Propositional calculus

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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.

Now, let;s get into these logical connectives and atomic formulae; what are they?

There are two sets in the language of propositional calculus: the first is a set of primitive symbols, that are variously known as atomic formulas, placeholders, proposition letters, or variables; the second set is a set of operator symbols, that are variously interpreted as logical operators or logical connectives.

Let’s also figure out what a well-formed formula is, right? A well-formed formula is essentially an atomic formula, or any formula that you can build up from atomic formulas through the use of operator symbols according to the rules of the grammar.

Sometimes mathematicians might point out the differences between propositional constants, propositional variables, and schemata. Propositional constants symbolize some particular proposition, while propositional variables, on the other hand, range over the entire set of all 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.

Propositional calculus
Source: www.cs.otago.ac.nz

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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What are the 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 ⊃). 

What is an argument in propositional calculus?

Propositional calculus defines an argument as a set of propositions. Essentially, a valid argument is a list of propositions in which the last proposition follows from or is implied by the rest of the propositions that preceed it. All other arguments are not considered valid. Modus ponens is  the simplest valid argument there is. One instance of modus ponens is this  list of propositions:

  1. P ⟶ Q
  2. P

_______________

∴ Q

This is a list of three propositions. Every line is a proposition, and the last one follows from the rest. The first two lines are the premises and the last line is the conclusion. You could say that any proposition C follows from any set of propositions (P1,..., Pn) if C has to be true whenever every member of the set (P1,..., Pn) is true. In the arguement mentioned above,  for any P and Q, whenever P → Q and P are true, then Q also has to be true.  When P is true, you can’t consider cases 3 and 4 from the truth table. When P → Q is true, you can’t consider case 2 from the truth table. That means that only case 1 is left, in which Q is also true. This means that Q is implied by the premises.

This generalizes in a schematic manner. That means, where φ and ψ may be any propositions at all,

  1. φ ⟶ ψ
  2. Φ

______________

∴ ψ

 

Other argument forms are convenient, but they aren’t necessary. With  a complete set of axioms, modus ponens is good enough to prove all other argument forms in propositional logic, which means that they could be considered to be a derivative. However, you need to know that this is not true of the extension of propositional logic to other logics like first-order logic. In first-order logic, at least one additional rule of inference is needed for the purpose of obtaining completeness.

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