Interpretation of a truth-functional propositional calculus

An interpretation of a truth-functional propositional calculus \({\mathcal {P}}\) is an assignment to each propositional symbol of \({\mathcal {P}}\) of one or the other (but not both) of the truth values truth (T) and falsity (F), and an assignment to the connective symbols of \({\mathcal {P}}\) of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of truth tables.

For \(n\) distinct propositional symbols there are \(2^{n}\) distinct possible interpretations. For any particular symbol \(a\), for example, there are \(2^{1}=2\) possible interpretations:

  1.  \(a\) is assigned T, or
  2.  \(a\) is assigned F.

For the pair \(a\), \(b\) there are \(2^{2}=4\) possible interpretations:

  1. both are assigned T,
  2. both are assigned F,
  3. \(a\) is assigned T and \(b\) is assigned F, or
  4. \(a\) is assigned F and \(b\) is assigned T.

Since \({\mathcal {P}}\) has \(\aleph _{0}\), that is, denumerably many propositional symbols, there are \(2^{\aleph _{0}}={\mathfrak {c}}\), and therefore uncountably many distinct possible interpretations of \({\mathcal {P}}\).


Interpretation of a sentence of truth-functional propositional logic

If φ and ψ are formulas of \({\mathcal {P}}\) and \({\mathcal {I}}\) is an interpretation of \({\mathcal {P}}\) then the following definitions apply:

  • A sentence of propositional logic is true under an interpretation \({\mathcal {I}}\) if \({\mathcal {I}}\) assigns the truth value T to that sentence. If a sentence is true under an interpretation, then that interpretation is called a model of that sentence.
  • φ is false under an interpretation \({\mathcal {I}}\) if φ is not true under \({\mathcal {I}}\).
  • A sentence of propositional logic is logically valid if it is true under every interpretation.
    \(\models \) φ means that φ is logically valid.
  • A sentence ψ of propositional logic is a semantic consequence of a sentence φ if there is no interpretation under which φ is true and ψ is false.
  • A sentence of propositional logic is consistent if it is true under at least one interpretation. It is inconsistent if it is not consistent.

Some consequences of these definitions:

  • For any given interpretation a given formula is either true or false.
  • No formula is both true and false under the same interpretation.
  • φ is false for a given interpretation iff \(\neg \phi \) is true for that interpretation; and φ is true under an interpretation iff \(\neg \phi \) is false under that interpretation.
  • If φ and \((\phi \to \psi )\) are both true under a given interpretation, then ψ is true under that interpretation.
  • If \(\models _{\mathrm {P} }\phi \) and \(\models _{\mathrm {P} }(\phi \to \psi )\), then \(\models _{\mathrm {P} }\psi \).
  • \( \neg \phi\)  is true under \({\mathcal {I}}\) iff φ is not true under \({\mathcal {I}}\).
  • \((\phi \to \psi )\) is true under \({\mathcal {I}}\) iff either φ is not true under \({\mathcal {I}}\) or ψ is true under \({\mathcal {I}}\).
  • A sentence ψ of propositional logic is a semantic consequence of a sentence \((\phi \to \psi )\) is logically valid, that is, \(\phi \models _{\mathrm {P} }\psi \) iff \(\models _{\mathrm {P} }(\phi \to \psi )\).