4. Symbolic Logic
Symbolic logic, which is a development of formal logic, is an attempt to apply the correct method of judgment by using mathematical symbols. Symbolic logic contrasts with formal logic in certain important ways. In formal logic, the subject matter is the relationship of implications between terms, that is, the relationship of implications between the subject and the predicate in a proposition. In contrast, symbolic logic focuses on the connection between terms, or between propositions, and its subject-matter is the study of the laws of thought through the use of mathematical symbols.
The five basic forms of connection between propositions are as follows (where p and q are two propositions):
| ⅰ) Negation | “not-p” . . . . ∼p (or p̄) |
| ⅱ) Disjunction | “p or q” . . . . p ⋎ q |
| ⅲ) Conjunction | “p and q” . . . . p ● q |
| ⅳ) Implication | “IF p then q” . . . . p ⊃ q |
| ⅴ) Equivalence | “p equals q” . . . . p ≡ q |
Through the combination of these five basic forms, any complicated deductive inference can be accurately expressed. For example, the basic laws of formal logic, namely, the law of identity, law of contradiction, and law of the excluded middle, can be symbolized as follows:
| Law of identity | p ⊃ q or p ≡ q |
| Law of contradiction | ∼(p ● ∼p) or (p ● p̄) |
| Law of excluded middle | p ⋎ ∼p or p ⋎ p |
Philosophers often proposed extensive thought systems, but the question is whether or not their logical constructions are correct. In order to ascertain their correctness, we need to use mathematical symbols and make calculations. Symbolic logic came into being from such a point of view.