See UNISA – Summary of 2010 Posts for a list of related UNISA posts. This post is one of three summary posts I will be building up over the next couple of months, so if you are following this topic or completing the same course as I this year, you may want to bookmark this post and come back occasionally for a peek and to give “candid” feedback.
** UNDER CONSTRUCTION ** Last change: 2010-01-27
This post contains a collection of examples, sorted alphabetically by title, referred to from UNISA Chatter – Formal Logic: Propositional Logic Summary.
ØÙÚ … these are copy|paste placeholders while we are working on this post.
Conjunctive Normal Form (CNF)
A sentence is in conjunctive normal form (CNF) is it is a conjunction of one or more disjunctions of one or more literals.
(A Ù B) Ú (C Ù D)
- [(A Ù B) Ú C] Ù [(A Ù B) Ú D]
- (A Ú C) Ù (B Ú C) Ù [(A Ù B) Ú D]
- (A Ú C) Ù (B Ú C) Ù (A Ú D) Ù (B Ú D) … CNF
Disjunctive Normal Form (DNF)
A sentence is in disjunctive normal form (DNF) is it is a disjunction of one or more conjunctions of one or more literals.
(A Ú B) Ù (C Ú D)
- [(A Ú B) Ù C] Ú [(A Ú B) Ù D]
- (A Ù C) Ú (B Ù C) Ú [(A Ú B) Ù D]
- (A Ù C) Ú (B Ù C) Ú (A Ù D) Ú (B Ù D) … DNF
Informal Proof of an argument
- We get the desired goal or conclusion if Carol is happy
- Either Max or Claire are home
- Max is not home or Carl is happy
- Claire is not home or Carl is happy
- --> So it must be that Carl is happy
X is logically true | necessary if and only if it is impossible for X to be false.
a=a is a great example.
The sentence Ø(Larger(a,b) Ù Larger (b,a)) cannot possibly be false, i.e. it is logically necessary, but not a tautology.
X and Y are logically equivalent if and only if it is impossible for either of them to be true and the other false.
Consider the sentences:
- a = b Ù Cube(a)
- a = b Ù Cube(b)
Proof 1 for logical equivalence:
- Suppose a = b Ù Cube(a) is true.
- Then, a = b is true and Cube(a) is true.
- Using indiscernibility of identicals or identity elimination (If b = c, then anything is true of b is also true of c) we know that Cube(b) is also true.
- Therefore a = b Ù Cube(a) logically implies the truth of a = b Ù Cube(b)
Proof 2 for logical equivalence:
- Suppose a = b Ù Cube(b) is true.
- Then, a = b is true and Cube(b) is true.
- Using symmetry of identity (If b = c then c = b) we know that b = a
- From this and Cube(b) we can conclude that Cube(a) is also true.
Test for tautology:
As shown, a logical equivalence is not necessarily tautological equivalent, although the reverse always applies.
X is a logical consequence of Y1, Y2, …Yn if and only if it is impossible for Y1, Y2, … Yn to be true and X is false.
Is A Ú C is a consequence of A Ù ØB and B Ú C?
In rows 1, 2, 3 and 7 the last sentence is true, together with the first and second. Therefore the last is a tautological consequence and therefore logical consequence of (1) and (2).
Negative Normal Form
Ø((A Ú B) Ù ØC) is not a NNF sentence.
- Ø(A Ú B) Ú ØØC
- Ø(A Ú B) Ú C
- (ØA Ù ØB) Ú C … is NNF
X and Y are tautologically equivalent if and only if there is no row of their joint table in which one value is true and the other is false.
Is A Ù B a consequence of A Ú B?
We note that the second sentence is a consequence of the first, but the first is not a consequence of the second.