Truth Tables (negation, conjunction, disjunction)

Negation and the truth table

For every proposition $p$p there exists another proposition $\sim p$~p meaning the negation of $p$p.

As an example we might write:

$p$p	$:$:	The plate is dirty
$\sim p$~p	$:$:	The plate is not dirty

Note that the negation of a negation becomes the original proposition. In symbols we could write that $\sim\left(\sim p\right)$~(~p) is the same as (or more formally, is equivalent to) $p$p.

There is a complementary relationship between the truth values of $p$p and $\sim p$~p. When $p$p is true, $\sim p$~p is false and when $p$p is false, $\sim p$~p is true. We can summarise this relationship into what is known as a truth table:

Truth Table Negation

$p$p	$\sim p$~p
$T$T	$F$F
$F$F	$T$T

Note that in logic there is no 'in between' state. A proposition is either true or not true. There can be no half truths. The English language is sometimes not so clear cut. For example, saying that you are not unhappy about leaving your school has a slightly different meaning to saying you are happy to leave your school. In any logical system we cannot allow for vague notions to creep in like this. Definitions must be precise so that our understandings are clear.

The symbol $\sim$~ is an example of an operator and there are many operators defined in the study of logic. 

 

Compound propositions

Two or more propositions can be joined with a connective (a connective is a type of operator) to become a compound proposition.

The conjunction connective

An example of a connective is what logicians call a conjunction, often symbolised as $\wedge$∧. Its meaning is the same as the English word "and" when used to indicate the conjoining of two ideas, such as "It's raining and I'm cold".

We have to be a little bit careful with the word "and" when used in other circumstances. For example the word "and" used in the phrase "I tripped and fell" is not a conjunction because we need $p\wedge q$p∧q to have the same sense as $q\wedge p$q∧p and the phrase "I fell and tripped" doesn't really make any sense. 

In addition, a phrase like "Paul left the party and Ed arrived at the party", if taken as a compound proposition, can imply no causality, so that we cannot assume that the reason Ed arrived is because Paul left.

The truth table for the precise meaning of $p\wedge q$p∧q is shown here. Because $p$p and $q$q each have two possible truth values, then we must consider all four combinations.

Truth Table Conjunction

$p$p	$q$q	$p\wedge q$p∧q
$T$T	$T$T	$T$T
$T$T	$F$F	$F$F
$F$F	$T$T	$F$F
$F$F	$F$F	$F$F

 

Note that the only instance where $p\wedge q$p∧q is true is when both $p$p and $q$q are both true. In the other three instances, $p\wedge q$p∧q is false.

This accords with our own understanding of the connective word "and" as used in an English sentence. For example we understand that both of the separate propositions "the sky is blue" and "the grass is green" need to be true if the compound proposition "the sky is blue and the grass is green" is to be true. If either one or both of the separate propositions is false, then the compound proposition is false also. 

The disjunction connective

Another important connective is the disjunction connective, symbolised by $\vee$∨. It is equivalent to the English word "or" in the inclusive sense. By inclusive we mean that the compound proposition $p\vee q$p∨q will be true if either $p$p or $q$q or both are true. In fact the only way $p\vee q$p∨q will not be true if $p$p and $q$q are both false. 

For example the compound proposition "the number is prime or the number is odd" means that either the number is prime, or else the number is odd, or else the number is both prime and odd.  

The English language use of the word or can be ambiguous. For example the word "or" in the sentence "I'll go to university or I'll get a low paying job" is used in an exclusive sense. What is really implied is contained in the sentence "I'll go to university or else I'll get a low paying job" - it's either one or the other but not both. In this case the disjunction connective $\vee$∨ doesn't apply (Incidentally there is another special connective for the "exclusive or" ).

Here is the truth table for the disjunction connective:

Truth Table Disjunction

$p$p	$q$q	$p\vee q$p∨q
$T$T	$T$T	$T$T
$T$T	$F$F	$T$T
$F$F	$T$T	$T$T
$F$F	$F$F	$F$F

 

Commutation and the effect of bracketing

In mathematics speak, and as we would expect, both the conjunction and disjunction connectives exhibit the commutative property - namely that $p\wedge q$p∧q is equivalent to $q\wedge p$q∧p and $p\vee q$p∨q is equivalent to $q\vee p$q∨p.  Swapping the order of the individual propositions should make no difference to the sense of the compound proposition. 

