User:Wvbailey/Predicate logic

Purpose of predicate logic

Some sentences cannot be analyzed effectively with the propositional calculus. The deductive form called modus ponens requires a first premise (assertion) of the form " IF...THEN..." and a second statement of existence of an specific object of discourse, such as in the following:

Example: "IF this pig #34 has wings THEN it is blue". "This pig #34 has wings." THEREFORE "This pig is blue".

The following sentence on the other hand has no IF... THEN structure and only a single premise. And yet we know that its reasoning yields a truth-value of TRUTH:

Example: "All blue pigs have wings. Therefore this blue pig #34 has wings."

The predicate logic allows us to study the

The predicate logic is part sentence-deconstruction method, part algebra, and part evaluation. Its purpose is the analysis of sentences down to their overall truth-value evaluation (TRUTH and FALSITY), given the truth values of their parts. The outcome of predicate logic will be a formula, and/or a truth-evaluation of a formula, that can be inserted into a propositional formula.

Thus predicate logic is an extension of the propositional logic (calculus), and as such the propositional logic becomes a component of the predicate calculus. For example, to the propositional calculus the predicate calculus adds four new axioms that use a new symbol ∀, a new symbol string " P(x) ", and expands the definition of a well-formed formula.

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 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 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 decomposes compound sentences (and strings of sentences) into simple sentences that are linked by the propositional connectives typically symbols & (AND), V (OR), ~(NOT), → (LOGICAL IMPLICATION), ≡ (LOGICAL EQUIVALENCE alternately ↔), and parentheses (, ). These and 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). Each unique simple sentence is assigned a new 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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w

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F

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T

"All blue pigs have wings."

"All blue pigs have wings" = a, "we found a blue pig" = b, "pig is blue" = w

"Pig is blue" AND "Pig has wings"

Predicate logic extends the analysis to the inside of propositional formulas. In the simplest analysis, it breaks a simple sentence such as "pigs have wings" into two parts: the subject "pig", i.e. the object 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"

Here ___ indicates an empty slot for a particular pig under discussion. the notion of "have wings" is now generalized to a template or notion of "wingedness", i.e. "winged thing".

Generalization to the infinite

The sentence All blue pigs have wings, when expressed about a pen of pigs containing those with wings and withou