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Formulas

There are three types of formulas in NDL: atomic formulas (or simply atoms); sentential formulas; and quantified formulas. We explain each in turn:
  1. An atom is either the constant true; or the constant false; or else it is of the form $R$($t_1$,$\ldots$,$t_n$), where $R$ is a relation symbol of arity $n$ and $t_1,\ldots,t_n$ are terms. An example of an atom is between(x,y,z). Infix notation is allowed in the case of binary relation symbols, for instance
    foo = 2
    x * 3 < 15
    father(joe) loves mary
  2. A sentential formula is either:
    1. a negation for any formula $F$;
    2. a conjunction for any formulas $F_1,F_2$;
    3. a disjunction for any formulas $F_1,F_2$;
    4. a conditional $F_1$ ==> $F_2$ for any formulas $F_1,F_2$; or
    5. a biconditional $F_1$ <==> $F_2$ for any formulas $F_1,F_2$.
    The negation sign ~ binds tighter than all other four connectives. Conjunctions and disjunctions have equal precedence, bind tigher than conditionals and biconditionals, and associate to the right. Conditionals and biconditionals have equal precedence and associate to the right. Accordingly,

    \begin{displaymath}\mbox{\tt A \symbol{38} \symbol{126}B \symbol{92}\symbol{47} C ==> D \symbol{38} E}\end{displaymath}

    is parsed as

    \begin{displaymath}\mbox{\tt [A \symbol{38} ((\symbol{126}B) \symbol{92}\symbol{47} C)] ==> [D \symbol{38} E]}\end{displaymath}

    To override these conventions, use parentheses or square brackets.
  3. A quantified formula is either of the form (forall $x$ $F$) (a universal quantification); or (exists $x$ $F$) (a existential quantification) for any variable $x$ and formula $F$. Brackets can also be used for grouping, so we can write [exists $x$ $F$] instead of (exists $x$ $F$).
We will use the letters \bgroup\color{red}$F$\egroup and \bgroup\color{red}$G$\egroup as typical formulas. We assume the reader is familiar with notions such as free and bound variable occurrences. These are defined in the standard way. Two formulas are said to be alphabetically equivalent iff each can be obtained from the other by consistently renaming bound variables. Alphabetically equivalent formulas will be viewed as identical. We write \bgroup\color{red}$F[x\mapsto t]$\egroup to denote the formula obtained from \bgroup\color{red}$F$\egroup by replacing every occurrence of \bgroup\color{red}$x$\egroup in it by the term \bgroup\color{red}$t$\egroup, renaming bound variables if necessary in order to avoid variable capture.
next up previous contents
Next: Proof syntax Up: Natural Deduction Language Previous: Terms   Contents
2004-08-06