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Definition: Syntax of PL0 - Propositions as Boolean Terms
The variables and functions defined in signature of propositional logic $PL0$ are called Boolean, named after the English mathematician George Bool.
First, we agree that the formal language \(L\subseteq \Sigma^* \) of $PL0$ is defined over an alphabet \(\Sigma\) containing the following letters:
- space character " ",
- Boolean constants: the characters "$1$" and "$0$",
- Boolean variables: small Latin letters \("a","b","c",\ldots,"x","y","z"\), and letters indexed with natural numbers, e.g. \("a_0","a_1","a_2",\ldots,"b_0","b_1","b_2",\ldots,\),
- the parentheses "$($" and "$)$", and
- the following additional letters "\(\neg\)", "\(\wedge\)", "\(\vee\)", "\(\Rightarrow\)", "\(\Leftrightarrow\)".
We will now specify the syntax of $PL0$:
- A Boolean constant is a proposition (we also will use the notion Boolean term synonymously for "proposition").
- Boolean variables are propositions.
- If "$\phi$" and "$\psi$" are propositions, then "$\neg \phi$," "$\phi\wedge\psi$", "$\phi\vee\psi$", "$\phi\Rightarrow\psi$", "$\phi\Leftrightarrow\psi$" are propositions.
- If "$\phi$" is a proposition, then the concatenation "$(\phi)$" is a proposition.
Examples
This syntax enables us to construct propositions (Boolean terms):
* $x$,
* $(x)$,
* $\neg a$,
* \((\neg(\neg y))\),
* \(((x\Rightarrow(y\vee(1\wedge(\neg w))))\wedge 0)\),
* \(((a\Leftrightarrow b)\vee(0\Rightarrow y))\).
Please note that we did not yet define any meaning (semantics) of propositions. We will catch up on this now.
Mentioned in:
Branches: 1
Chapters: 2
Corollaries: 3 4 5 6
Definitions: 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Examples: 21 22 23
Lemmas: 24 25 26 27 28 29 30 31 32 33 34 35 36 37
Parts: 38
Proofs: 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
Propositions: 57 58
Sections: 59
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References
Bibliography
- Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
- Mendelson Elliott: "Theory and Problems of Boolean Algebra and Switching Circuits", McGraw-Hill Book Company, 1982
