2005 PropositionalAndPredicateCalculus

• (Goldrei, 2005) ⇒ Derek Goldrei. (2005). “Propositional and Predicate Calculus: A Model of Argument.” In: Springer.

Subject Headings: Propositional Logic System, Predicate Logic System.

Notes

• It defines Propositional Variable, Propositional Formula.

Quotes

2.1 Introduction

• 'Proposition' is often used to mean a statement about which it is sensible to ask whether it is true of false.

2.2 The construction of propositional formulas

• In this section we shall describe the formal language which we shall use to represent statements.
• With considerations like these in mind, we shall define our formal statements as follows. First we shall specify the formal language, that is, the symbols from which strings can be formed. We shall always allow brackets - these will be needed to avoid ambiguity. We shall specify a set [math]\displaystyle{ P }[/math] of basic statements, called propositional variables. From these we can build more complex statements by joining statements together using brackets and symbols in a set [math]\displaystyle{ S }[/math] of connectives, which are going to represents ways of connecting statements to each other, like V for 'or' and other symbols mentioned earlier. ...
  • Convention for variables We shall normally use individual lower case letters like p, q, r, s, ... and subscripted letters like p₀,p₁,p₂,...p_n,... for our propositional variables. Distinct letters or subscripts give us distinct symbols. When we don't specific the set [math]\displaystyle{ P }[/math] of propositional variables in a precise way, we shall use p,q,r and so on to represent different members of the set.
• 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.

3.2 A formal system for propositional calculus

• We are about to describe a formal proof system and say what is meant by a formal derivation of a formula. Our aim is that the formal system should match logical consequence. For a set Γ of formulas and a formula ϕ, we write Γ |= ϕ to express that ϕ is a logical consequence of Γ. We can record Γ as a set of assumptions from which ϕ follows. We shall use the similar notation Γ |- ϕ to express that there is a formal derivation of ϕ from Γ.

References

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	Author	volume	Date Value	title	type	journal	titleUrl	doi	note	year
2005 PropositionalAndPredicateCalculus	Derek Goldrei			Propositional and Predicate Calculus: A Model of Argument			http://www.springer.com/math/book/978-1-85233-921-0