Editing Propositional formula (section)

=== Usefulness of propositional formulas ===
Analysis: In [[deductive reasoning]], philosophers, rhetoricians and mathematicians reduce arguments to formulas and then study them (usually with [[truth table]]s) for correctness (soundness). For example: Is the following argument sound?
: "Given that consciousness is sufficient for an [[artificial intelligence]] and only conscious entities can pass the [[Turing test]], before we can conclude that a robot is an artificial intelligence the robot must pass the Turing test."

Engineers analyze the [[logic circuits]] they have designed using synthesis techniques and then apply various reduction and minimization techniques to simplify their designs.

Synthesis: Engineers in particular synthesize propositional formulas (that eventually end up as circuits of symbols) from [[truth table]]s. For example, one might write down a truth table for how [[binary addition]] should behave given the addition of variables "b" and "a" and "carry_in" "ci", and the results "carry_out" "co" and "sum" Σ:
* Example: in row 5, ( (b+a) + ci ) = ( (1+0) + 1 ) = the number "2". written as a binary number this is 10<sub>2</sub>, where "co"=1 and Σ=0 as shown in the right-most columns.
{| class="wikitable" style="text-align:center; margin-left: auto; margin-right: auto; border: none;"
|-
! row
! b !! a !! ci !! !! (b+a)+ci !! co !! Σ
|-
! 0
| 0 || 0 || 0 || || 0 || 0 || 0
|-
! 1
| 0 || 0 || 1 || || 1 || 0 || 1
|-
! 2
| 0 || 1 || 0 || || 1 || 0 || 1
|-
! 3
| 0 || 1 || 1 || || 2 || 1 || 0
|-
! 4
| 1 || 0 || 0 || || 1 || 0 || 1
|-
! 5
| 1 || 0 || 1 || || 2 || 1 || 0
|-
! 6
| 1 || 1 || 0 || || 2 || 1 || 0
|-
! 7
| 1 || 1 || 1 || || 3 || 1 || 1
|}