81 lines
2.6 KiB
Markdown
81 lines
2.6 KiB
Markdown
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---
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tags:
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- propositional-logic
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- logic
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---
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# Syntax of propositional logic
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## Syntax of formal languages versus semantics
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> The syntactical study of a language is the study of the expressions of the
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> language and the relations among them _without regard_ to the possible
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> interpretations or 'meaning' of these expressions.
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Syntax is talking about the order and placement of propositions relative to
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connectives and what constitutes a well-formed expression in these terms.
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Semantics is about what the connectives mean, in other words: truth-functions
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and truth-values and not just placement and order.
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## Formal specification of the syntax of the language of Sentential Logic
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### Vocabulary
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Propositions in SL are capitalised Roman letters (non-bold) with or without
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natural number subscripts. We may call these proposition letters. For example:
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$$
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P, Q, R,... P_{1}, Q_{1}, R_{1}, ...
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$$
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The connectives of SL are the five truth-functional connectives:
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$$
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\lnot, \land, \lor, \rightarrow, \leftrightarrow
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$$
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The punctuation marks of SL consist in the left and right parentheses:
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$$
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( )
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$$
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### Grammar
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1. Every letter in a statement is a proposition.
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2. If $P$ is a proposition then $\lnot P$ is a proposition.
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3. If $P$ and $Q$ are propositions, then $P \land Q$ is a proposition
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4. If $P$ and $Q$ are propositions, then $P \lor Q$ is a proposition
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5. If $P$ and $Q$ are propositions, then $P \rightarrow Q$ is a proposition
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6. If $P$ and $Q$ are propositions, then $P \leftrightarrow Q$ is a proposition
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7. Nothing is a proposition unless it can be formed by repeated application of
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rules 1-6
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### Additional syntactic concepts
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We also distinguish:
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- the **main connective**
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- **immediate sentential components**
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- **sentential components**
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- **atomic components**
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These definitions provide a formal specification of the concepts of
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[atomic and molecular propositions](Atomic_and_molecular_propositions.md)
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introduced previously.
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1. If $P$ is an atomic proposition, $P$ contains no connectives and hence does
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not have a main connective. $P$ has no immediate propositional components.
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1. If $P$ is of the form $\lnot Q$ where $Q$ is a proposition, then the main
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connective of $P$ is the negation symbol that occurs before $Q$ and $Q$ is
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the immediate propositional component of $P$
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1. If P is of the form:
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1. $Q \land R$
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1. $Q \lor R$
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1. $Q \rightarrow R$
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1. $Q \leftrightarrow R$
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where $Q$ and $R$ are propositions, then the main connective of $P$ is the
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connective that occurs between $Q$ and $R$ and $Q$ and $R$ are the immediate
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propositional components of $P$.
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