1Logic

Review of Basic Concepts in Logic
Perhaps the greatest paradox of all is that there are paradoxes in mathematics.
—E. Kasner and J. Newman Mathematics and the Imagination (1940)

Some Mathematical Symbols

for all	such that	implies
there exists	there does not exist	does not imply
		iff if and only if

Definition. A proposition is a statement that is either true or false, but cannot be both true and false. Propositions must be decidable in a given context; that is, one must be able to determine whether the proposition is true or false.

Examples.

Propositions	Not Propositions
An equilateral triangle is isosceles.	Which quadrilateral is a rectangle?
Two lines perpendicular to the same line are parallel.	x² = 36.
MSU Moorhead is in Minnesota.	The person is tall.

Definition. A statement that is neither true nor false is called a paradox.

Example. This sentence is false.

In logic, we often represent simple propositions by letters such as p or q. Frequently, these simple propositions are connected by logical connectives to form compound statements.

Definitions and Examples.

Symbol	Name	Read	Truth Value	Example
	conjunction	p and q	True when both conjunctives p, q are true, otherwise false.	1 + 1 = 2 and 3 < 4.
	disjunction	p or q	True when at least one of the disjunctives p, q is true.	3 > 2 or 3 < 2.
~p	negation	not p	True iff p is false.	It is not true that 1 + 1 = 1.

Note the disjunction is inclusive, also called inclusive or, i.e. both p and q may be true for to be true. In the English language, we do not distinguish between inclusive and exclusive or, but in mathematics we need to have greater clarity; hence, mathematicians needed to choose between inclusive or and exclusive or. Examples from spoken English are: (1) Is the answer true or false? (exclusive or since only one answer may be given.) (2) Do you want ketchup or mustard? (inclusive or since a person may choose one or the other or both.)

Examples.
(a) 1 + 1 = 2 and π is rational. (false)
(b) or . (false)

(d) It is false that 0 is negative. (true)