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Take ordinary material logical implication on $\{0,1\}$, designate $1$, and interpret negation by the constant-zero operation $N$. Then $A\to (B\to A)$ and $(NB\to A)\to (NA\to B)$ are always designated, and modus ponens rule and logical implication discharge preserve consequence. But $(NB\to NA)\to (A\to B)$ is false for $A=1,B=0$. Thus the first two schemes and the deduction rule do not derive the last scheme. This detects the loss of a genuine classical-negation principle even though all the displayed schemes are tautologies under the usual classical negation.

In the Löb's theorem proof, $T⊢\square D\to \square A$ is obtained before any reflection assumption. Put $H=(\square A\to A)$. Propositional reasoning and the diagonal equation give $T⊢H\to D$. Necessitation and distribution yield $T⊢\square H\to \square D$, hence the displayed implication. This is the arithmetic interpretation of the Löb axiom in Gödel-Löb logic, and avoids necessitating a temporary assumption.

For standard provability satisfying positive introspection, suppose $T⊢{\square }^{j}A\to A$ for some positive integer $j$. Positive introspection gives $T⊢\square A\to {\square }^{j}A$, hence $T⊢\square A\to A$. Löb's theorem yields $T⊢A$. The same argument applies with an already iterated first-order sentence in place of $A$.

William Nicol developed the Nicol prism, an optical device for producing polarized light. A surviving instrument contains two pieces of calcite joined with Canada balsam, using the crystal's birefringence to separate the optical responses.^[ref]

Used as a polarizer and analyser in microscopy, such optics make mineral behaviour under polarized light observable. Rotation and transmission changes help distinguish crystalline properties that ordinary illumination conceals. This connects instrument construction with practical mineralogy and petrology.

A first-order formula is Delta-one modulo ZF if that theory proves it equivalent both to a Sigma-one formula in set theory and to a Pi-one formula in set theory, with the same free variables. Equivalently its assertion and negation each have existential-bounded definitions over ZF. A witness gives upward absoluteness for the first definition; the universal form gives downward absoluteness. Thus the formula has Delta-one absoluteness between nested transitive models of ZF.