In addition, we can create compound propositions that use both connectives. As an example, consider the following propositions:

$p$p	$:$:	John is having roast beef for dinner
$q$q	$:$:	John is having Yorkshire pudding for dinner
$r$r	$:$:	John is having dessert for dinner

From these we could form a number of compound propositions that involve all three simple propositions.

For example consider the compound proposition  $\left(p\vee q\right)\wedge w$(p∨q)∧w.

It would read:

 John is having roast beef or Yorkshire pudding, and having dessert for dinner

Note that the main connective is a conjunction. The expression is a conjunction of two propositions, the first of which is a compound proposition (formed by a disjunction). Remember that the disjunction is inclusive in that John could end up eating the roast beef, the Yorkshire pudding and the dessert.  

We could also write a proposition with $p$p,$q$q and $r$r in the same order, the disjunction and conjunction connectives in the same order, but the bracketing grouped around the last two propositions, so that our compound proposition becomes  $p\vee\left(q\wedge r\right)$p∨(q∧r). 

It now reads differently:

John is having roast beef, or he is having Yorkshire pudding and dessert

What has happened is that the main connective is now a disjunction. The expression is a disjunction of two propositions with the second proposition a compound proposition formed by a conjunction.

This example illustrates how bracketing may have a profound effect on the meaning of the statement. 

 

Determining truth tables for other expressions 

With the truth tables for negation ($\sim$~), conjunction ($\wedge$∧) and disjunction ($\vee$∨) we are now in a position to construct truth tables for compound propositions that include one, two or all three connectives. We give two examples to illustrate the process. 

 

Examples

There are short cut methods to find the truth values of this compound proposition, but for now our strategy will be to progressively create the column that we want.

Example 1

Suppose we wanted to determine the truth table for the compound proposition given by $\sim p\vee q$~p∨q. 

We start with columns for $p$p and $q$q, then we need a column for the negation of $p$p and then we need a column for the disjunction of the negation of $p$p with the proposition $q$q. 

So, in the expanded version, we create this table:

$p$p	$q$q	$\sim p$~p	$\sim p\vee q$~p∨q
$T$T	$T$T	$F$F	$T$T
$T$T	$F$F	$F$F	$F$F
$F$F	$T$T	$T$T	$T$T
$F$F	$F$F	$T$T	$T$T

To check this, look at how $\sim p$~p has truth values that are the opposite of $p$p. Then columns 2 and 3 are used to generate the truth values of the column we need. Note that for a disjunction to be false, both $\sim p$~p and $q$q must be false.

Example 2

As another example, we could determine the truth table for $p\vee\left(\sim q\wedge r\right)$p∨(~q∧r)  with the following columns. Note that we now need $2^3=8$23=8 rows to cover all the possibilities of truth values for $p$p, $q$q and $r$r:

$p$p	$q$q	$r$r	$\sim q$~q	$\sim q\wedge r$~q∧r	$p\vee\left(\sim q\wedge r\right)$p∨(~q∧r)
$T$T	$T$T	$T$T	$F$F	$F$F	$T$T
$T$T	$T$T	$F$F	$F$F	$F$F	$T$T
$T$T	$F$F	$T$T	$T$T	$T$T	$T$T
$T$T	$F$F	$F$F	$T$T	$F$F	$T$T
$F$F	$T$T	$T$T	$F$F	$F$F	$F$F
$F$F	$T$T	$F$F	$F$F	$F$F	$F$F
$F$F	$F$F	$T$T	$T$T	$T$T	$T$T
$F$F	$F$F	$F$F	$T$T	$F$F	$F$F

Remember that for a conjunction proposition to be true, both parts must be true, and for a disjunction proposition to be true at least one of the parts must be true. Check each column very carefully to make sure you understand how it was constructed.

Worked Examples

Question 1

Question 2

Question 3

Question 4