User:Wvbailey/Predicate logic

Predicate versus propostional logic

Both logics are carried out with rules (sometimes called "laws") that operate on strings of discrete, distinguishable, and with the exception of the variables, finite collection of symbols. In a fully axiomitized calculus these strings have no meanings whatsoever -- they are manipulated by formal rules (e.g. a Turing machine, a mathematician or philosopher blindly following the rules). In the informal logic the symbols have distinct meanings.

An analysis of a compound sentence (or strings of sentences) begins with the broad disassembly of the propositional logic and then continues deeper into individual sentences with the predicate logic. As the predicate logic resides "inside" the broader structure of the propositional logic, for each sentence that it evaluates the predicate logic must return {TRUE, FALSE} if it to be compatible with the propositional logic.

Propositional logic

Propositional logic starts with compound sentences (and strings of sentences). It decomposes them into simple sentences, globally abbreviates each unique simple sentence with a symbol called a variable, and links these "atoms" with a small number of propositional connectives, typically symbols & (AND), V (OR), ~(NOT), → (LOGICAL IMPLICATION), ≡ (LOGICAL EQUIVALENCE alternately ↔), and parentheses (, ). These symbols plus an unbounded set of variable symbols (e.g. a, b, c, ..., x, y, z, ... ) constitute the entire symbol set of the propositional calculus (or logic).

Example: "Piglet number #34 is blue. Blue piglets develop wings. This piglet is an animal. Wingled animals can fly. Therefore 'we surmise that this pig will fly."

'Pig #34 is blue' & 'Blue piglets develop wings' & 'this piglet is an animal' & 'Animals that develop wings can fly.' Therefore 'Blue piglets are animals.' Therefore 'this then the pig can fly."

Each unique simple sentence, now assigned its unique symbol (variable) that is "global" with respect to the sentence -- the sentence is not further broken down. Then formulas are created by linking the variables by use of the propositional connectives together with parenthesis formation-rule. The resulting symbol string can then be manipulated by an algebra of the propositional logic. But for the logic to be useful the final stage is one of evaluation -- to each variable the assignment of "truth" or "falsity" ( {T, F}, {1, 0} ) and then a "calculation" of the truth value of the formula for that particular set of truth-value assignments. A typical analysis will use a truth table to evaluate all 2ⁿ possible combinations of truth-value assignments. Is the following good reasoning (NO, it is not good reasoning):

"We are searching for a winged pig. We know from previous experience that if a pig is blue it will have wings. Ergo: we will search for blue pigs."

The analysis tears the compound sentences down into the simplest sentences possible that can still express the content of the sentence (this analysis takes practice):

IF 'pig has wings' AND "IF 'pig is blue' THEN 'pig has wings' THEN 'Pig is blue' "

Assign variables to the simple sentences. In the example the symbol => is still implication → but the sentence form is called modus ponens or deduction from two premises.

• 'pig has wings' ≡ w
• 'pig is blue' ≡ b

(IF ( w AND (IF b THEN w)) THEN b) ≡ (w & (b → w)) => b)

If the above were "good reasoning" the yellow column under "arg" would be all TRUTHs for all four combinations of truth-value assignment of "b" and "w". But the reasoning is flawed: the red "F" indicates that quite possibly we will encounter winged pigs that are not blue; if we go winged-pig hunting we'd better open our search to pigs of other colors.

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===Predicate logic===:

One variable relations

Predicate logic extends the analysis to the inside of propositions (i.e. sentences0. In the simplest analysis, the logic breaks a simple sentence into two parts: the subject i.e. the object or objects of discussion (assertion, description), and the predicate, at minimum a verb and more likely a verb-clause that expresses a quality or attribute of the subject "has wings":

"pigs have wings" becomes "___ have wings"

Attempts are made to express the verb as "is" or "was", i.e. This structural form is called a sentence-skeleton . One author (Reichenbach) makes a distinction about the form.

Here ___ indicates an empty slot for a particular pig under discussion to be considered in relation to the notion "have wings". The notion of "have wings" is now generalized to a template or notion of "wingedness", that is, the collection templates that define all "winged things": { airplane, bird, bat, pterodactyl, flying_insect, unicorn, winged_pig }. More than likely these templates are the names of collections with further templates: set of birds: { chicken, turkey, goldfinch, chickadee, robin, etc }.

Relations as functions

Predicate relations, of the single-variable membership kind ("Pig #34 is a member of 'winged things'") or of the relational kind ("Pig #34 has longer wings than pig #52"), are actually functions because ... why [??] certainly true that, when evaluated, that the truth-value of the sentence will be either TRUTH or FALSITY but not both.

Example: Truth-value of "Pig #34 is a member of the class of objects 'winged things' " is TRUTH or FALSITY by inspection of the particular pig and comparing it to the 'winged thing' template. Whereas "Pig #34 has longer wings than pig #52" requires a measurement of the wingspan of both pigs (given that they both have wings at all) and thereby receives a truth value of TRUTH or FALSITY..

Multi-variable relations

A slightly more complex case is the one of relationship between two objects with the skeleton-structure " ____ RELATION ____ ".

Examples: "Pig #34's wings are longer than pig #52's wings". "The number '5' is greater than the number '3'." "Ann is the aunt of Nathan."