Scope of Logic

Scope of logic

As it has developed, many distinctions have been introduced into logic. 
These distinctions serve to help formalize different forms of logic as a 
science. Here are some of the more important distinctions.


Deductive and inductive reasoning

Originally, logic consisted only of deductive reasoning which concerns what 
follows universally from given premises. However it is important to note that 
inductive reasoning—the study of deriving a reliable generalization from 
observations—has sometimes been included in the study of logic. 
Correspondingly, we must distinguish between deductive validity and inductive 
validity. An inference is deductively valid if and only if there is no 
possible situation in which all the premises are true and the conclusion 
false. The notion of deductive validity can be rigorously stated for systems 
of formal logic in terms of the well-understood notions of semantics. 
Inductive validity on the other hand requires us to define a reliable 
generalization of some set of observations. The task of providing this 
definition may be approached in various ways, some less formal than others; 
some of these definitions may use mathematical models of probability. For 
the most part our discussion of logic deals only with deductive logic.


Formal and informal logic

The study of logic is divided into formal and informal logic.

Formal logic (sometimes called "symbolic logic") attempts to capture the 
nature of logical truth and inference in formal systems, which consist of a 
formal language, a set of rules of derivation (often called "rules of 
inference"), and sometimes a set of axioms. The formal language consists of 
a (often small) set of discrete symbols, a syntax, and (often) a semantics, 
and expressions in this language are often called "formulas". The rules of 
derivation and potential axioms then operate with the language to specify a 
set of theorems, which are formulas that are either axioms or are derivable 
using the rules of derivation. In the case of formal logical systems, the 
theorems are often interpretable as expressing logical truths (tautologies), 
and in this way can such systems be said to capture at least a part of 
logical truth and inference. Formal logic encompasses a wide variety of 
logical systems. For instance, propositional logic and predicate logic are 
a kind of formal logic, as well as temporal logic, modal logic, Hoare logic, 
the calculus of constructions, etc. Higher-order logics are logical systems 
based on a hierarchy of types.

Informal logic is the study of logic as used in natural language arguments. 
Informal logic is complicated by the fact that it may be very hard to tease 
out the formal logical structure embedded in an argument. Informal logic is 
also more difficult because the semantics of natural language assertions is 
much more complicated than the semantics of formal logical systems, due to 
the presence of such phenomena as defeasibility.