Propositions as variables

Logic: Propositional logic

Propositions as variables

Now that we have seen all basic ingredients of logic we introduce a more efficient way to write down propositions. We have seen that compound propositions can become very long sentences, which is not ideal. Earlier we already used the letters #\blue p# and #\blue q# to abbreviate propositions. Here we introduce proposition letters, which are variables that represent propositions.

Proposition letters for simple propositions

Example

In general, we use the indexed letters #\blue{p_1}, \blue{p_2},\ldots# to denote propositions. We call these proposition letters or variables.

If the number of propositions we work with is a relatively small number, we may also use the letters #\blue p, \blue q, \blue r,\ldots# to denote propositions.

Consider the proposition \(\blue p\rightarrow (\blue q\land\blue r)\), where #\blue p#, #\blue q#, #\blue r# are variables. If we put

\[\begin{array}{rcl}\blue p&=& \blue{\text{"Chris is ill."}}\\ \blue q&=&\blue{\text{"Chris stays at home."}}\\ \blue r&=&\blue{\text{"Chris eats soup."}}\end{array}\] then the proposition reads #\blue{\textit{"If}\textrm{ Chris is ill, }}#
#\blue{\textit{then}\textrm{ he stays at home and eats soup."}}#

Proposition letters are not only useful for abbreviating concrete propositions, they also enable us to work with abstract propositions that do not represent concrete statements.

But if instead we put
\[\begin{array}{rcl}\blue p&=& \blue{\text{"Denise stays with her friend."}}\\ \blue q&=& \blue{ \text{"Denise plays games."}} \\ \blue r&=& \blue{\text{"Denise eats soup."}}\end{array}\] then the proposition reads #\blue{ \textit{"If}\textrm{ Denise stays with her friend, }}#
#\blue{\textit{then}\textrm{ she plays games and eats soup."}}#

Proposition letters with indices

Proposition letters with indices, such as #\blue{p_1}, \blue{p_2},\ldots, \blue{p_n}#, can be useful in a couple of situations.

They enable us to refer to #n# as the number of propositions, and we can make general statements about propositions in which multiple simple propositions occur.

When we want to state the truth of a compound proposition using an arbitrary number #n# of simple propositions, then we can say "For all propositions #\blue{p_1},\ldots,\blue{p_n}# the compound proposition is true". This would not be possible if we used #\blue p, \blue q, \blue r,\ldots#

Proposition letters with indices can also be used to connect propositions to indices, as the following example shows.

Example

The following case illustrates how proposition letters with indices can be useful. We define
\[\begin{array}{cclcl}
\blue{p_1}&=&\blue{\text{"Julie has }}&\blue{1}&\blue{\text{ child"}}\\
\blue{p_2}&=&\blue{\text{"Julie has }}&\blue{2}&\blue{\text{ children"}}\\
\vdots &&& \vdots&\\
\blue{p_n}&=&\blue{\text{"Julie has }}&\blue{n}&\blue{\text{ children"}}\end{array}\]

By using this notation with indices, it is immediately clear which proposition is meant by #\blue{p_{11}}#. When using #\blue p, \blue q, \blue r,\ldots# as proposition letters, this same proposition would be denoted by #\blue z#. It is much harder to see what proposition is meant by #\blue z#.

Other proposition letters

We can also use other letters if we feel that is more useful for a specific context.

In the example below, we use the first letter of each name as the corresponding proposition letter. This makes it easy to remember which proposition belongs to which proposition letter.

\[\begin{array}{rcl}
\blue a&=& \blue{\text{"Amy works."}}\\
\blue b&=&\blue{\text{"Ben works."}}\\
\blue c&=&\blue{\text{"Chris works."}}\end{array}\]

Boolean

The proposition letters can be thought of as Boolean variables. This means that they can take precisely two values. In our case these values are true and false. In various other settings, the values are denoted by #0# and #1#.

We prefer to use the term proposition letter to Boolean variable because it reminds us of the fact that we can substitute concrete propositions for these variables.

Now that we have variables for propositions, we introduce some more notation for compound propositions.

Variables for compound propositions

We also use Greek capital letters like #\green{\Phi},\green{\Psi}, \green{\Theta},\ldots# to denote propositions. These are are primarily meant for compound propositions.

Let #\green{\Phi}# be a proposition constructed from #n# different propositions #\blue{p_1}, \blue{p_2},\ldots,\blue{p_n}# by means of logical operators. Then we also write \[\green{\Phi}(\blue{p_1},\ldots, \blue{p_n} )\] instead of #\green{\Phi}# if we want to make the dependence on #\blue{p_1},\blue{p_2},\ldots,\blue{p_n}# clear.

Example

Let #\green{\Phi}= \blue p\rightarrow (\blue{q}\land\blue{r})#.
Then we also write #\green{\Phi} = \green{\Phi}(\blue p,\blue q,\blue r)# to indicate the dependence on #\blue p#, #\blue q#, #\blue r#.

This makes substitutions easy. For example, we may substitute the concrete propositions from the example with Chris and Denise for #\blue p#, #\blue q#, and #\blue r#. Also, we could substitute #\blue p# for #\blue q# and for #\blue r# so as to obtain the proposition
\[\green{\Phi}(\blue p, \blue p, \blue p) = \blue p\rightarrow(\blue{p}\land\blue p) \] which is always true.

Proposition letters
The propositions #\blue{p_1}, \blue{p_2},\ldots,\blue{p_n}# occurring in #\green{\Phi}# need not be simple propositions. Yet they behave like simple propositions when we regard them as variables. In other words: when the variable #\blue{p_1}# occurs in a compound proposition #\green{\Phi}#, it reveals nothing but the proposition letter #\blue{p_1}# in #\green{\Phi}#. But if we substitute \[\blue{ \textrm{"Today I am going to school }\textit{and}\textrm{ I will }\textit{not}\textrm{ do my homework."}}\] for #\blue{p_1}#, we will have logical operators appearing in #\blue{p_1}#.

Simple propositions
Although Greek capitals are mostly meant for compound propositions, we sometimes use them for simple propositions too.

For instance, if we want to compare the propositions #\blue p# and #\blue p\lor(\blue{q}\land \neg\blue q)#, we can define \[\begin{array}{rcl}\green{\Phi}&=& \blue p\\\green{\Psi}&=& \blue p\lor(\blue{q}\land \neg\blue q)\end{array}\]Now #\blue q\land \blue{\neg q}# is always false, so #\blue p\lor(\blue{q}\land\neg\blue q)# gives the same value (true or false) as #\blue p#, whatever the values of #\blue p# and #\blue q# are. In the definition of the bi-implication we saw that #\green{\Phi}# and #\green{\Psi}# are therefore equivalent. Later we will come back to the concept of equivalence.

We define
\[\begin{array}{rcl}p&=&\text{"Julia goes to school by bike."}\\q&=&\text{"It is rainy."}\\r&=&\text{"It is sunny."}\end{array}\]Write the compound proposition #\Phi# given by

#\Phi=# "Julia goes to school by bike if and only if it is rainy or sunny."
with the proposition letters #p#, #q#, and #r#, using each exactly once.

#\Phi=# #p\leftrightarrow(q\lor r)#

We substitute the letters #p#, #q#, and #r# for the corresponding simple propositions within #\Phi#. We then find "#p# if and only if #q# or #r#". We now place brackets and substitute the symbols #\leftrightarrow# and #\lor# for the correct phrases or words in #\Phi#. We then find #\Phi=# #p\leftrightarrow(q\lor r)#.

New example