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 and to abbreviate propositions. Here we introduce proposition letters, which are variables that represent propositions.
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Example
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In general, we use the indexed letters 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 to denote propositions.
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Consider the proposition , where , , are variables. If we put
then the proposition reads
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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 then the proposition reads
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Proposition letters with indices, such as , can be useful in a couple of situations.
They enable us to refer to 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 of simple propositions, then we can say "For all propositions the compound proposition is true". This would not be possible if we used
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
By using this notation with indices, it is immediately clear which proposition is meant by . When using as proposition letters, this same proposition would be denoted by . It is much harder to see what proposition is meant by .
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.
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 and .
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.
We also use Greek capital letters like to denote propositions. These are are primarily meant for compound propositions.
Let be a proposition constructed from different propositions by means of logical operators. Then we also write instead of if we want to make the dependence on clear.
Example
Let .
Then we also write to indicate the dependence on , , .
This makes substitutions easy. For example, we may substitute the concrete propositions from the example with Chris and Denise for , , and . Also, we could substitute for and for so as to obtain the proposition
which is always true.
The propositions occurring in need not be simple propositions. Yet they behave like simple propositions when we regard them as variables. In other words: when the variable occurs in a compound proposition , it reveals nothing but the proposition letter in . But if we substitute for , we will have logical operators appearing in .
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 and , we can define Now is always false, so gives the same value (true or false) as , whatever the values of and are. In the definition of the bi-implication we saw that and are therefore equivalent. Later we will come back to the concept of equivalence.
We define
Write the compound proposition given by
"Paulo drives his car if and only if it is sunny and warm."
with the proposition letters , , and , using each exactly once.
We substitute the letters , , and for the corresponding simple propositions within . We then find " if and only if and ". We now place brackets and substitute the symbols and for the correct phrases or words in . We then find .