67 lines
2.5 KiB
Markdown
67 lines
2.5 KiB
Markdown
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---
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categories:
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- Logic
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tags: [propositional-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 language and the relations among them _without regard_ to the possible interpretations or 'meaning' of these expressions.
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Syntax is talking about the order and placement of propositions relative to connectives and what constitutes a well-formed expression in these terms. Semantics is about what the connectives mean, in other words: truth-functions 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 natural number subscripts. We may call these proposition letters. For example:
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```plain
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P, Q, R...P1, Q1, R1...
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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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~, &, v, ⊃, ≡
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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 proposition letter is a proposition.
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1. If **P** is a proposition then **~P** is a proposition.
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1. If **P** and **Q** are propositions, then **(P & Q)** is a proposition
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1. If **P** and **Q** are propositions, then **(P v Q)** is a proposition
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1. If **P** and **Q** are propositions, then **(P ⊃ Q)** is a proposition
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1. If **P** and **Q** are propositions, then **(P ≡ Q)** is a proposition
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1. Nothing is a proposition unless it can be formed by repeated application of clauses 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 atomic and molecular propositions _introduced earlier_.
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1. If **P** is an atomic proposition, **P** contains no connectives and hence does not have a main connective. **P** has no immediate sentential components.
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1. If **P** is of the form **~Q** where **Q** is a proposition, then the main connective of **P** is the tilde that occurs before **Q** and **Q** is the immediate sentential component of **P**.
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1. If P is of the form:
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1. **Q & R**
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1. **Q v R**
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1. **Q ⊃ R**
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1. **Q ≡ R**
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where **Q** and **R** are propositions, then the main connective of **P** is the connective that occurs between **Q** and **R** and **Q** and **R** are the immediate sentential components of **P**.
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