Aristotelian logic
Aristotelian logic was pioneered by Aristotle. Although it is possible that Aristotle was taught by someone else, the earliest study of reasoning can be attributed to Aristotle. Aristotle and his followers held that two of the most important principles of logic are the law of non-contradiction and the law of excluded middle. This kind of logic is now given various names to distinguish it from more recent systems of logic, e.g., Aristotelian logic or classical two-valued logic.
The law of non-contradiction states that no proposition is both true and false and law of excluded middle states that a proposition must either be true or false. In combination, these laws require two truth values that are mutually exclusive. A proposition can be either true or false, but cannot be both at the same time.
Some have considered classical logic to be just like a mathematical theory, and in particular the laws of non-contradiction and the excluded middle to be simply axioms of the theory, which have to be assumed without proof. In fact this is not so:
- Assume the law of non-contradiction is false. This means it can still be true, so therefore it is true (it is only the law of non-contradiction that prevents " can be" from necessarily becoming "is"). Therefore classical logic still remains valid.
- Assume the law of the excluded middle is not true. It does not follow that the law of the excluded middle is false, or indeed that any other proposition of classical logic which was true is now false.
- More generally, consider the proposition: "The validity of Rule X is fundamental to the validity of logic. Unless you assume the validity of Rule X, logic is not valid". Now assume that Rule X (whatever it might be) is false. The conclusion that logic is not valid has to follow by logical reasoning. But if logic is not valid, this reasoning is also invalid, and the conclusion cannot be drawn. Thus, the validity of logic is independent of the assumption of validity of any of its supposed laws. (This is an argument by self-reference.)
A better way to look at these laws is that, without them, the logic still remains valid, but a whole lot of illogic becomes valid as well. Thus, those laws are simply filters for stripping away the illogic, and leaving only the part that doesn't depend on them—the logic.
Formal logic
See also Propositional calculus
Formal logic, also called symbolic logic, is concerned primarily with the structure of reasoning. Formal logic deals with the relationships between concepts and provides a way to compose proofs of statements. In formal logic, concepts are rigorously defined, and sentences are translated into a precise, compact, and unambiguous symbolic notation.
Some examples of symbolic notation are:
- p: 1 + 2 = 3
This statement defines p is 1 + 2 = 3 and that is true.
Two propositions can be combined using the operations of conjunction, disjunction or conditional. These are called binary logical operators. Such combined propositions are called '\compound proposition's. For example,
- p: 1 + 1 = 2 and "logic is the study of reasoning."
In this case, and is a conjunction. The two propositions can differ totally from each other.
In mathematics and computer science, one may want to state a proposition depending on some variables:
- p: n is an odd integer.
This proposition can be either true or false according to the variable n.
A proposition with free variables is called propositional function with domain of discourse D. To form an actual proposition, one uses quantifiers. For every n, or for some n, can be specified by quantifiers: either the universal quantifier or the existential quantifier. For example,
- for all n in D, P(n).
This can be written also as:
The standard situation in