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:
- Sentence Input: Logic Literals, Logical Operators
- Sentence Output: Well-Formed Formulas, Truth Values
- Sentence Measure: Logical Validity, Syntactic Correctness, Semantic Soundness
- ...
- 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 range from being a Satisfiable Logic Sentence to being an Unsatisfiable Logic Sentence.
- It can be in Disjunctive Normal Form.
- It can be in Conjunctive Normal Form.
- ...
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.
  - a Contract Law-Related Logic Rule.
- …
Counter-Examples:
- Boolean Logic Proposition:
  - Y ⇒ True.
  - X And Y ⇒ False.
- Socrates was a man ⇒ True.
- “I have seen a white swan. ⇒ True.

See: Mathematical Sentence, Semantic Relation, Predicate Formula Variable, Rule Antecedent, Formal Logic, Logic Formula, Logical Expression.

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.

Logic Sentence

References

2009

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