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As the title says, What is the difference between validity and satisfiability?
Suppose I have a sentence If the sun is made of blue cheese, then cats fly.
How do I tell if its valid or satisfiable?

Helosy
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1 Answers1

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In mathematical logic, satisfiability and validity are elementary concepts of semantics.

A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true.

A formula is valid if all interpretations make the formula true.

A formula $\varphi$ is valid iff its negation : $\lnot \varphi$ is unsatisfiable.

Examples from propositional logic :

$\lnot p \lor p$ is valid;

$p \lor q$ and $p \to q$ are examples of satisfiable formulas (but not valid);

$p \land \lnot p$ is unsatisfiable.

The above concepts appy to formulas; a single statement of natural language is either true or false.

Specifically, the statement :

"If the sun is made of blue cheese, then cats fly",

if we read the connective "if..., then ..." in the truth-functional way (i.e. as the material conditiona), is true, because the antecedent : "the sun is made of blue cheese" is false.

The sentence is an instance of the formula $p \to q$, that is satisfiable (but not valid).

Community
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Mauro ALLEGRANZA
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  • I think the line "The above concepts appy to formulas; a single statement of natural language is either true or false." is misleading, since it's easy to read the emphasis as being on formulas vs. sentences (and of course a sentence is not true or false in the absence of an interpretation either); in fact, even the reference to natural language is misleading, since the real point is the role of "the world" as an intended interpretation. – Noah Schweber Sep 09 '18 at 15:56