Logic , Form and Validity and Fallacies

Logic

The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basic of logic is the syllogism, consisting of a major and a minor premise and a conclusion

• Major Premise: Thirty men can do a piece of work thirty times as quickly as one man.
• Minor Premise: One man can dig a posthole in thirty seconds;
• therefore-Conclusion: Thirty men can dig a posthole in one second.

Syllogisms are arguments that take several parts, typically with two statements which are assumed to be true (or premises) that lead to a conclusion. There are three major types of syllogism:

• Conditional syllogism: If A is true then B is true (If A then B).
• Categorical syllogism: If A is in C then B is in C.
• Disjunctive syllogism: If A is true, then B is false (A or B).

The Structure of Syllogism: A categorical syllogism is an argument consisting of exactly three categorical propositions (two premises and a conclusion) in which there appear a total of exactly three categorical terms, each of which is used exactly twice. One of those terms must be used as the subject term of the conclusion of the syllogism, and we call it the minor term of the syllogism as a whole. The major term of the syllogism is whatever is employed as the predicate term of its conclusion. The third term in the syllogism doesn't occur in the conclusion at all, but must be employed in somewhere in each of its premises; hence, we call it the middle term.

Since one of the premises of the syllogism must be a categorical proposition that affirms some relation between its middle and major terms, we call that the major premise of the syllogism. The other premise, which links the middle and minor terms, we call the minor premise.

Consider, for example, the categorical syllogism:
No geese are felines.
Some birds are geese.
Therefore, Some birds are not felines.

Clearly, "Some birds are not felines" is the conclusion of this syllogism. The major term of the syllogism is "felines" (the predicate term of its conclusion), so "No geese are felines" (the premise in which "felines" appears) is its major premise. Simlarly, the minor term of the syllogism is "birds," and "Some birds are geese" is its minor premise. "geese" is the middle term of the syllogism.

Standard Form of A Syllogism -
A categorical syllogism in standard form always begins with the premises, major first and then minor, and then finishes with the conclusion.

The mood of a syllogism is simply a statement of which categorical propositions (A, E, I, or O) it comprises, listed in the order in which they appear in standard form. Thus, a syllogism with a mood of OAO has an O proposition as its major premise, an A proposition as its minor premise, and another O proposition as its conclusion; and EIO syllogism has an E major premise, and I minor premise, and an O conclusion; etc.

In addition to mood, a syllogism is characterized by its figure which is solely determined by the position in which its middle term appears in the two premises: in a first-figure syllogism, the middle term is the subject term of the major premise and the predicate term of the minor premise; in second figure, the middle term is the predicate term of both premises; in third, the subject term of both premises; and in fourth figure, the middle term appears as the predicate term of the major premise and the subject term of the minor premise.
There are exactly 256 distinct forms of syllogisms, out of which only 24 are valid.

Form and Validity

Validity of a categorical syllogism depends solely upon its logical form. The rules for deciding the validity of syllogism are:

Rule 1: The middle term must be distributed at least once.
Rule 2: If a term is distributed in the conclusion, then it must be distributed in a premise.
Rule 3: Two negative premises are not allowed.
Rule 4: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise
Rule 5: If both premises are universal, the conclusion cannot be particular.

Fallacies

Fallacies arise when one of the rules are broken, for example, following fallacies arise when these rules are broken (and the explanation given in the brackets

Rule 1: Undistributed middle (The middle term is what connects the major and the minor term. If the middle term is never distributed, then the major and minor terms might be related to different parts of the M class, thus giving no common ground to relate S and P.)
example - All sharks are fish
All dolphins are fish
Therefore All dolphins are sharks

Rule 2:
Illicit major and illicit minor (When a term is distributed in the conclusion, let’s say that P is distributed, then that term is saying something about every member of the P class. If that same term is NOT distributed in the major premise, then the major premise is saying something about only some members of the P class. Remember that the minor premise says nothing about the P class. Therefore, the conclusion contains information that is not contained in the premises, making the argument invalid.)
example - All horses are animals
Some dogs are not horses
therefore, Some dogs are not animals

and, All tigers are mammalians
All mammalians are animals
Therefore All animals are tigers

Rule 3: Exclusive premises (If the premises are both negative, then the relationship between S and P is denied. The conclusion cannot, therefore, say anything in a positive fashion. That information goes beyond what is contained in the premises.)
example - No fish are mammals
Some dogs are not fish
Therefore Some dogs are not mammals

Rule 4:
Drawing an affirmative conclusion from a negative premise, or drawing a negative conclusion from an affirmative premise. (Two directions, here. Take a positive conclusion from one negative premise. The conclusion states that the S class is either wholly or partially contained in the P class. The only way that this can happen is if the S class is either partially or fully contained in the M class (remember, the middle term relates the two) and the M class fully contained in the P class. Negative statements cannot establish this relationship, so a valid conclusion cannot follow. Take a negative conclusion. It asserts that the S class is separated in whole or in part from the P class. If both premises are affirmative, no separation can be established, only connections. Thus, a negative conclusion cannot follow from positive premises.Note: These first four rules working together indicate that any syllogism with two particular premises is invalid.)

example - All parrots are birds
Some tigers are not parrots
Therefore Some tigers are birds

Rule 5: Existential fallacy (On the Boolean model, Universal statements make no claims about existence while particular ones do. Thus, if the syllogism has universal premises, they necessarily say nothing about existence. Yet if the conclusion is particular, then it does say something about existence. In which case, the conclusion contains more information than the premises do, thereby making it invalid.)
example - All mammals are animals
All tigers are mammals
Therefore Some tigers are animals