Universe Of Discourse

In formal logic, an argument is defined over a universe of discourse. Every argument and every statement made in that universe applies to all entities in the universe. There are no subdivisions of any kind. Every sentence in formal logic is a context-free sentence. In natural language, this is different as a matter of convenience only. Every sentence could be made context-free by including the context as part of the sentence. For example, suppose a sentence talks about cars, with the understanding that the discussion is limited to vehicles. One can remove these constraints by starting all sentences with "Among all vehicles, ...". There is nothing wrong with this and there is nothing fundamentally different about that part of the sentence than about any other part of the sentence. The only difference is that it's inconvenient, but, of course, that has no place in formal logic.

Allowing people to play with the bounds of the universe of discourse (restricting it arbitrarily) allows them to pass subtle fallacies unnoticed. For example, if one restricts the concept of 'earning' only to human artifacts then this means that land, trees, forests, gold mines, oil reserves, fish, wheat, et cetera on end can never be earned. This is an entirely valid point of view accepted by many left and right-wingers alike. On the other hand, if one removes all restrictions on 'earning' then the entity which we most depend on for our survival (the sun) is an unearned resource. This is also a valid point of view and nearly all philosophers (who spend their entire careers arguing and reasoning about such things) accept this as a powerful argument against the proposition that 'earning' serves as the sole basis of justice. This forces people to come up with alternative bases of justice and most philosophers are happy enough with the alternatives that they end up rejecting the concept of 'earning' altogether. But both of these viewpoints are unusual and only occur when people are on the up and up.

What usually happens is that people resort to shifting the bounds of the universe of discourse so that it excludes any counter-examples to their argument but includes what they have a vested interest in defending. For example right-wingers support the concept of 'earning' in order to defend their property by excluding the unearned sun but including whatever arbitrary natural resources society puts under the purview of private property. If people start deploying orbiting solar panels large enough to cast shadows.... This kind of fallacious reasoning is hard to detect. It is always best to pick the widest possible universe of discourse since the cost is trivial (eg, one is forced to disambiguate between Red as Communist and Red as Colour) while the benefits are great.

It is trivial to apply all adjectives to all possible objects. Is the sun evil? No. Are electrons happy? No. Are smiles electrically charged? No. And so on. If any ambiguity is involved, it only serves to pinpoint ambiguity in the adjective in question. For example, are rocks stupid? If one defines stupidity as unwillingness to learn then rocks are stupid (because they possess no will). If one defines it as deliberate willingness to avoid learning then rocks are not stupid (because they lack deliberation). The second outcome seems rather strange and thus allows people to discriminate in favour of the first definition.

The process of applying qualities to "inappropriate" objects has been used again and again to great advantage in the refinement of formal definitions. See DefinitionOfLife for an example. In contrast, EdwardKiser on ConceptsOutOfContext would argue that "alive" does not apply to inorganic matter in the first place and would have made coming up with a formal definition of the term extremely difficult, if not outright impossible. -- RichardKulisz

Argument on TragedyOfTheCommons moved to that page.

I'm a great fan of promiscuous models of existence (not my term!). If you couple them with paraconsistency then you've not got a problem with paradoxes of negation, and if you're careful with Curry Paradoxes neither. The problem I have with the concept of the Universe of Discourse is people never seem to pin down if this is in the realm of epistemology or ontology, or in other (vaguer) words, beliefs or truth.

I'm not a great fan of ontology it seems like an exercise for metaphysicians and quacks to talk loudly in University bars. I'd be much happier if the Universe of Discourse were explicitly epistemic, I guess then we'd need a universe to ourselves. :)

Priest has a cool arithmetic based on church numerals and a paraconsistent successor function, apparently! -- DanSheppard

Do you think you can be imperialistic in a universe of discourse? Or territorial, or enclose and evict? I don't know if the metaphors are useful. But I do wonder how we're always doing good stuff to it like pushing back its frontiers, and leaping around in it, we never seem to be smiting or oppressing in it, which I'm sure we probably actually do regularly. -- DanSheppard

Imperialism would entail propaganda and ideological redefinition. Sure, why not?

As for the UniverseOfDiscourse, it is not a strictly epistemic notion. I don't think it's epistemic at all. I have to accept ontology because I don't accept paraconsistency. Of course, this is the first time I've heard of it so perhaps you could explain its attraction? I'd especially like to know how one removes ((p or not p) then q) given that it's a simple application of addition and disjunctive syllogism. -- RichardKulisz

There are a lot of different ways of building a paraconsistent logic, one of the simplest is a modification of an idea proposed by Belnap. The logic is four-valued, with bottom, top, true, false. bottom means 'neither true nor false' and top 'both true and false' (effectively replacing logical values with sets of same, and extending logical operations to operate on those sets in the natural way). You can see this logic as either being epistemic or as ontological in a Dialthesic [http://plato.stanford.edu/ Philosophy terms] universe.

(Assuming you meant [if (p and not p) then q] above), in this logic you have (p and not p) having a valid model (p=top) and you have no ECSQ (proving nothing from a paradox). What you're left with is for MP [p->q q->r |= p->r] becomes [... |= p->r (q and not q)], and for MT [(p or q) ((not q) or r) |= p or r] you have [(... |= (p or r) (q and not q)]. 'I can't do C right now, cos it's really late, and I don't have my thesis notes to hand. I've probably made loads of mistakes already'. But the idea is you have all the standard rules but with extra (x and not x) alternatives. (x and not x) has the model that x=top, the logic always allows for the possibility that there are paradoxes at intermediate values.

Whether you believe the classical solution or the 'or it could be a paradox' part is a matter for heuristics. (You could believe constantly that everything will be shown to be a paradox in the future, and so refuse to accept anything, but you wouldn't be very useful).

I'll perhaps move this onto another page when I get time and be less terse.

Priest is working on an arithmetic for paraconsistent successor functions, apparently. Part of my thesis is (will be) on a paradigm for knowledge representation and manipulation in AI based around a liberal universe of discourse avoiding Russell paradoxes, the Liar, and so on, by being paraconsistent. There's also been talk as to whether the Godel proof would apply in a paraconsistent universe, as it relies on proof by contradiction, which probably has a get out clause in many paraconsistent logics.

-- DanSheppard

It seems, that the UniverseOfDiscourse used in an argument (among non-logicians) is formed in the same way, that any context needed to understand a sentence, argument, speech is, namely by something like NearestFittingContext (see there).

-- GunnarZarncke