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252:"if anything is such that X then it is such that Y". Also, the quantifiers are given their usual objectual readings, so that a positive existential statement has existential import, while a universal one does not.) An analogous case concerns the empty conjunction and the empty disjunction. The semantic clauses for, respectively, conjunctions and disjunctions are given by
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In other words, an existential quantification of the open formula φ is true in a model iff there is some element in the domain (of the model) that satisfies the formula; i.e. iff that element has the property denoted by the open formula. A universal quantification of an open formula φ is
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Logics whose theorems are valid in every, including the empty, domain were first considered by
Jaskowski 1934, Mostowski 1951, Hailperin 1953, Quine 1954, Leonard 1956, and Hintikka 1959. While Quine called such logics "inclusive" logic they are now referred to as
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is the empty set having no members. In traditional and classical logic domains are restrictedly non-empty in order that certain theorems be valid. Interpretations with an empty domain are shown to be a trivial case by a convention originating at least in 1927 with
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true in a model iff every element in the domain satisfies that formula. (Note that in the metalanguage, "everything that is such that X is such that Y" is interpreted as a
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358:{\displaystyle A\models \phi _{1}\land \dots \land \phi _{n}\iff \forall \phi _{i}(1\leq i\leq n),A\models \phi _{i}}
469:{\displaystyle A\models \phi _{1}\lor \dots \lor \phi _{n}\iff \exists \phi _{i}(1\leq i\leq n),A\models \phi _{i}}
154:{\displaystyle A\models \exists x\phi (x){\text{ iff there is an }}a\in A{\text{ such that }}A\models \phi }
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It is easy to see that the empty conjunction is trivially true, and the empty disjunction trivially false.
233:{\displaystyle A\models \forall x\phi (x){\text{ iff every }}a\in A{\text{ is such that }}A\models \phi }
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32:red areas are nonempty.)
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539:-related article is a
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486:free logic
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492:See also
248:of the
58:Quine's
50:Bernays
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66:truth,
44:domain
42:empty
40:, the
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