Propositions

PROPOSITIONS: Mathematics is the study of the properties of mathematical structures. A mathematical structure is defined by a set of “axioms”. An “axiom” is a true statement about the properties of the structure.

“Logic” is the discipline that deals with the methods of reasoning. It gives a set of rules and techniques to determine whether a given argument is valid or not. True assertions which can be inferred from the truth of axioms are called “theorems”. A “proof” of a theorem is an argument that establishes that the theorem is true for a specified mathematical structure.

A “proposition” or “statement” is any declarative sentence which is true (T) or false (F). We refer to T or F as the truth value of the statement.

Propositional calculus is the calculus of propositions. Some illustrations below explain the concept well.

(a)    The sentence “3 3 = 6” is a statement, since it can be either true or false. Since it happens to be a true statement, its truth value is T.

(b)    The sentence “2 = 0” is also a statement, but its truth value is F.

(c)    “It will rain tomorrow” is a proposition. For its truth value, we shall have wait for tomorrow.

(d)    “Solve the following equation for y” is not a statement, since it cannot be assigned any truth value whatsoever. It is an imperative, or command, rather than a   declarative statement.

(e)    The Liar’s Paradox: “This statement is false” gets us into a bind:

If it were true, then since it is declaring itself to be false, it must be false. On the other hand, if it were false, then its declaring itself false is a lie, so it is true! In other words, if it is true, then it is false, and if it is false, then it is true, and we go around in circles. We get out of this bind by refusing to accord it the privileges of statementhood. In other words it is not a statement. An equivalent pseudo statement is “I am lying”, so we call this liar’s paradox. Such sentences are called “self-referential” sentences, since they refer to themselves.

We use the letters p, q, r, s, ...... for propositions. Thus for example, we might decide that P should stand for the proposition “The earth is round”. Then we shall write

p: “the earth is round” to express this. We read this

p is the statement “the earth is round”.