In propositional logic, a propositional formula is a type of syntactic formula, well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may 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 such as NOT, AND, OR, or IMPLIES. In mathematics, a propositional formula is more referred to as a "proposition", more 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. For the purposes of the propositional calculus, propositions 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 ". A sequence of discrete sentences are considered to be linked by "AND"s, formal analysis applies a recursive "parenthesis rule" with respect to sequences of simple propositions. For example: The assertion: "This cow is blue; that horse is orange but this horse here is purple." is a compound proposition linked by "AND"s:. 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!". Thus the simple "primitive" assertions must be about specific states of mind; each must have at least a subject, a verb, an adjective or adverb. "Dog!" 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 be reworded into a series of simple sentences, although the result will sound stilted; 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 its subject and a predicate.
The predicate calculus generalizes the "subject|predicate" form into a form with the following blank-subject structure " ___|predicate", 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, "has wings" as the predicate, thus it asserts that object "this pig" is a member of the class 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|BThe generalization of "this pig" to a 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", either we find p to be a member of this domain or not.
Thus we have a relationship W between p and, W evaluates to where is the set of the boolean values "true" and "false". For B and p and: B evaluates to. So we now can analyze the connected assertions "B AND W" for its overall truth-value, i.e.: evaluates to In particular, simple sentences that employ notions of "all", "some", "a few", "one of", etc. are treated by the predicate calculus. Along with the new function symbolism "F" two new symbols are introduced: ∀, ∃; 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". Tarski asserts; some authors refer to "predicate logic with identity" to emphasize this extension. See more about this below. An algebra, 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 *, +, ~, &, ∨, =, ≡, ∧, ￢ are manipulated within a system of rules.
These symbols, 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 i