Editing Propositional formula (section)

=== 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".