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, abbreviates each unique simple sentence (globally i.e. in its entirety) with a unique symbol called a variable, and links these "atoms" with a small number of propositional connectives, most-commonly the symbols & (AND), V (OR), ~(NOT), → (IMPLIES; IF...THEN...), ≡ (LOGICALLY EQUIVALENT 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). The bold-faced connectives indicate the final level of the formula.

Example: A propositional formula will look something like this: ( (c & d) V (p & ~(c & ~(d)) ). It is just a string of symbols that is given an interpretation.

'This piglet is blue' & (IF 'this piglet is blue' THEN 'this piglet sprouts wings') & IF 'this piglet sprouts wings' THEN this piglet will fly": b & (b → w) & (w → f). Formally: ( (b & (b → w)) & (w → f) )

Each unique simple sentence is assigned a unique variable that is "global" with respect to the sentence -- the sentence is not further broken down. Then formulas are created by linking the variables with the appropriate 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.

arg

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T

T

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T

Predicate logic

Partition of a domain of discourse into a co-domain (range) into a "Truth-set" [??] ( versus "TRUE or FALSE evaluation.

"'Blue pigs have wings' is a TRUTH", [??]

Predicate logic extends the analysis to the inside of propositions (i.e. simple sentences).

Single-variable relations, one-place functions

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/phrase that expresses a quality or attribute or property of the subject:

'This pig is blue' becomes 'object|attribute' becomes 'argument|function'

'blue pigs have wings' becomes IF '___ is blue THEN '___ have wings', or '___ is winged' meaning " '___' is a member of the set of 'winged things' ".

Path to this reasoning is a bit obscure. Boole. Venn diagrams, viewing a quality-attribute-property as required for set-membership.

Could be written as W( ), eventually to be filled in with a particular variable such as 'p' for 'pig'.

Attempts are made to express the verb as "is" or "was" and the verb-phrase as a modifier (expressing a quality or attribute) of the object(s) of discourse, such as "blue", "winged", 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 'has wings' or 'is winged'. The notion of 'is winged' 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}, {etc} }. More than likely these templates represent further templates: e.g. {bird} { chicken, turkey, goldfinch, chickadee, robin, etc }.

Function as variable: No reason not to include a further generalization such as '____ is ____' to be filled in with the appropriate argument (object) and

"...the form of the sentence cited, therefore, the symbolization f(x) has now been generally accepted. The property is here conceived as a function of the thing, which in turn is regarded as the argument of the function." (Reichenbach p. 81)

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."