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Propositional calculus

What is propositional calculus?

Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.

Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

What is propositional calculus used for?

As a formal system the propositional calculus is concerned with determining which formulas (compound proposition forms) are provable from the axioms. Valid inferences among propositions are reflected by the provable formulas, because (for any A and B) A ⊃ B is provable if and only if B is always a logical consequence of A. The propositional calculus is consistent in that there exists no formula in it such that both A and ∼A are provable. It is also complete in the sense that the addition of any unprovable formula as a new axiom would introduce a contradiction. Further, there exists an effective procedure for deciding whether a given formula is provable in the system.

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Are truth symbols in propositional calculus?

 Simple (atomic) propositions are denoted by letters, and compound propositions are formed using the standard symbols: · for “and,” ∨ for “or,” ⊃ for “if . . . then,” and ∼ for “not.”

The propositional calculus is based on the study of well-formed formulas, or wff for short. New wff of the form (∼A), (A ∨ B), (A ∧ B), (A ⊃ B), (A ≡ B), (IF A THEN B ELSE C)are formed from given wff A, B, and C using logical connectives; respectively they are called negation, disjunction, conjunction, implication, equivalence, and conditional. 

(∼A), (A ∨ B), (A ∧ B),

(A ⊃ B), (A ≡ B),

(IF A THEN B ELSE C)

Proofs and theorems within the propositional calculus are conducted in a formal and rigorous manner: certain basic axioms are assumed and certain rules of inference are followed. In particular these rules must deal with the various connectives.

The rules of inference are stated using a form such as:

  • α
  • β

The rule should be interpreted to mean that on the assumption that α is true, it can be deduced that β is then true. Logicians often use the notation α|-β. In writing the rules it is convenient to employ a notation such as Γ, A ⇒ B Γ is some set of wff whose truth has been established; A and B are some other wff highlighted for the purposes of the rule; ⇒ denotes implication (to avoid confusion with the symbol ⊃). 


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Propositional calculus

October 14, 2020

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What is propositional calculus?

Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.

Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

What is propositional calculus used for?

As a formal system the propositional calculus is concerned with determining which formulas (compound proposition forms) are provable from the axioms. Valid inferences among propositions are reflected by the provable formulas, because (for any A and B) A ⊃ B is provable if and only if B is always a logical consequence of A. The propositional calculus is consistent in that there exists no formula in it such that both A and ∼A are provable. It is also complete in the sense that the addition of any unprovable formula as a new axiom would introduce a contradiction. Further, there exists an effective procedure for deciding whether a given formula is provable in the system.

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Are truth symbols in propositional calculus?

 Simple (atomic) propositions are denoted by letters, and compound propositions are formed using the standard symbols: · for “and,” ∨ for “or,” ⊃ for “if . . . then,” and ∼ for “not.”

The propositional calculus is based on the study of well-formed formulas, or wff for short. New wff of the form (∼A), (A ∨ B), (A ∧ B), (A ⊃ B), (A ≡ B), (IF A THEN B ELSE C)are formed from given wff A, B, and C using logical connectives; respectively they are called negation, disjunction, conjunction, implication, equivalence, and conditional. 

(∼A), (A ∨ B), (A ∧ B),

(A ⊃ B), (A ≡ B),

(IF A THEN B ELSE C)

Proofs and theorems within the propositional calculus are conducted in a formal and rigorous manner: certain basic axioms are assumed and certain rules of inference are followed. In particular these rules must deal with the various connectives.

The rules of inference are stated using a form such as:

  • α
  • β

The rule should be interpreted to mean that on the assumption that α is true, it can be deduced that β is then true. Logicians often use the notation α|-β. In writing the rules it is convenient to employ a notation such as Γ, A ⇒ B Γ is some set of wff whose truth has been established; A and B are some other wff highlighted for the purposes of the rule; ⇒ denotes implication (to avoid confusion with the symbol ⊃). 


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