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 and to abbreviate propositions. Here we introduce proposition letters, which are variables that represent propositions.
Proposition letters for simple propositions
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. |
Consider the proposition , where , , are variables. If we put then the proposition reads |
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 |
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 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.
We see an '...if...' in the proposition. We rewrite the proposition in such a way that it has the form 'if..., then...'. We find ''If I have sugar and flour, then I bake cookies''. We substitute the letters , , and for the corresponding simple propositions within . We then find "If and , then ". We now place brackets and substitute the symbols and for the correct phrases or words in . We then find .
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