Syntax of propositional logic

Syntax of formal languages versus semantics

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.

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.

Formal specification of the syntax of the language of Sentential Logic

Vocabulary

Propositions in SL are capitalised Roman letters (non-bold) with or without natural number subscripts. We may call these proposition letters. For example:

$$ P, Q, R,... P_{1}, Q_{1}, R_{1}, ... $$

The connectives of SL are the five truth-functional connectives:

$$ \lnot, \land, \lor, \rightarrow, \leftrightarrow $$

The punctuation marks of SL consist in the left and right parentheses:

$$ ( ) $$

Grammar

  1. Every letter in a statement is a proposition.
  2. If \(P\) is a proposition then \(\lnot P\) is a proposition.
  3. If \(P\) and \(Q\) are propositions, then \(P \land Q\) is a proposition
  4. If \(P\) and \(Q\) are propositions, then \(P \lor Q\) is a proposition
  5. If \(P\) and \(Q\) are propositions, then \(P \rightarrow Q\) is a proposition
  6. If \(P\) and \(Q\) are propositions, then \(P \leftrightarrow Q\) is a proposition
  7. Nothing is a proposition unless it can be formed by repeated application of rules 1-6

Additional syntactic concepts

We also distinguish:

  • the main connective
  • immediate sentential components
  • sentential components
  • atomic components

These definitions provide a formal specification of the concepts of atomic and molecular propositions introduced previously.

  1. If \(P\) is an atomic proposition, \(P\) contains no connectives and hence does not have a main connective. \(P\) has no immediate propositional components.
  2. If \(P\) is of the form \(\lnot Q\) where \(Q\) is a proposition, then the main connective of \(P\) is the negation symbol that occurs before \(Q\) and \(Q\) is the immediate propositional component of \(P\)
  3. If P is of the form:
    1. \(Q \land R\)
    2. \(Q \lor R\)
    3. \(Q \rightarrow R\)
    4. \(Q \leftrightarrow R\)

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 propositional components of \(P\).

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