- published: 15 Sep 2009
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Substitution is a fundamental concept in logic. A substitution is a syntactic transformation on formal expressions. To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols by other expressions. The resulting expression is called a substitution instance of the original expression.
Where Ψ and Φ represent formulas of propositional logic, Ψ is a substitution instance of Φ if and only if Ψ may be obtained from Φ by substituting formulas for symbols in Φ, always replacing an occurrence of the same symbol by an occurrence of the same formula. For example:
is a substitution instance of:
and
is a substitution instance of:
In some deduction systems for propositional logic, a new expression (a proposition) may be entered on a line of a derivation if it is a substitution instance of a previous line of the derivation (Hunter 1971, p. 118). This is how new lines are introduced in some axiomatic systems. In systems that use rules of transformation, a rule may include the use of a substitution instance for the purpose of introducing certain variables into a derivation.
Substitution may refer to:
Logic (from the Ancient Greek: λογική, logike) is the branch of philosophy concerned with the use and study of valid reasoning. The study of logic also features prominently in mathematics and computer science.
Logic was studied in several ancient civilizations, including Greece, India, and China. In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric. Logic was further extended by Al-Farabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication. In the East, logic was developed by Hindus, Buddhists and Jains.
Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning.
The concept of logical form is central to logic. The validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic.
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