Propositional Logic Sentence

A Propositional Logic Sentence is a formal logic sentence composed of propositional variables and propositional relations.

• AKA: ϕ, Propositional Formula, Well-Formed Boolean Formula.
• Context:
  • It can (typically) abide by a Propositional Logic Grammar.
  • It can be a Falsifiable Propositional Formula (if there is an interpretation that maps the formula to false).
  • It can range from being a Tautological Propositional Formula (if it becomes true for every interpretation) to being a Contradictory Propositional Forumla (if it becomes false for every interpretation).
  • It can range from being an Abstract Propositional Formula to being a Propositional Formula Structure, such as a truth table.
• Example(s):
  • [math]\displaystyle{ A }[/math], where [math]\displaystyle{ A }[/math] is a propositional variable.
  • [math]\displaystyle{ ¬ A }[/math], with a logical negation relation.
  • [math]\displaystyle{ A ∨ B }[/math], with a logical disjunction relation.
  • [math]\displaystyle{ A ∧ B }[/math], with a logical conjunction relation.
  • [math]\displaystyle{ A → B }[/math], with a Logical Implication Relation.
  • [math]\displaystyle{ A ↔ B }[/math], with a Logical Bi-implication Relation.
  • [math]\displaystyle{ A → B ↔ ¬A ∨ B }[/math].
  • [math]\displaystyle{ f → B }[/math] is satisfiable (choose [math]\displaystyle{ B }[/math] = true or false)
  • [math]\displaystyle{ A → f }[/math] is falsifiable (choose A = true)
  • [math]\displaystyle{ (A → B) ↔ (¬A ∨ B) }[/math], is a Tautology.
  • [math]\displaystyle{ (A → B) ∧ A ∧ ¬B }[/math], is a Contradiction.
  • …
• Counter-Example(s):
  • a First-Order Logic Sentence with logic functions, such as:
    • [math]\displaystyle{ C }[/math] = P(a,x) ∨ Q(b) ∨ ¬R(x).
    • [math]\displaystyle{ D }[/math] = P(x,a) ∨ Q(x) ∧ P(y,a) ∨ Q(b).
• See: Logic Literal, Logic Clause, Predictate Logic Sentence, Mathematical Sentence.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Propositional_formula
  • In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula.

    A propositional formula is constructed from simple propositions, such as "x is greater than three" or propositional variables such as P and Q, using connectives such as NOT, AND, OR, and IMPLIES; for example:

    (x = 2 AND y = 4) IMPLIES x + y = 6.

    In mathematics, a propositional formula is often more briefly referred to as a “proposition", but, more precisely, a propositional formula is not a proposition but a formal expression that denotes a proposition, a formal object under discussion, just like an expression such as "x + y" is not a value, but denotes a value. In some contexts, maintaining the distinction may be of importance.

2005

• (Goldrei, 2005) ⇒ Derek Goldrei. (2005). “Propositional and Predicate Calculus: A mOdel of Argument." Springer.
  • QUOTE: Our formal version of statements, which we'll call formulas, is given by the following definition.
    • Definition: Formula Let [math]\displaystyle{ P }[/math] be a set of propositional variables and let [math]\displaystyle{ S }[/math] be the set of connectives {...}. A formula is a member of the set Form(P,S) of strings of symbols involves elements of P, S and brackets (and ) formed according to the following rules.
      • (i) Each propositional variable is a formula.
      • (ii) If theta and ψ are formuals, then so are::
        • ¬θ
        • (θ ∧ ψ)
        • (θ ∨ ψ)
        • (θ → ψ)
        • (θ ↔ ψ)
      • (iii) All formulas arise from finitely many applications of (i) and (ii).
      • If we use a different set [math]\displaystyle{ S }[/math] of connectives, for instance just {Or, implies}, then clause (ii) is amended accordingly to cover just these symbols.
    • In many books the phrase well-formed formula is used instance of formula. These 'well-formed' emphasizes that the string has to obey special construction rules.

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