Editing Propositional formula (section)

==Propositions==
For the purposes of the propositional calculus, '''propositions''' (utterances, sentences, assertions) are considered to be either simple or compound.<ref>Hamilton 1978:1</ref> Compound propositions are considered to be linked by sentential connectives, some of the most common of which are "AND", "OR", "IF ... THEN ...", "NEITHER ... NOR ...", "... IS EQUIVALENT TO ..." . The linking semicolon ";", and connective "BUT" are considered to be expressions of "AND". A sequence of discrete sentences are considered to be linked by "AND"s, and formal analysis applies a [[Recursion|recursive]] "parenthesis rule" with respect to sequences of simple propositions (see more [[#Well-formed formulas (wffs)|below]] about well-formed formulas).
: For example: The assertion: "This cow is blue. That horse is orange but this horse here is purple." is actually a compound proposition linked by "AND"s: ( ("This cow is blue" AND "that horse is orange") AND "this horse here is purple" ) .

Simple propositions are declarative in nature, that is, they make assertions about the condition or nature of a ''particular'' object of sensation e.g. "This cow is blue", "There's a coyote!" ("That coyote IS ''there'', behind the rocks.").<ref>[[Principia Mathematica]] (PM) p. 91 eschews "the" because they require a clear-cut "object of sensation"; they stipulate the use of "this"</ref> Thus the simple "primitive" [[Logical assertion|assertion]]s must be about specific objects or specific states of mind. Each must have at least a subject (an immediate object of thought or observation), a verb (in the active voice and present tense preferred), and perhaps an adjective or adverb. "Dog!" probably implies "I see a dog" but should be rejected as too ambiguous.

: Example: "That purple dog is running", "This cow is blue", "Switch M31 is closed", "This cap is off", "Tomorrow is Friday".

For the purposes of the propositional calculus a compound proposition can usually be reworded into a series of simple sentences, although the result will probably sound stilted.

=== Relationship between propositional and predicate formulas ===
The [[predicate calculus]] goes a step further than the propositional calculus to an "analysis of the ''inner structure'' of propositions"<ref>(italics added) Reichenbach{{clarify|reason=There is no 'Reichenbach' entry under 'References', not even a link to an article about Reichenbach.|date=October 2013}} p.80.</ref> It breaks a simple sentence down into two parts (i) its subject (the object ([[singular term|singular]] or plural) of discourse) and (ii) a [[Predicate (grammar)|predicate]] (a verb or possibly verb-clause that asserts a quality or attribute of the object(s)). The predicate calculus then generalizes the "subject|predicate" form (where | symbolizes [[concatenation]] (stringing together) of symbols) into a form with the following blank-subject structure " ___|predicate", and the predicate in turn generalized to all things with that property.

: Example: "This blue pig has wings" becomes two sentences in the ''propositional calculus'': "This pig has wings" AND "This pig is blue", whose internal structure is not considered. In contrast, in the predicate calculus, the first sentence breaks into "this pig" as the subject, and "has wings" as the predicate. Thus it asserts that object "this pig" is a member of the class (set, collection) of "winged things". The second sentence asserts that object "this pig" has an attribute "blue" and thus is a member of the class of "blue things". One might choose to write the two sentences connected with AND as:
:: p|W AND p|B

The generalization of "this pig" to a (potential) member of two classes "winged things" and "blue things" means that it has a truth-relationship with both of these classes. In other words, given a [[domain of discourse]] "winged things", p is either found to be a member of this domain or not. Thus there is a relationship W (wingedness) between p (pig) and { T, F }, W(p) evaluates to { T, F } where { T, F } is the set of the [[Boolean data type|Boolean values]] "true" and "false". Likewise for B (blueness) and p (pig) and { T, F }: B(p) evaluates to { T, F }. So one now can analyze the connected assertions "B(p) AND W(p)" for its overall truth-value, i.e.:
: ( B(p) AND W(p) ) evaluates to { T, F }

In particular, simple sentences that employ notions of "all", "some", "a few", "one of", etc. called [[logical quantifier]]s are treated by the predicate calculus. Along with the new function symbolism "F(x)" two new symbols are introduced: ∀ (For all), and ∃ (There exists ..., At least one of ... exists, etc.). The predicate calculus, but not the propositional calculus, can establish the formal validity of the following statement:
: "All blue pigs have wings but some pigs have no wings, hence some pigs are not blue".

=== Identity ===
Tarski asserts that the notion of IDENTITY (as distinguished from LOGICAL EQUIVALENCE) lies outside the propositional calculus; however, he notes that if a logic is to be of use for mathematics and the sciences it must contain a "theory" of IDENTITY.<ref>Tarski p.54-68. Suppes calls IDENTITY a "further rule of inference" and has a brief development around it; Robbin, Bender and Williamson, and Goodstein introduce the sign and its usage without comment or explanation. Hamilton p. 37 employs two signs ≠ and = with respect to the '''valuation''' of a formula in a formal calculus. Kleene p. 70 and Hamilton p. 52 place it in the predicate calculus, in particular with regards to the arithmetic of natural numbers.</ref> Some authors refer to "predicate logic with identity" to emphasize this extension. See more about this below.