The Paraconsistency Tax

The Paraconsistency Tax is the additional cost of having to tolerate inconsistency. What is that cost? Where does it come from? Why is it a tax? 

“Contemporary logical orthodoxy has it that, from contradictory premises, anything follows.” [1] In other words, classical logic is explosive. The underlying intuition is that information that conclusions are drawn from, the set of propositions, should always be logically consistent; if there are contradicting propositions, then any conclusion can be drawn, including propositions which have nothing to do with those that are in the inconsistent set: e.g., “it is sunny” and “it isn’t sunny” can result in the conclusion that “moon is made of green cheese”. 

Classical logic isn’t intended to model realistic human reasoning, but a closely related idea in psychology is that of cognitive dissonance, or stress experienced from perceiving contradictory information, or inconsistency: 

“In the field of psychology, cognitive dissonance is the perception of contradictory information. Relevant items of information include a person’s actions, feelings, ideas, beliefs, values, and things in the environment. Cognitive dissonance is typically experienced as psychological stress when persons participate in an action that goes against one or more of those things.” [2]

A system of reasoning rules is called “paraconsistent” if the set of allowed conclusions from an inconsistent set of propositions is not explosive. Loosely speaking, a paraconsistent logic tolerates paradox. 

The relative benefit of explosive reasoning rules is that they ignore the specifics of the contradiction: they fail if there is contradiction, that is, they treat all contradictions equally. This isn’t sophisticated at all, but is simple. In contrast, reasoning rules that tolerate contradiction can do so in multiple ways, but every time, they require more information from the individual doing the thinking. 

For example, one way to tolerate contradictions is to remove from the available information all that leads to inconsistency. The problem with this is that you need to find all combinations of available propositions that result in concluding inconsistency; i.e., you need to pay for computing all ways to conclude inconsistency, in order to identify propositions to ignore. 

Another approach is to ask an additional question every time contradiction is encountered, namely, which of the contradicting propositions is more important. The question is intended to elicit preference, so that if we have propositions p and its negation, the person interested in drawing the conclusion needs to say which of the two they prefer, and therefore, which of the two should be concluded from when the two are premises.

A third, and cheapest approach relative to the two above, is to randomly pick one of the contradicting propositions to be the conclusion from the contradiction. 

A fourth approach is to allow multiple conclusions, and treat them separately. This resembles having multiple acceptable extensions in an argumentation framework [3].  

Even in the cheapest of the four, where you pick randomly one of the contradicting propositions, there is some effort that is absent when anything can be concluded. 

If any approach to paraconsistency requires more information than failure in case of inconsistency, then the former requires more effort in any case, and is thus a tax, or a cost that cannot be avoided, it is a tax for being paraconsistent.

References

1. Priest, Graham, Koji Tanaka, and Zach Weber, “Paraconsistent Logic”, The Stanford Encyclopedia of Philosophy (Summer 2018 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/sum2018/entries/logic-paraconsistent/>
2. https://en.wikipedia.org/wiki/Cognitive_dissonance
3. Dung, Phan Minh. “On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games.” Artificial intelligence 77.2 (1995): 321-357.