Rational Mathematics

Logic Algebra

A compound proposition consists of other propositions combined in such a way that the truth value of the compound can be unambiguously determined if the truth values of its component propositions are known. The ways in which propositions can be compounded are known as logical connectives. I call a logical connective L genuine if the truth value of L(p, q, r, ...) depends on the truth values of all of its components. A proposition that is true whatever the truth values of its components, or is false whatever the truth values of its components, is known as a tautology. A tautology is ungenuine.

The four possible logical connectives on a single proposition p can be shown in truth table form as follows:

p		T(p)	P(p)	N(p)	F(p)
1		1	1	0	0
0		1	0	1	0

The propositions T(p) and F(p), which are respectively true and false regardless of the value of p, are tautologies; they remain true or false if any other proposition is substituted for p. There are only two genuine logical operators: P and N. The proposition P(p), which is true when p is true and false when p is false, is the proposal of p. Symbolically P(p) = p. The proposition N(p), which is true when p is false and false when p is true, is the negation of p. It says ‘p is false’. We cannot have N(p) = p, since a proposition by definition cannot be both true and false. A statement that is both true and false is known as a self-contradiction. Two negations cancel out to produce an assertion: N(N(p)) = p.

On two propositions, p and q, there are 16 possible logical operations: Of these 16 there are 8 that are true and 8 that are false when p and q are both true, we will call these positive and negative operations respectively. The truth tables for the negative operations are found by replacing 0 by 1 and 1 by 0 in the truth tables for the corresponding positive operations.

Of the 8 positive operations on two components three are not genuine operations (one is true regardless of the values of p and q, one has the same truth values as p, and one has the same truth values as q). The 5 genuine positive operations are conveniently denoted by the vowels A, E, I, O, U. The 5 genuine negative operations can be correspondingly denoted by the same letters with hats: Â, Ê, Î, Ô, Û (or bars but I could't find how to do those in HTML.) Their values can be exhibited in the following truth tables:

p	q		pAq	pEq	pIq	pOq	pUq		pÂq	pÊq	pÎq	pÔq	pÛq
1	1		1	1	1	1	1		0	0	0	0	0
1	0		0	0	0	1	1		1	1	1	0	0
0	1		0	0	1	1	0		1	1	0	0	1
0	0		0	1	1	0	1		1	0	0	1	0

These operations can be translated into corresponding ordinary English words, as indicated below, although care is of course needed in the converse operation of translating English arguments into the logical notation, since there are many variations of phrasing and ambiguities in the normal English usage.
pAq = ‘p and q’ (conjunction)
pEq = ‘p is logically equivalent to q’ or ‘p if and only if q’ (biconditional).
pIq = ‘p implies q’ or ‘if p then q’ (implication or conditional)
pOq = ‘p or q or both’ (disjunction or the inclusive or).
pUq = ‘p is implied by q’ (converse implication)
pÂq = ‘not both p and q’ (nand).
pÊq = ‘p or q but not both’ (the exclusive or).
pÎq = ‘not p implies not q’ (inverse implication).
pÔq = ‘neither p nor q’ or ‘not p and not q’.
pÛq = ‘p is false unless q is true’ (contrapositive implication).

Some more long-winded phrases are sometimes used to express the above propositions. For example pIq = ‘p is a sufficient condition for q’, pUq = ‘p is a necessary condition for q’, and pEq = ‘p is a necessary and sufficient condition for q’.