In propositional logic, a propositional formula is a type of syntactic formula which is well formed. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula. A propositional formula is constructed from simple propositions, such as "five is greater than three" or propositional variables such as p and q, using connectives or logical operators such as NOT, AND, OR, or IMPLIES; for example:
(p AND NOT q) IMPLIES (p OR q). In mathematics, a propositional formula is often more briefly referred to as a "proposition", but, more precisely, a propositional formula is not a proposition but a formal expression that denotes a proposition, a formal object under discussion, just like an expression such as "x + y" is not a value, but denotes a value. In some contexts, maintaining the distinction may be of importance.
Propositions For the purposes of the propositional calculus, propositions (utterances, sentences, assertions) are considered to be either simple or compound. 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 recursive "parenthesis rule" with respect to sequences of simple propositions (see more 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."). Thus the simple "primitive" assertions 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" It breaks a simple sentence down into two parts (i) its subject (the object (singular or plural) of discourse) and (ii) a 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 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 quantifiers 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. Some authors refer to "predicate logic with identity" to emphasize this extension. See more about this below.
An algebra of propositions, the propositional calculus
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