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
- Premises
- Either Max or Claire are home
- Max is not home or Carl is happy
- Claire is not home or Carl is happy
- If we assume in premise 1, that Max is home, then Claire is not.
- In premise 2 we are saying that either Max is not home or Carl is happy.
- --> So it must be that Carl is happy

## Logical Truth

*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.

## Logical Equivalence

*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.

## Logical Consequence

*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. *

Example:

Is A **Ú** C is a consequence of A **Ù** **Ø**B 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** **Ø**C) is not a NNF sentence.

…

**Ø**(A**Ú**B)**Ú****Ø****Ø**C**Ø**(A**Ú**B)**Ú**C**(ØA**… is NNF**Ù**ØB) Ú C

## Tautological Consequence

*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.*

Example:

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.

## Tautology

* (1) is a tautology if and only if every row of its truth table is true.*

## Tautology Equivalence

*(1) and (2) 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.*

show without using truth table:

~(pv(~p and q))=~p and q

I took Formal Logic 3 2011 semester 1 (still waiting for results). After googeling for days (read "hours") for an explanation of Logical Consequence, Tautological Consequence, Tautology Equivalence I only finally understood it after reading your explanation. Thank you.