfair
advantage in the debate) or deny having committed a fallacy
(using sophistry) even after being pointed out.