Editing Propositional formula (section)

== More complex formulas ==
As shown above, the CASE (IF c THEN b ELSE a ) connective is constructed either from the 2-argument connectives IF ... THEN ... and AND or from OR and AND and the 1-argument NOT. Connectives such as the n-argument AND (a & b & c & ... & n), OR (a &or; b &or; c &or; ... &or; n) are constructed from strings of two-argument AND and OR and written in abbreviated form without the parentheses. These, and other connectives as well, can then be used as building blocks for yet further connectives. Rhetoricians, philosophers, and mathematicians use truth tables and the various theorems to analyze and simplify their formulas.

Electrical engineering uses drawn symbols and connect them with lines that stand for the mathematicals act of substitution and replacement. They then verify their drawings with truth tables and simplify the expressions as shown below by use of [[Karnaugh map]]s or the theorems. In this way engineers have created a host of "combinatorial logic" (i.e. connectives without feedback) such as "decoders", "encoders", "mutifunction gates", "majority logic", "binary adders", "arithmetic logic units", etc.

=== Definitions ===
A definition creates a new symbol and its behavior, often for the purposes of abbreviation. Once the definition is presented, either form of the equivalent symbol or formula can be used. The following symbolism =<sub>Df</sub> is following the convention of Reichenbach.<ref>Reichenbach p. 20-22 and follows the conventions of PM. The symbol =<sub>Df</sub> is in the [[metalanguage]] and is not a formal symbol with the following meaning: "by symbol ' s ' is to have the same meaning as the formula '(c & d)' ".</ref> Some examples of convenient definitions drawn from the symbol set { ~, &, (, ) } and variables. Each definition is producing a logically equivalent formula that can be used for substitution or replacement.
:* definition of a new variable: (c & d) =<sub>Df</sub> s
:* OR: ~(~a & ~b) =<sub>Df</sub> (a &or; b)
:* IMPLICATION: (~a &or; b) =<sub>Df</sub> (a → b)
:* XOR: (~a & b) &or; (a & ~b) =<sub>Df</sub> (a ⊕ b)
:* LOGICAL EQUIVALENCE: ( (a → b) & (b → a) ) =<sub>Df</sub> ( a ≡ b )

===Axiom and definition ''schemas''===
The definitions above for OR, IMPLICATION, XOR, and logical equivalence are actually [[axiom schema|schema]]s (or "schemata"), that is, they are ''models'' (demonstrations, examples) for a general formula ''format'' but shown (for illustrative purposes) with specific letters a, b, c for the variables, whereas any variable letters can go in their places as long as the letter substitutions follow the rule of substitution below.
: Example: In the definition (~a &or; b) =<sub>Df</sub> (a → b), other variable-symbols such as "SW2" and "CON1" might be used, i.e. formally:
:: a =<sub>Df</sub> SW2, b =<sub>Df</sub> CON1, so we would have as an ''instance'' of the definition schema (~SW2 &or; CON1) =<sub>Df</sub> (SW2 → CON1)

=== Substitution versus replacement ===
'''Substitution''': The variable or sub-formula to be substituted with another variable, constant, or sub-formula must be replaced in all instances throughout the overall formula.
: Example: (c & d) &or; (p & ~(c & ~d)), but  (q1 & ~q2) ≡ d. Now wherever variable "d" occurs, substitute (q<sub>1</sub> & ~q<sub>2</sub>):
:: (c & (q<sub>1</sub> & ~q<sub>2</sub>)) &or; (p & ~(c & ~(q<sub>1</sub> & ~q<sub>2</sub>)))

'''Replacement''': (i) the formula to be replaced must be within a tautology, i.e. ''logically equivalent'' ( connected by ≡ or ↔) to the formula that replaces it, and (ii) unlike substitution its permissible for the replacement to occur only in one place (i.e. for one formula).
: Example: Use this set of formula schemas/equivalences: 
:# ( (a &or; 0) ≡ a ). 
:# ( (a & ~a) ≡ 0 ). 
:# ( (~a &or; b) =<sub>Df</sub> (a → b) ). 
:# <li value="6"> ( ~(~a) ≡ a )</li>
:{{ordered list|list-style-type=lower-alpha
| start with "a": a
| Use 1 to replace "a" with (a &or; 0): (a &or; 0)
| Use the notion of "schema" to substitute b for a in 2: ( (a & ~a) ≡ 0 )
| Use 2 to replace 0 with (b & ~b): ( a &or; (b & ~b) )
| (see below for how to distribute "a &or;" over (b & ~b), etc.)
}}