Editing Raven paradox (section)

====Rejecting material implication====
The following propositions all imply one another: "Every object is either black or not a raven", "Every raven is black", and "Every non-black object is a non-raven." They are therefore, by definition, logically equivalent. However, the three propositions have different domains: the first proposition says something about "every object", while the second says something about "every raven".

The first proposition is the only one whose domain of quantification is unrestricted ("all objects"), so this is the only one that can be expressed in [[first-order logic]]. It is logically equivalent to:

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

and also to

<math display="block">\forall\ x, \overline{Bx}\ \rightarrow\ \overline{Rx}</math>

where <math>\rightarrow</math> indicates the [[material conditional]], according to which "If <math>A</math> then {{nobr|<math>B</math>"}} can be understood to mean {{nobr|"<math>B</math> or <math>\overline{A}</math>".}}

It has been argued by several authors that material implication does not fully capture the meaning of "If <math>A</math> then {{nobr|<math>B</math>"}} (see the [[paradoxes of material implication]]). "For every object, {{nobr|<math>x</math>,}} <math>x</math> is either black or not a raven" is ''true'' when there are no ravens. It is because of this that "All ravens are black" is regarded as true when there are no ravens. Furthermore, the arguments that Good and Maher used to criticize Nicod's criterion (see {{slink||Good's baby}}, above) relied on this fact – that "All ravens are black" is highly probable when it is highly probable that there are no ravens.

To say that all ravens are black in the absence of any ravens is an empty statement. It refers to nothing. "All ravens are white" is equally relevant and true, if this statement is considered to have any truth or relevance.

Some approaches to the paradox have sought to find other ways of interpreting "If <math>A</math> then {{nobr|<math>B</math>"}} and "All <math>A</math> are {{nobr|<math>B</math>,"}} which would eliminate the perceived equivalence between "All ravens are black" and "All non-black things are non-ravens."

One such approach involves introducing a [[many-valued logic]] according to which "If <math>A</math> then {{nobr|<math>B</math>"}} has the [[truth value]] {{nobr|<math>I</math>,}} meaning "Indeterminate" or "Inappropriate" when <math>A</math> is false.<ref name=Farell1979>{{cite journal| last=Farrell |first=R. J.| title=Material Implication, Confirmation and Counterfactuals| journal=Notre Dame Journal of Formal Logic|date=April 1979| volume=20| number=2| pages=383–394| url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ndjfl/1093882546 |doi=10.1305/ndjfl/1093882546| doi-access=free}}</ref> In such a system, [[contraposition]] is not automatically allowed: "If <math>A</math> then {{nobr|<math>B</math>"}} is not equivalent to "If <math>\overline{B}</math> then {{nobr|<math>\overline{A}</math>".}} Consequently, "All ravens are black" is not equivalent to "All non-black things are non-ravens".

In this system, when contraposition occurs, the [[Linguistic modality|modality]] of the conditional involved changes from the [[indicative]] ("If that piece of butter ''has been'' heated to 32&nbsp;°C then it ''has'' melted") to the counterfactual ("If that piece of butter ''had been'' heated to 32&nbsp;°C then it ''would have'' melted"). According to this argument, this removes the alleged equivalence that is necessary to conclude that yellow cows can inform us about ravens:

{{quote|
In proper grammatical usage, a contrapositive argument ought not to be stated entirely in the indicative. Thus:

{{quote|style=font-size:inherit|From the fact that if this match is scratched it will light, it follows that if it does not light it was not scratched.}}

is awkward. We should say:

{{quote|style=font-size:inherit|From the fact that if this match is scratched it will light, it follows that if it ''were'' not to light it ''would'' not have been scratched.&nbsp;...}}

One might wonder what effect this interpretation of the Law of Contraposition has on Hempel's paradox of confirmation. "If <math>a</math> is a raven then <math>a</math> is black" is equivalent to "If <math>a</math> were not black then <math>a</math> would not be a raven". Therefore whatever confirms the latter should also, by the Equivalence Condition, confirm the former. True, but yellow cows still cannot figure into the confirmation of "All ravens are black" because, in science, confirmation is accomplished by prediction, and predictions are properly stated in the indicative mood. It is senseless to ask what confirms a counterfactual.<ref name=Farell1979/>}}