Does the claim of an is/ought gap presuppose relevance logic (at least for morality-talk)?

Question

Imagine Hume's remarks but with reference to the usual disjunction introduction:

In every system of conjunction, which I have hitherto met with, I have always remarked, that the author proceeds for some time in the ordinary way of reasoning, and establishes the being of a primal unity, or makes observations concerning the integrity of human affairs; when of a sudden I am surprised to find, that instead of the usual copulations of propositions, and, and not and, I meet with no proposition that is not connected with an or, or an or not. This change is imperceptible; but is, however, of the last consequence. For as this or, or or not, expresses some new relation or affirmation, it's necessary that it should be observed and explained; and at the same time that a reason should be given, for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it.

In other words, would inferring "ought" from "is" be similar to going from, "I am a narwhal," to, "I am a narwhal or Jupiter is larger than the sun," etc.? Because otherwise, I'm at a loss now as to what Hume's point is (I know what it's supposed to be, but for now, he seems to fail in making it):

1. (Hume's guillotine): A ⊬ OBB unless A = OBC (trivially if OBB = OBC)

But then:

2. (Hume's antiknife?): A ⊬ (A ∨ B) unless A = A ∨ B

... so that (for those of us who are not relevance logicians) going from an "is" to an "ought" can just be an axiom of some system, and a sense of these axiomatics underscored whichever targets Hume had in mind (retrospectively, as with natural-law theories (I suppose), or prospectively, as with Rawls' "acting from our true nature"):

3. ∃ABC((A ∧ B) ⊢ OBC) and A, B are stipulated to not be OB-sentences
4. Alternatively: ∃ABC((A ∧ B) → OBC) (this would probably be a better format for an axiom)

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Note: I tried doing a search "is-ought gap and relevance logic" on Google, and nothing clearly, exactly came up that was relevant(!). To be sure, some of the results were for extended discussions (essays) in which both topics were analyzed, but from the snippets it wasn't evident how directly these matters were related, if directly enough at all.

Answer 1

You are being a little cheeky here. It took me a few minutes to realise what you had done. The passage you quote is derived from David Hume's Treatise on Human Nature, Book 3, Part I, Section 1. Hume is talking there of how writers on moral philosophy tend to start by speaking of what 'is' and then proceed by subtle degrees to speaking of what 'ought' without explaining the relationship between the two. You have changed Hume's text by substituting conjunction for what is, and disjunction for what ought.

The analogy is not a good one. Hume's point is not that ought-statements are unrelated to is-statements, but rather that they are not deducible from them. No logic takes you from is to ought. Some theory of human nature might do the job, and Hume himself is happy to supply one. Hume's criticism is directed against writers who purport to be able to derive moral obligations from reason alone. In modern parlance, we might say that the gap between is and ought is a distinction between different modalities. It is more like an inability to move from A to □A.

In the case of disjunctive addition, there are logical grounds for arguing that A ∨ B is deducible from A. For one thing, it is truth-preserving, i.e. it is not possible for A to be true and A ∨ B to be false. It is also dual to conjunction elimination. More generally, it is a feature of an entire logic that is simple, elegant and widely used. One may object to the logic and propose to replace it with one that lacks disjunctive addition, but attempts to do so have not received wide support, and bluntly, they are just messy and unsatisfactory.

Answer 2

Let's imagine you have a box of LEGOs. These are your facts or "is" statements. You also have a picture of a castle you want to build - these are your "ought" statements, or what you believe should happen. It's like you're trying to build that castle using only the LEGOs you have. But sometimes, you don't have the right pieces. In other words, just because you have certain facts, you can't always figure out what should happen next.

Now, let's imagine that you have another box of different building blocks, called Relevance Logic blocks. These blocks have different rules about how they can be put together. The person asking the question is trying to use these Relevance Logic blocks to build the LEGO castle.

Here are some issues with that:

1. Mixing Up Blocks: It's like trying to use your Relevance Logic blocks in the same way you use your LEGOs. These are two different kinds of blocks with different rules. Just like you can't use LEGO instructions to build with other blocks, you can't use relevance logic in the same way as classical logic.

   • Question: Do you think you can use the Relevance Logic blocks the same way you use LEGOs?

2. Misunderstanding the Picture: The person is looking at the picture of the castle (the "ought" statement) and thinking it's just another LEGO block (an "is" statement). But a picture of a castle and a LEGO block are two different things.

   • Question: Can you build a castle just by looking at a picture?

3. Making Up New Rules: The person is trying to make a new rule that lets them use a LEGO block to create a castle. But making a new rule doesn't change the fact that you might not have the right blocks to build the castle.

   • Question: Can you build a castle with a single LEGO block if you make a new rule?

4. Adding Unrelated Pieces: The person is trying to add a piece from a different game (like a chess piece) to the LEGO castle. But adding a chess piece doesn't help you build a LEGO castle. This is like trying to derive an "ought" statement from an "is" statement.

   • Question: Can you use a chess piece to build a LEGO castle?

So, while it's a fun idea to think about using different blocks and rules, but what we're really trying to do: build a LEGO castle. And for that, we need the right LEGO pieces.