Mathematical Logics
There are two type of reasoning: Verbal reasoning
and non verbal reasoning. In verbal reasoning we use numbers alphabets to solve
the problems, but in non verbal reasoning we use some diagrams, figures,
pictures, etc. Mathematical logic is concerned with methods of reasoning. The
Greek philosopher and scientist “Aristotle” is said to be the first person to
have studied logical reasoning. Logical reasoning is the essence of mathematics
and is therefore an important starting point for study of discrete mathematics.
Logics among other things, have provided the theoretical basis for many areas
of computer science such as digital logic design, Automata theory, and computability,
and artificial intelligence etc. The one component of logic is proposition
calculus, which deals with statements with value true or false and is concerned
with analysis of propositions. And the other part is predicates calculus, which
deals with the predicates which are propositions containing variables. In this blog
we discuss some propositions and their applications in daily life.
Propositions
Before define propositions, a reader should know
the sentence and statements.” A number of words making a complete grammatical
structure having a sense and meaning in logical mathematics is called a
sentence. The mathematical formulation of sentence is.
Sentence = Subject + verb + object. The sentence
may be classified in to two categories in the
view of logic, Declarative or non declarative sentence. ‘A proposition or statement
is a declarative sentence that is either true or false, for example,” four plus five equal to nine” And “ three plus
three is equal to six” are both statements because both statements has true
value. If” two plus four is equal to seven” And “seven plus three is equal to
twelve “are both statements because they have false value. Therefore a
statement and proposition is a sentence having values. This value may be true
or false. Similarly “x + y> 1 “ is not a statement because for some values
of x and y the sentence is true, whereas for others it is false.. For Instance
if x =1 and y =2, the sentence is true, If x =-3 and y =1, this is false. The
truth or falsity of a statement is truth value. Since only two possible truth
values are possible in this logic called two valued logic. Questions
exclamations and commands are not propositions or statements. For Example.
(b) 3 +2 = 5
(c) (5,6) is a subset of (7,6,5)
(d) Do you speak Hindi ?.
(e) Close the door.
(f) What the hot day.
(g) We shall have chicken for dinner.
Here the sentence
(a), (b), and (c), are propositions or statements, the first is false and
second and third are true. (d) is a question, not a declarative sentence, hence
it is not a statement. (e) is a declarative sentence, but not a statement,
since it is true or false depends on the value of x.
(f), is not a statement,
because it is a command.
(g), is not a
statement, it is exclamation.
(h), is a
statement, since it is either true or false but not both, although one has to
wait until dinner to find out if it is true or false.
Dr. Hakimuddin Khan
AssociateProfessor
Dept. of Management Studies
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