Is Logic Arbitrary?

Is Logic Arbitrary?

Propositional logic is the logic of truth functions: negation takes us from true to false and false to true; conjunction takes us from double true to true and otherwise to false; disjunction takes us from single true or double true to true and otherwise to false. But are these the only truth functions there are? No: what about the truth function that takes us from double true to false, or from single true to true but not from double true? That is, suppose we had a symbol for a truth function that is like conjunction except that it requires both conjuncts to be false for it to be true, or one that gives truth only if one of the disjuncts is true but not both. These might be written “nand” and “nor”, as in “p nand q” and “p nor q”. These truth functions exist and we can define them. We don’t seem to have any natural language words for them, but why should logic care about that? If we are interested in truth-functional logic as a general theory, we should make room for these non-standard truth functions and study their properties—they certainly generate well-defined entailments (for example, “not p” follows from “p nand q”). It would be arbitrary to exclude these truth functions from logical theory. And yet they are not mentioned in standard logic texts.

Predicate calculus introduces two quantifiers: “all” and “some”. It studies the entailments thereof. But in natural language there are many more quantifiers (and no doubt others could be defined): “most”, “many”, “a few”, “several”, “nearly all”, etc. They all serve to indicate quantity, and they all feature in valid inferences. For example, although “Most F are G” does not entail “This F is G”, it does entail “This F is probably G”; and we can infer from “Several F are G” that not just one F is G. Yet quantifier logic, as standardly presented, does not include these non-standard quantifiers, as they are called (though they are perfectly standard outside of standard logic textbooks). Surely a general theory of quantifiers should include the full range of quantifiers; it is arbitrary to exclude them. It leaves quantifier logic incomplete. It is true that such quantifiers are not found in arithmetic, with “all” and “some” ruling the roost, but they are common elsewhere and should be accorded the respect that is their due.[1] Just because predicate calculus historically arose from the desire to formalize arithmetic is no reason to ignore the logic of other quantifiers. There is thus an arbitrariness built into the logic that is customarily taught to students and thought to define the subject. Like propositional calculus, predicate calculus is too confined, too exclusive, too focused on one region of the logical landscape. We need a more inclusive logic open to the historically marginalized. Call this deviant logic if you like, but notice the pejorative connotations of that term.[2]

Colin

[1] No one seems interested in Goldbach’s sister’s conjecture that most numbers greater than 2 are the sum of two primes.

[2] We might see this as an essay in logical politics.