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, }}# 
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."}}# 
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.
\[\begin{array}{rcl}p&=&\text{"Julia goes to school by bike."}\\q&=&\text{"It is rainy."}\\r&=&\text{"It is sunny."}\end{array}\]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.
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)#.
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