9

Natural language is context-dependent, like the statement “My uncle is a plumber”, which is true or false depending on who asserts it.

There has been lots of discussion about fictional entities and their place in classical logic (predicate logic), with Gottlob Frege even claiming that the statement “Pegasus does not exist” is neither true or false; similarly there is the problem of the inference (existential generalization) from “Pegasus is a flying horse” – which is regarded as true – to “A flying horse does exist”.

How is this different and more problematic than context-dependence of natural language in all the many other situations and the difficulties this poses to logically formalize such statements?

  • Numbers are contextual too, 1: int is different from 1:real... – Mozibur Ullah Oct 2 '17 at 15:37
  • Modal logic is the logical language that deals with fictional and counterfactual situations. Possible world semantics is key in how logic deals with counterfactuals. In terms of specific difficulties in trying to formulate a logical language for this topic, quantification in modal logic is one of the most contentious topics in the subject. People like Quine said that it is nonsensical to ever allow quantification to take place in modal logic. – Not_Here Oct 2 '17 at 16:07
  • People like Ruth Barcan Marcus took the opposite view, and she specifically laid a majority of the groundwork for the actual logical language, in the same way that Kripke did a lot of the groundwork for giving modal logic its first complete semantics with possible world semantics. Quine's main attack as to why it is nonsensical to quantify modal logic is in his paper Quantifiers and Propositional Attitudes, Kripke gives a great (contemporary) response here. – Not_Here Oct 2 '17 at 16:11
  • On the standard interpretation of quantifiers in first order languages all statements that existentially quantify over fictional entities are false. Those that do not quantify but name, like “Pegasus does not exist,” can be paraphrased a la Russell by converting fictional names into predicates, "there is not a thing which is Pegasus" is plainly true. See here for other ways of dealing with even inconsistent fictions. I am not quite sure why context dependence would be a problem. – Conifold Oct 2 '17 at 19:59
  • @Conifold classical logic is context-independent, right? So … this means it's truth-preserving even in the weirdest combinations of assertions? But if one formalizes nat-lang statements like “My uncle is a plumber”, “my uncle” appears as variable u and refers context-independent to (say) “Bob J. Miller sr.”? So there doesn't seem any work involved here. Question: is “Pegasus is a flying horse” (= TRUE in some sense!) more problematic for formalization? Because the context (if that's the right term) ‘we're just talking about Greek mythology’ applies to the whole statement? – wolf-revo-cats Oct 2 '17 at 20:59
2

Natural languages and formal languages are very different things, with different functions. Any formal language can at best cover a small subset of the domain of discourse of natural language, but with the advantage of eliminating ambiguity and providing strict definitions for validity and invalidity. There are formal logics, like formal mathematics, that have proven valuable for applications quite distant from their origins (for example, computer programming).

It is not so much the case that possibility, fictionality and counterfactuality are necessarily more difficult, it is that they are outside what can be fluently covered by traditional logics. New and different kinds of formal languages called modal logics needed to be created in order to translate such concepts. How well they succeeded is a matter of opinion, but keep in mind a formal language concept is never an exact translation of a natural language concept, but only at best a plausible approximation.

1

If we have an object x in front of us, we can ask many questions about it such as "is x red?", "does x have a mass of more than 5kg?", "is x warmer than 300K?".

Of course, you could argue these questions aren't well-defined. For example, how much of x must be red? All of it? More than half of it? Any portion of it? And what do we mean by red? When does red become orange or pink or violet? And, since color depends on reflected light, what lighting conditions are we using? Sunlight? "Black" (ultraviolet) light? Orange fluorescent light from mercury lamps? Total darkness? (in which case we'd be looking at emitted, not reflected, light).

However, we generally accept that we could find a reasonable definition of "red", and decide whether a given object is red or not.

However, we can't ask the question "does x exist"? Why not? Because the fact you can refer to x means that x must exist in some sense. For example, if we ask "do flying horses exist", we've already created the concept of flying horses. In contrast, if we ask "do sl6eyun7el exist?", we have no idea what sl6eyun7el means, so it doesn't exist even in our minds.

In our first paragraph example above, we would need to have flying horses standing in front of us to ask "do flying horses exist", in which case it's fairly obvious they do.

There is a mathematically precise way to address this issue. Although mathematicians often say "there exists x such that P(x)" or "for all x, P(x)", where P(x) is some property, they are actually being a little sloppy.

Formally, any existential ("there exists") or universal quantification ("for all") must have a "universe of discussion", or more formally, a set.

The correct forms of the earlier statements are "there exists x in set S such that P(x)" or "for all x in set S, P(x)".

How does this help? It now means we can regard the existence of x as a property of the set S, instead of as a property of x itself.

In other words, we can ask "does S have the property that one or more of its elements is a flying horse?".

This makes the answer simple: if S is the world of fiction, it is true that one or more of its elements is a flying horse; if S is the world of reality it is not true (as far as we know) that one or more of its elements is a flying horse.

And, just to be nitpicky, I realize you could put a horse on an airplane or that flying horses may exist in reality but we haven't seen them yet, but you get the idea.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.