Using PL to Describe Properties of Systems

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 $\models _{\mathrm {P} }(\phi \to \psi )$.