Formal Logic/Predicate Logic/Informal Conventions

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Informal Conventions Edit

Our predicate language,  , like  , is cumbersome and difficult to read. The superscripts and subscripts are distracting, and there are more parentheses than are needed. As with sentential logic, we will informally use simplified variants. The official language, however, is still used for definitions and other formalities. Most of the rules for generating informal variants will be familiar from sentential logic.

Transformation rules Edit

We create informal variants of official   formulae as follows. The examples are cumulative.

  • We drop a subscript if it is '0'. Thus we write   instead of   and write   instead of  . We cannot drop the subscript from  , however.
  • The official grammar requires operation letters and predicate letters to have the superscript indicating the number of places. For example,   is a two-place operation letter and   is a three place predicate letter. In most cases, we can drop the superscript and let the context show the number of places. For example, we can write
Here we observe from the context that the operation letter has two places, thus leaving it understood that   is an informal variant of  . Similarly, we observe from the context that the predicate letter in
has three places. This makes  , as used in this context, an informal variant of  . This convention assumes our   to be grammatically correct. In general, we will avoid grammatically incorrect expressions. We will also try to avoid, for example, using   and   in close proximity. Otherwise, their informal variant could cause confusion.
  • We will omit outermost parentheses. For example, we will write
instead of
  • We will let a series of the same binary connective associate on the right. For example, we can transform the official
into the informal
However, the best we can do with
  • We will use precedence rankings to omit internal parentheses when possible. For example, we will regard   as having lower precedence (wider scope) than  . This allows us to write
instead of
However, we cannot remove the internal parentheses from
Our informal variant of this latter formula is

Precedence and scope Edit

Precedence rankings indicate the order that we evaluate the sentential connectives and quantifier phrases.   has a higher precedence than  . Thus, in evaluating


we start by evaluating


first. Scope is the length of expression that is governed by the connective. The occurrence of   in (1) has a wider scope than the occurrence of  . Thus the occurrence of   in (1) governs the whole sentence while the occurrence of   in (1) governs only the occurrence of (2) in (1).

The full ranking from highest precedence (narrowest scope) to lowest precedence (widest scope) is:

      highest precedence (narrowest scope)
      lowest precedence (widest scope)

Quantifier phrases have the same precedence as negation signs.

Examples Edit

The following examples are predicate logic variants of the examples at the sentential logic Informal Conventions page. First,


can be written informally as




can be written informally as


Some unnecessary parentheses may prove helpful. In the two examples above, the informal variants may be easier to read as




Note that the informal formulae


are restored to its official form as


By contrast, the informal formula


is restored to its official form as