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Formal Logic/Sentential Logic/Informal Conventions

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Informal Conventions

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In The Sentential Language, we gave an informal description of a sentential language, namely  . We have also given a Formal Syntax for  . Our official grammar generates a large number of parentheses. This makes formal definitions and other specifications easier to write, but it makes the language rather cumbersome to use. In addition, all the subscripts and superscripts quickly get to be unnecessarily tedious. The end result is an ugly and difficult to read language.

We will continue to use official grammar for specifying formalities. However, we will informally use a less cumbersome variant for other purposes. The transformation rules below convert official formulae of   into our informal variant.


Transformation rules

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

  • The official grammar required sentence letters to have the superscript '0'. Superscripts aren't necessary or even useful until we get to the predicate logic, so we will always omit them in our informal variant. We will write, for example,   instead of  .
  • We will omit the subscript if it is '0'. Thus we will write   instead of  . However, we cannot omit all subscripts; we still need to write, for example,  .
  • 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
 
is
 
  • 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
 
Full precedence rankings are given below.

Precedence and scope

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Precedence rankings indicate the order that we evaluate the sentential connectives.   has a higher precedence than  . Thus, in calculating the truth value of

 

we start by evaluating the truth value of

 

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)

Examples

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Let's look at some examples. First,

 

can be written informally as

 


Second,

 

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

 

and

 


Note that the informal formula

 

is restored to its official form as

 

By contrast, the informal formula

 

is restored to its official form as

 


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