# Logic Sentence

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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*). - ForNo
*x*, [math]\displaystyle{ f }[/math](*x*). - ∃
*x*, ¬*f*(*x*). - ∀
*x*, ¬*f*(*x*). - ForNo
*x*, ¬*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.

- ∃
- …

- Propositional Logic Sentence/Boolean Logic Sentence.
**Counter-Example(s):**- Boolean Logic Proposition.
*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.

## 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
- docs.rinet.ru/KofeynyyPrimer/ch38.htm
- expression: Results in a value of true or false.