Logic Sentence

A logic sentence is a formal sentence that abides by a logic language.

• AKA: Logic Expression, Well-Formed Formula, WFF, Formal Logic Sentence, Deductive Logic Formula.
• Context:
  • It must contain one or more Logic Literals.
  • It can contain Logical Operators.
  • It can be a part of a Logic Statement (along with a truth value)
  • It can range from being a Propositional Logic Formula to being a First-Order Logic Formula.
  • It can be in Disjunctive Normal Form.
  • It can be in Conjunctive Normal Form.
  • It can range from being a Satisfiable Logic Sentence to being an Unsatisfiable Logic Sentence.
• Example(s):
  • Propositional Logic Sentence/Boolean Logic Sentence.
    • [math]\displaystyle{ y }[/math]
    • [math]\displaystyle{ x ∧ y }[/math].
    • [math]\displaystyle{ x ∧ ¬x }[/math], an Unsatisfiable Logic Sentence.
    • (X ∧ Y) ∨ ¬(W ∧ Z).
    • (X ∧ Y) ∨ W.
    • (X ∧ ¬ Y) ∨ (Not $W$ ∧ Z). (in Disjunctive Normal Form)
    • [math]\displaystyle{ ¬(X ∨ Y)/\lt math\gt . *** \lt math\gt X }[/math] ∨ (Y ∧ (W Or Z)).
    • (X ∨ ¬Y ∨ Not W) ∧ (X ∨ [math]\displaystyle{ Y }[/math] ∨ Z), in Conjunctive Normal Form with two Logic Clauses, four Logic Literals, and 3 Literals per Clause.
  • First-Order Logic Sentence.
    • ∃x, [math]\displaystyle{ f }[/math](x).
    • ∀x, [math]\displaystyle{ f }[/math](x).
    • ForNox, [math]\displaystyle{ f }[/math](x).
    • ∃x, ¬f(x).
    • ∀x, ¬f(x).
    • ForNox, ¬f(x).
    • ∀x (CHILD(x) → LOVES(x,Santa)); “Every child loves Santa.”
    • ∀x (LOVES(x,Santa) → ∀ y (REINDEER(y) → LOVES(x,y))); “Everyone who loves Santa loves any reindeer.”
    • REINDEER(Rudolph) ∧ REDNOSE(Rudolph); “Rudolph is a reindeer, and Rudolph has a red nose.”
    • ∀x (REDNOSE(x) → WEIRD(x) ∨ CLOWN(x)); “Anything which has a red nose is weird or is a clown.”
    • ¬ ∃x (REINDEER(x) ∧ CLOWN(x)); “No reindeer is a clown.”
    • ∀x (WEIRD(x) → ¬ LOVES(Scrooge,x)); “Scrooge does not love anything which is weird.”
    • ¬ CHILD(Scrooge); “Scrooge is not a child.”
    • a DL Rule.
  • …
• Counter-Example(s):
  • Boolean Logic Proposition.
    • Y ⇒ True.
    • X And Y ⇒ False.
  • Socrates was a man ⇒ True.
  • "I have seen a white swan. ⇒ True.
  • “Person X will likely choose Y with 85% likelihood”. (A Probabilistic Statement).
  • “Who are you?” (a Query).
  • “Run!” (a Command).
  • “Greenness perambulates”
  • “I had one grunch but the eggplant over there.”
• See: Mathematical Sentence, Semantic Relation, Predicate Formula Variable, Rule Antecedent.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
  • In mathematical logic, a sentence of a predicate logic is a well formed formula with no free variables. A sentence is viewed by some as expressing a proposition. It makes an assertion, potentially concerning any structure of L. This assertion has a fixed truth value with respect to the structure. In contrast, the truth value of a formula (with free variables) may be indeterminate with respect to any structure. As the free variables of a formula can range over several values (which could be members of a universe, relations or functions), its truth value may vary.
• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Well-formed_formula
  • In computer science and mathematical logic, a well-formed formula or simply formula[1] (often abbreviated WFF, pronounced "wiff" or "wuff") is a symbol or string of symbols that is generated by the formal grammar of a formal language. To say that a string \ S is a WFF with respect to a given formal grammar \ G is equivalent to saying that \ S belongs to the language generated by \ G. A formal language can be identified with the set of its WFFs.
  • A key use of WFFs is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated.
  • In formal logic, proofs can be represented by sequences of WFFs with certain properties, and the final WFF in the sequence is what is proven. This final WFF is called a theorem when it plays a significant role in the theory being developed, or a lemma when it plays an accessory role in the proof of a theorem.
• http://en.wiktionary.org/wiki/well-formed_formula
  • A statement that is expressed in a valid, syntactically correct, manner
• http://www.coli.uni-saarland.de/projects/milca/courses/comsem/xhtml/d0e1-gloss.xhtml
  • FOL: formulae that are built from a vocabulary, the logical symbols of FOL and first-order variables according to the syntax rules of FOL.
• http://www.earlham.edu/~peters/courses/logsys/glossary.htm
  • Wff. Acronym of "well-formed formula", pronounced whiff. A string of symbols from the alphabet of the formal language that conforms to the grammar of the formal language. See decidable wff, formal language.
• http://www.earlham.edu/~peters/courses/logsys/glossary.htm
  • Closed wff. In predicate logic, a wff with no free occurrences of any variable; either it has constants in place of variables, or its variables are bound, or both. Also called a sentence. See bound variables; free variables; closure of a wff.
• http://www.earlham.edu/~peters/courses/logsys/glossary.htm
  • Open wff. In predicate logic, a wff with at least one free occurrence a variable. See free variables; propositional function. Some logicians use the terms, 1-wff, 2-wff,...n-wff for open wffs with 1 free variable, 2 free variables, ...n free variables. (Others call these 1-formula, 2-formula,...n-formula.)
• CYC Glossary http://www.cyc.com/cycdoc/ref/glossary.html
  • well-formed formula: A formula in CycL is well-formed if it conforms to the syntax of CycL and passes all the restrictions on arity and argument types of the relations that are used in it.
• docs.rinet.ru/KofeynyyPrimer/ch38.htm
  • expression: Results in a value of true or false.

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