For the purposes of this post I will be dealing mainly with one of the simpler and most commonly used forms of logic, Predicate Logic.
First some terms:
Valid. A logical argument is valid if the conclusion follows from the premises. That is, the structure of the argument is such that if the premises are true the conclusion must be true. Any valid logical argument is tautologous.
Sound A logical argument is sound if it is valid and then premises are true.
Logic, then, is a formal method of drawing out the necessary implications of the premises. And for that it requires that all the premises are stated clearly and precisely. Any equivocation on the meaning of the premises invalidates the argument. Leaving premises unstated invalidates the argument.
We must also be careful of conflating logic in the strict sense with rational argument. Induction is a rational argument but it falls short of the standards of logic. Appealing to a genuinely knowledgable authority is a rational argument but it is not logcal in the strict sense.
The basics of Predicate Logic are as follows.
Predicate logic can be described as an algebra of truth. As with algebra the operations of Predicate Logic can be worked out using symbols, substituting in the premises of a particular argument as required.
Predicate logic has the following operations
1) Conjunction or "AND". The conjunction of two statements is TRUE if and only if both statements are TRUE.
2) Disjunction or "OR". The disjunction of two statements is TRUE if and only if either or both are TRUE.
3) Negation or "NOT". The negation of a statement is TRUE if and only iff the statement is FALSE.
4) Implicationor "IF...THEN". "A implies B" is FALSE only if A is TRUE and B is FALSE. This leads to the intuitively odd result that a falsehood implies anything.
Predicate logic also includes quantification - we can say "For all" or "For some".
Predicate Logic has 3 axioms, sometimes called the "laws of logic".
1) Identity - a statement has the same truth value as itself
2) The excluded middle. For any statement A "A OR (NOT A)" is TRUE.
3) Non-Contradiction For any statement A "A AND (NOT A)" is FALSE
[Note to Admins. I'm not sure where this should go. Please use your best judgement]
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