Talk:Predicate logic

For Wikipedia Editors, Where is Predicate Calculus?

Why is predicate calculus redirected to predicate logic, the two are not really the same thing at all. Predicate Calculus should be under the section on AI not maths or logic. Tautologies are the same - no article. - There seems to be a general lack of articles on AI, one of the very few weak areas of Wikipedia - .. Lucien86 21:05, 26 September 2007 (UTC) (I could work as an author in this area but there must be university people who know a lot more about the intricacies of the subject on a formal level than I do, and of course have much better (+free) access to reference information.)

Lucien86 21:05, 26 September 2007 (UTC)

some content possibilities

• Notion of "predicate logic" versus "predicate calculus" (the axiomatic/axiomatized form of "the predicate logic")

• Like propositional logic/calculus, either has to do with { TRUTH, FALSITY } of "sentences" (objects standing, or not standing, in relations; objects as members, or not members, of collections/sets)

• (Restrict the presentation to first order logic ? Refer reader elsewhere for higher-order logics ?)
• Why predicate logic?

• Extension of the propositional logic by (treating, examining, analyzing) simple sentences into two concatenated components (parts) (where | repesents concatenation) : "subject|predicate".

• Propositional logic applied to the finite is sufficient, predicate logic is not necessary (cf Stolyar 1970) the finite assertions. HOWEVER, extension to infinite sets requires [?? is this true] the LoEM law of Excluded Middle. From this we can infer that the interesting stuff w.r.t. "predicate logic" occurs around its applicatation to infinite domains. Thus the notion of generality (universality). Stolyar is a good reference for this.

• ~

• propositions regarding generality: "propositions expressing a property or relationship of objects of an entire set " (p. 149)
• propositions regarding existence: "propositions on the existence of objects belonging to a given set possessing a certain property or having a certain relationship with other objects " (p. 149)

• Subject: object or objects of discourse
• Predicate: asserts an attribute or quality to the object(s) of discourse
• Sentence skeleton (Kleene 1952:144): " x has attribute f " "The predicate f is then a function of one variable x. This variable ranges over some domain of objects".
• Notion of the subject as a member of a class (set) of objects discribed by the predicate, so there is a truth relationship between an instance of the subject (that pig over there) as being a member of a class (winged objects) with a { truth, falsity } outcome of the assertion.
• Expression of a relationship of belonging ∈ (element of), so that p ∈ W. (this pig is a member of the set/class of winged objects" as opposed to ("all pigs are members of set/class of winged objects)
• Generalizes this into an algebraic form into subject = variable e.g. "p", predicate e.g. "W" written as "W(p)"

• Expressions of relationship:, sentence skeletons such as: "___ is equal to ___"; "___ is the brother of ___ ". " x R y " where R is a relationship between x and y.

• Formula formation, nuts and bolts:

• P: predicate of zero variables, i.e. a propositon
• R(x): predicate of 1 variable: "x is in relation with attribute R"
• R(x, y): predicate of two variables: "x is in relation R with y"

• define wff
• parsing: symbol rank (seniority) cf propositional formula

• Predicate logic involves the notion of function with a range { T, F }
• Predicate functions partition the domain of discourse into a truth set { subset of domain that evaluates to } => T
• Example of a finite collection of objects plus Venn diagrams helps to illustrate the ideas of the "generalization" quantifier ∀ and existence quantifier ∃ when "generalized" to an infinite collection
• Bound versus free variables

• The quantifiers ∀x and ∃x bind the variable x

• Universal validity (over the domain of definition/discourse), extends notion of tautology
• The basic equivalences

• Generality ∀ is similar to logical AND, Existence ∃ is similar to logical OR
• (a V b V c) ≡ ~(~a & ~b & ~c): ∃p:W(p) ≡ ~∀p:~W(p); "Wigned pigs exists." ≡ "It's not the case that gvien all pigs there are no winged pigs."
• (a & b & c) ≡ ~(~a V ~b V ~c): ∀p:W(p) ≡ ~∃p:~W(p)  ; "All pigs have wings." ≡ "NOT even one NOT-winged pig exists."
• ~(a & b & c) ≡ (~a V ~b V ~c): ~∀p:W(p) ≡ ∃p:~W(p)
• etc
• (a & q) & (b & r) & (c & s) ≡ (a & b & c) & (q & r & s) : ∀p:( W(p) & B(p) ) ≡ ∀p:W(p) & ∀p:B(p)
• etc (Stolyar 1970:164-165)

• Additional axioms that turn of predicate logic into the predicate calculus (Stolyar's distinction between a logic and a calculus) (p. 166). Here "y" is a variable (or also a constant in some theories e.g. Goedel 1931)

• ∀p:W(p) → W(y) ; "For all pigs, "pig|has wings" is true" implies "winged pigginess" as a constant in the universe ??
• W(y) → ∃p:W(p) ; "Winged pigginess" implies that "Some pigs have wings".
• From Q → W(p) INFER Q → ∀p:W(p)
• From W(p) → Q INFER ∃p:W(p) → Q

• Prenex normal form: All the quantifiers ∃ and ∀ can be placed "up front" (on the far left of) any formula
• Predicate logic with equality
• Elimination of ∃x:A(x) ≡ ~∀x:~A(x)

• ( (pig #1 is winged) & (pig #2 is winged) & (pig #3 is winged) ) can be evaluated and handled by the existing rules. The equivalence "There exists at least one pig that is winged, when extended over a domain of three pigs, becomes: EITHER ( (pig #1 is winged) OR (pig #2 is winged) OR (pig #3 is winged) ) which by de Morgan's laws is NOT-( NOT-(pig #1 is winged) AND NOT-(pig #2 is winged) AND NOT-(pig #3 is winged)≡ It is NOT the case that ( (pig #1 is NOT winged) AND (pig #2 is NOT winged) AND (pig #3 is not winged)) i.e. ∃p:W(p) ≡ ~∀p:~W(p) over finite sets. Generality extends de Morgan's law to the infinite.

• First order logics
• Higher-order logics, reduction to first order logic ??

• HISTORY: where did this come from? Aristotle syllogism ... esp. Boole, calculus of sets, Venn, then Whitehead-Russell (PM)

wvbailey Wvbailey (talk) 21:50, 17 November 2007 (UTC)

The "syntax" section seems to be extracted from a documentation of SWI-Prolog

Which is quite obvious after some careful reading. --Xiaq (talk) 14:30, 19 August 2011 (UTC)