Content

Propositions

There are two aims for the rest of this module. These are to find tautologies and to simplify logical statements. When are linking statements true for all truth values of the propositions involved and how can we make the statements less obfuscating? Can we simplify the statements? We move simultaneously toward these goals.

In https://www.iep.utm.edu/prop-log/, we have the following definition: Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form other propositions, statements or sentences. Hopefully these new propositions are of some value to us.

Reading between the lines you will see that propositions are sometimes called sentences or statements. Elsewhere they may be called cases. Anyway by a proposition we mean a statement that can be assigned a truth value of $T$, if it is true and $F$, if it is false. Because we want to combine these from time to time in algebraic type ways, we'll mostly use the value 1 for $T$ and 0 for $F$.

Going back to the last section we know that the following are propositions:

  1. $\hspace{10px}$These lines are parallel;
  2. $\hspace{7px}$This angle is $30^{\circ}$;
  3. $\hspace{3px}$Jake has a knife;
  4. $\hspace{3px}$Jake killed the victim;
  5. $\hspace{7px}$All men are liars;
  6. $\hspace{4px}$Helen is a liar;
  7. Helen is a man.

Not all sentences are propositions though. Look at these. Do any of them lack an important property?

  1. Go home!
  2. $\hspace{8px}$Can I have a drink?
  3. $\hspace{12px}$Advance Australia fair.
  4. $\hspace{8px}$This sentence is false.
  5. $\hspace{4px}$These peas are horrible.

Questions

  1. Consider the sentences (viii) to (xii) above and say why each is a proposition or not.

It may already have entered your head that there is a similarity here with Boolean algebra. The emphasis that we have so far on zeros and ones may have got your antennae going. Let us move further in this direction by defining the negation of a proposition. If $p$ is a proposition, then the negation of $p$, written $p'$, is true when $p$ is false and false when$ p$ is true. This can be shown in a truth table, see Table 5. A truth table is one where we give all possible truth vales to all propositions and determine the resulting truth values of all of the operations on these truth values.

Table 5: The truth table for negation
$p$ $p'$
0 1
1 0

Questions

  1. Write down the negations of propositions (i) to (vii) and (xii) in the text above.

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