Editing Propositional formula (section)

==An algebra of propositions, the propositional calculus==
{{essay-like|section|date=June 2021}}
An algebra (and there are many different ones), loosely defined, is a method by which a collection of symbols called variables together with some other symbols such as parentheses (, ) and some sub-set of symbols such as *, +, ~, &, &or;, =, ≡, &and;, ￢ are manipulated within a system of rules. These symbols, and well-formed strings of them, are said to represent objects, but in a specific algebraic system these objects do not have meanings. Thus work inside the algebra becomes an exercise in obeying certain laws (rules) of the algebra's [[syntax]] (symbol-formation) rather than in [[semantics]] (meaning) of the symbols. The meanings are to be found outside the algebra.

For a well-formed sequence of symbols in the algebra —a formula— to have some usefulness outside the algebra the symbols are assigned meanings and eventually the variables are assigned values; then by a series of rules the formula is evaluated.

When the values are restricted to just two and applied to the notion of simple sentences (e.g. spoken utterances or written assertions) linked by propositional connectives this whole algebraic system of symbols and rules and evaluation-methods is usually called the [[propositional calculus]] or the sentential calculus.

While some of the familiar rules of arithmetic algebra continue to hold in the algebra of propositions (e.g. the commutative and associative laws for AND and OR), some do not (e.g. the [[distributive law]]s for AND, OR and NOT).

=== 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
|}

=== Propositional variables ===
The simplest type of propositional formula is a '''[[propositional variable]]'''. Propositions that are simple ([[atomic formula|atomic]]), symbolic expressions are often denoted by variables named ''p'', ''q'', or ''P'', ''Q'', etc. A propositional variable is intended to represent an atomic proposition (assertion), such as "It is Saturday" = ''p'' (here the symbol = means " ... is assigned the variable named ...") or "I only go to the movies on Monday" = ''q''.

=== Truth-value assignments, formula evaluations ===
Evaluation of a propositional formula begins with assignment of a truth value to each variable. Because each variable represents a simple sentence, the truth values are being applied to the "truth" or "falsity" of these simple sentences.

'''Truth values in rhetoric, philosophy and mathematics'''

The truth values are only two: { TRUTH "T",  FALSITY "F" }. An [[empiricist]] puts all propositions into two broad classes: ''analytic''—true no matter what (e.g. [[tautology (logic)|tautology]]), and ''synthetic''—derived from experience and thereby susceptible to confirmation by third parties (the [[verification theory]] of meaning).<ref>Empiricits eschew the notion of ''a priori'' (built-in, born-with) knowledge. "Radical reductionists" such as [[John Locke]] and [[David Hume]] "held that every idea must either originate directly in sense experience or else be compounded of ideas thus originating"; quoted from Quine reprinted in 1996 ''The Emergence of Logical Empriricism'', Garland Publishing Inc. http://www.marxists.org/reference/subject/philosophy/works/us/quine.htm</ref> Empiricists hold that, in general, to arrive at the truth-value of a [[synthetic proposition]], meanings (pattern-matching templates) must first be applied to the words, and then these meaning-templates must be matched against whatever it is that is being asserted. For example, my utterance "That cow is ''{{blue|blue}}''!" Is this statement a TRUTH? Truly I said it. And maybe I ''am'' seeing a blue cow—unless I am lying my statement is a TRUTH relative to the object of my (perhaps flawed) perception. But is the blue cow "really there"? What do you see when you look out the same window? In order to proceed with a verification, you will need a prior notion (a template) of both "cow" and "{{blue|blue}}", and an ability to match the templates against the object of sensation (if indeed there is one).{{citation needed|date=October 2016}}

'''Truth values in engineering'''

Engineers try to avoid notions of truth and falsity that bedevil philosophers, but in the final analysis engineers must trust their measuring instruments. In their quest for [[Robust statistics|robustness]], engineers prefer to pull known objects from a small library—objects that have well-defined, predictable behaviors even in large combinations, (hence their name for the propositional calculus: "combinatorial logic"). The fewest behaviors of a single object are two (e.g. { OFF, ON }, { open, shut }, { UP, DOWN } etc.), and these are put in correspondence with { 0, 1 }. Such elements are called [[Digital data|digital]]; those with a continuous range of behaviors are called [[analog signal|analog]]. Whenever decisions must be made in an analog system, quite often an engineer will convert an analog behavior (the door is 45.32146% UP) to digital (e.g. DOWN=0 ) by use of a [[comparator]].<ref>[[Neural net]] modelling offers a good mathematical model for a comparator as follows: Given a signal S and a threshold "thr", subtract "thr" from S and substitute this difference d to a [[sigmoid function]]: For large "gains" k, e.g. k=100, 1/( 1 + e<sup>−k*d</sup> ) = 1/( 1 + e<sup>−k*(S-thr)</sup> ) = { ≃0, ≃1 }.{{clarify|What is the meaning of the curly braces here? Denoting set comprehension wouldn't make sense.|date=October 2016}} For example, if "The door is DOWN" means "The door is less than 50% of the way up", then a threshold thr=0.5 corresponding to 0.5*5.0 = +2.50 volts could be applied to a "linear" measuring-device with an output of 0 volts when fully closed and +5.0 volts when fully open.</ref>

Thus an assignment of meaning of the variables and the two value-symbols { 0, 1 } comes from "outside" the formula that represents the behavior of the (usually) compound object. An example is a garage door with two "limit switches", one for UP labelled SW_U and one for DOWN labelled SW_D, and whatever else is in the door's circuitry. Inspection of the circuit (either the diagram or the actual objects themselves—door, switches, wires, circuit board, etc.) might reveal that, on the circuit board "node 22" goes to +0 volts when the contacts of switch "SW_D" are mechanically in contact ("closed") and the door is in the "down" position (95% down), and "node 29" goes to +0 volts when the door is 95% UP and the contacts of switch SW_U are in mechanical contact ("closed").<ref>In actuality the digital 1 and 0 are defined over non-overlapping ranges e.g. { "1" = +5/+0.2/−1.0 volts, 0 = +0.5/−0.2 volts }{{clarify|Explain the meaning of curly braces and slash here.|date=October 2016}}. When a value falls outside the defined range(s) the value becomes "u" -- unknown; e.g. +2.3 would be "u".</ref> The engineer must define the meanings of these voltages and all possible combinations (all 4 of them), including the "bad" ones (e.g. both nodes 22 and 29 at 0 volts, meaning that the door is open and closed at the same time). The circuit mindlessly responds to whatever voltages it experiences without any awareness of TRUTH or FALSEHOOD, RIGHT or WRONG, SAFE or DANGEROUS.{{citation needed|date=October 2016}}