Formal Logic/Sentential Logic/Formal Syntax

← The Sentential Language ↑ Sentential Logic Informal Conventions →

Formal Syntax edit

In The Sentential Language, we informally described our sentential language. Here we give its formal syntax or grammar. We will call our language  .

Vocabulary edit

  • Sentence letters: Capital letters 'A' – 'Z', each with (1) a superscript '0' and (2) a natural number subscript. (The natural numbers are the set of positive integers and zero.) Thus the sentence letters are:
  • Sentential connectives:
  • Grouping symbols:

The superscripts on sentence letters are not important until we get to the predicate logic, so we won't really worry about those here. The subscripts on sentence letters are to ensure an infinite supply of sentence letters. On the next page, we will abbreviate away most superscripts and subscripts.

Expressions edit

Any string of characters from the   vocabulary is an expression of  . Some expressions are grammatically correct. Some are as incorrect in   as 'Over talks David Mary the' is in English. Still other expressions are as hopelessly ill-formed in   as 'jmr.ovn asgj as;lnre' is in English.

We call a grammatically correct expression of   a well-formed formula. When we get to Predicate Logic, we will find that only some well formed formulas are sentences. For now though, we consider every well formed formula to be a sentence.

Construction rules edit

An expression of   is called a well-formed formula of   if it is constructed according to the following rules.

  The expression consists of a single sentence letter
  The expression is constructed from other well-formed formulae   and   in one of the following ways:

In general, we will use 'formula' as shorthand for 'well-formed formula'. Since all formulae in   are sentences, we will use 'formula' and 'sentence' interchangeably.

Quoting convention edit

We will take expressions of   to be self-quoting and so regard


to include implicit quotation marks. However, something like


requires special consideration. It is not itself an expression of   since   and   are not in the vocabulary of  . Rather they are used as variables in English which range over expressions of  . Such a variable is called a metavariable, and an expression using a mix of vocabulary from   and metavariables is called a metalogical expression. Suppose we let   be   and   be   Then (1) becomes

 ' '  ' '  ' ' 

which is not what we want. Instead we take (1) to mean (using explicit quotes):

the expression consisting of ' ' followed by   followed by ' ' followed by   followed by ' ' .

Explicit quotes following this convention are called Quine quotes or corner quotes. Our corner quotes will be implicit.

Additional terminology edit

We introduce (or, in some cases, repeat) some useful syntactic terminology.

  • We distinguish between an expression (or a formula) and an occurrence of an expression (or formula). The formula

is the same formula no matter how many times it is written. However, it contains three occurrences of the sentence letter   and two occurrences of the sentential connective  .

  •   is a subformula of   if and only if   and   are both formulae and   contains an occurrence of  .   is a proper subformula of   if and only if (i)   is a subformula of   and (ii)   is not the same formula as  .
  • An atomic formula or atomic sentence is one consisting solely of a sentence letter. Or put the other way around, it is a formula with no sentential connectives. A molecular formula or molecular sentence is one which contains at least one occurrence of a sentential connective.
  • The main connective of a molecular formula is the last occurrence of a connective added when the formula was constructed according to the rules above.
  • A negation is a formula of the form   where   is a formula.
  • A conjunction is a formula of the form   where   and   are both formulae. In this case,   and   are both conjuncts.
  • A disjunction is a formula of the form   where   and   are both formulae. In this case,   and   are both disjuncts.
  • A conditional is a formula of the form   where   and   are both formulae. In this case,   is the antecedent, and   is the consequent. The converse of   is  . The contrapositive of   is  .
  • A biconditional is a formula of the form   where   and   are both formulae.

Examples edit


By rule (i), all sentence letters, including


are formulae. By rule (ii-a), then, the negation


is also a formula. Then by rules (ii-c) and (ii-b), we get the disjunction and conjunction


as formulae. Applying rule (ii-a) again, we get the negation


as a formula. Finally, rule (ii-c) generates the conditional of (1), so it too is a formula.


This appears to be generated by rule (ii-c) from


The second of these is a formula by rule (i). But what about the first? It would have to be generated by rule (ii-b) from




cannot be generated by rule (ii-a). So (2) is not a formula.

← The Sentential Language ↑ Sentential Logic Informal Conventions →