Compound propositions

Logic: Propositional logic

Compound propositions

Up to this point, we have seen propositions and logical operators. With these two ingredients, we can build compound propositions.

Simple and compound propositions

A proposition is called a simple proposition if it does not contain any of the logical operators #\neg#, #\lor#, #\land#, #\rightarrow#, and #\leftrightarrow#.

A proposition is called a compound proposition if it contains one or more of the logical operators #\neg#, #\lor#, #\land#, #\rightarrow#, and #\leftrightarrow#.

Examples of simple propositions

#\blue{\textrm{"I go to the gym."}}#
#\blue{\textrm{"I go to the swimming pool."}}#
#\blue{\textrm{"I take a shower."}}#

Example of a compound proposition
#\blue{\textit{"If}\textrm{ I go to the gym }\textit{or}\textrm{ the swimming pool,}} #
#\blue{\textit{then}\textrm{ I take a shower."}}#

No compound proposition without simple ones

Each compound proposition has at least one simple proposition. For if it has a logical operator the arguments of this operator must be propositions with fewer logical operators. Continuing down we must arrive at a simple proposition contained in the compound proposition.

In the example \[\blue{\textit{"If}\textrm{ I go to the gym }\textit{or}\textrm{ the swimming pool, }\textit{then}\textrm{ I take a shower."}}\] three simple propositions occur:

#\blue{\textrm{I go to the gym.}}#
#\blue{\textrm{I go to the swimming pool.}}#
#\blue{\textrm{I take a shower.}}#

Verum and falsum

We know that propositions may take both values true and false, like #\blue{\textrm{"It is snowing."}}#. For practical use we will fix the following two simple propositions.

The proposition #\top# always takes the value true.
The proposition #\bot# always takes the value false.

We write the propositions with special symbols in order to distinguish them from the values true and false.

For any proposition #\blue p#, the compound proposition #\blue p\lor \blue{\neg{p}}# always takes the value true. There are many other propositions that are always true. Later we will give a special name to such propositions: verum.

Similarly the compound proposition #\blue p\land \blue{\neg{p}}# always takes the value false. Such a proposition will be called a falsum.

Select all compound propositions from the options below.

Where can I find the bread and the soup?
If grass is not green, it is brown.
If my fingers become purple, I go to a doctor.
Elephants can fly.

The propositions "If grass is not green, it is brown" and "If my fingers become purple, I go to a doctor" are compound expressions.

The expression "Where can I find the bread and the soup?" is not a proposition, and therefore it is not a compound proposition either.

The expression "Elephants can fly" is a simple proposition.

New example