Using PL to Describe Properties of Systems
Interpretation of a truth-functional propositional calculus
An interpretation of a truth-functional propositional calculus is an assignment to each propositional symbol of 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 of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of truth tables.
For distinct propositional symbols there are distinct possible interpretations. For any particular symbol , for example, there are possible interpretations:
For the pair , there are possible interpretations:
- both are assigned T,
- both are assigned F,
- is assigned T and is assigned F, or
- is assigned F and is assigned T.
Since has , that is, denumerably many propositional symbols, there are , and therefore uncountably many distinct possible interpretations of .
Interpretation of a sentence of truth-functional propositional logic
If φ and ψ are formulas of and is an interpretation of then the following definitions apply:
- A sentence of propositional logic is true under an interpretation if 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 if φ is not true under .
- A sentence of propositional logic is logically valid if it is true under every interpretation.
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 is true for that interpretation; and φ is true under an interpretation iff is false under that interpretation.
- If φ and are both true under a given interpretation, then ψ is true under that interpretation.
- If and , then .
- is true under iff φ is not true under .
- is true under iff either φ is not true under or ψ is true under .
- A sentence ψ of propositional logic is a semantic consequence of a sentence is logically valid, that is, iff .