Sentence examples for has the same extension from inspiring English sources

A set A is regular iff it has the same extension with respect to both membership relations: x ∈ A ≡ x ε A. The comprehension axiom asserts that for any uniform formula φ (x) in which all parameters (free variables other than x) are regular, there is an object {x | φ (x)} such that ∀x(x ∈ A ≡ φ* & x ε A ≡ φ).

For consider the unary connective \(\dq\), defined by the clause Assuming that Kripke (1971 19800) is right that water is necessarily H2O, \(\dq\) has the same extension as \(\dq{\neg}\) in every possible world, and so satisfies the permutation invariance criterion as a matter of necessity (McGee 1996, 578).

On a domain \(D\) containing no two objects with exactly the same mass, \(\dq{\approx}\) has the same extension as \(\dq\)—the set \(\{ \langle x, x \rangle : x \in D\}\)—and as we have seen, this extension is invariant under every permutation of the domain.

Thus, 'Scott' and 'the author of Waverly' have the same extension (in the actual state-description).

Thus, being and goodness have the same extension while differing intensionally.

On her view, 'this small fragrant wild flower' and 'Clematis' have the same extension, yet they possess distinct intensions (Jones 1893 94, 36).

For example, 'creature with a heart' and 'creature with a kidney' have the same extension because they are true of the same individuals: all the creatures with a kidney are creatures with a heart.

It is likely that Philip encountered the fundamental notion of extensional equivalence and intensional difference in the work of the Arabic commentators of Aristotle; both Avicenna and Averroes argue that unity and being have the same extension while differing conceptually.

Applying the well-known technique of interpreting sentences as predicators of degree 0, he generalizes the fact that two predicators of degree n (say, P and Q) have the same extension if and only if ∀x1∀x2…∀xn(Px1x2…xn ↔ Qx1x2…xn) holds.

Medium coarse: Properties are identical just in case they necessarily have the same extension (the precise import of this condition depends on which notion of necessity is at play).

Then, analogously, two sentences (say, p and q), being interpreted as predicators of degree 0, must have the same extension if and only if p↔q holds, that is if and only if they are equivalent.