Editing Raven paradox (section)

====Existential presuppositions====
A number of authors have argued that propositions of the form "All <math>A</math> are <math>B</math>" presuppose that there are objects that are <math>A</math>.<ref>Strawson PF (1952) ''Introduction to Logical Theory'', Methuan & Co. London, John Wiley & Sons, New York</ref> This analysis has been applied to the raven paradox:<ref name=Cohen1987>{{cite journal |last=Cohen |first=Yael |date=March 1987 |title=Ravens and relevance |journal=[[Erkenntnis]] |volume=26 |issue=2 |pages=153–179 |doi=10.1007/BF00192194|s2cid=122284270 }}</ref>

{{quote|... <math>H_1</math>: "All ravens are black" and <math>H_2</math>: "All nonblack things are nonravens" are not ''strictly equivalent'' ... due to their different existential presuppositions. Moreover, although <math>H_1</math> and <math>H_2</math> describe the same regularity – the nonexistence of nonblack ravens – they have different logical forms. The two hypotheses have different senses and incorporate different procedures for testing the regularity they describe.}}

A modified logic can take account of existential presuppositions using the presuppositional operator, '*'. For example,

<math display="block">\forall\ x,\ *Rx\rightarrow Bx</math>

can denote "All ravens are black" while indicating that it is ravens and not non-black objects which are presupposed to exist in this example.

{{quote|... the [[logical form]] of each hypothesis distinguishes it with respect to its recommended type of supporting evidence: the possibly true [[substitution instance]]s of each hypothesis relate to different types of objects. The fact that the two hypotheses incorporate different kinds of testing procedures is expressed in the formal language by prefixing the operator '*' to a different predicate. The presuppositional operator thus serves as a relevance operator as well. It is prefixed to the predicate '<math>x</math> is a raven' in <math>H_1</math> because the objects relevant to the testing procedure incorporated in "All raven are black" include only ravens; it is prefixed to the predicate '<math>x</math> is nonblack', in <math>H_2</math>, because the objects relevant to the testing procedure incorporated in "All nonblack things are nonravens" include only nonblack things. ... Using [[Fregean]] terms: whenever their presuppositions hold, the two hypotheses have the same [[sense and reference|referent]] (truth-value), but different [[sense and reference|senses]]; that is, they express two different ways to determine that truth-value.<ref name=Cohen1987/>}}