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Introduction to Mathematical Logic!
The rules of mathematical logic specify methods of reasoning mathematical statements. Greek philosopher, Aristotle, was the pioneer of logical reasoning. Logical reasoning provides the theoretical base for many areas of mathematics and consequently computer science. It has many practical applications in computer science like design of computing machines, artificial intelligence, definition of data structures for programming languages etc.
Major Categories
Mathematical logics can be broadly categorized into three categories.
Propositional Logic − Propositional Logic is concerned with statements to which the truth values, "true" and "false", can be assigned. The purpose is to analyse these statements either individually or in a composite manner.
Predicate Logic − Predicate Logic deals with predicates, which are propositions containing variables. A predicate represents an expression of one or more variables.
Rules of Inference − To deduce new statements from the statements whose truth that we already know, Rules of Inference are used. Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have.
Propositional Logic
A proposition is a collection of declarative statements that has either a truth value "true" or a truth value "false". A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc). The connectives connect the propositional variables.
Some examples of Propositions are given below −
- "Man is Mortal", it returns truth value TRUE
- "12 + 9 = 3 – 2", it returns truth value FALSE
The following is not a Proposition −
"A is less than 2". It is because unless we give a specific value of A, we cannot say whether the statement is true or false.
Predicate Logic
A predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable.
The following are some examples of predicates −
- Let E(x, y) denote "x = y"
- Let X(a, b, c) denote "a + b + c = 0"
- Let M(x, y) denote "x is married to y"
Rules of Inference
Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements.
An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypothesis). The symbol “$\therefore$”, (read therefore) is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises.
Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have.
For Example, If P is a premise, we can use Addition Rule of Inference to derive $ P \lor Q $.
$$\begin{matrix} P \ \hline \therefore P \lor Q \end{matrix}$$
Example
Let P be the proposition, "He studies very hard" is true
Therefore − "Either he studies very hard Or he is a very bad student." Here Q is the proposition "he is a very bad student".
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