Impredicative Propositions
Given the following examples-as-definitions, what does one make of the subsequent reasoning:
- (1) "This sentence is simple." (2) "This sentence is complex, and it is conjoined by AND."
Then assign the variable "s" to the left-most sentence "This sentence is simple". Define "compound" c = "not simple" ~s, and assign c = ~s to "This sentence is compound"; assign "j" to "It is conjoined by AND". The second sentence can be expressed as:
- ( NOT(s) AND j )
If truth values are to be placed on the sentences c = ~s and j, then all are clearly FALSEHOODS: e.g. "This sentence is complex" is a FALSEHOOD (it is simple, by definition). So their conjunction (AND) is a falsehood. But when taken in its assembed form, the sentence a TRUTH.
This is an example of the paradoxes that result from an impredicative definition -- that is, when an object m has a property P, but the object m is defined in terms of property P. The best advice for a rhetorician or one involved in deductive analysis is avoid impredicative definitions but at the same time be on the lookout for them because they can indeed create paradoxes. Engineers, on the other hand, put them to work in the form of propositional formulas with feedback.
Read more about this topic: Propositional Formula
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