Here’s a funny logically valid argument:
The analytic/synthetic distinction between truths is the same as the a priori / a posteriori distinction.
The analytic/synthetic distinction between truths makes sense.
If 1 and 2, then every truth is knowable.
So, every truth is knowable. (1–3)
If every truth is knowable, then every truth is known.
So, every truth is known. (4–5)
If every truth is known, there is an omniscient being.
So, there is an omniscient being. (6–7)
I won’t argue for 1 and 2: those are big-picture substantive philosophical questions. I am sceptical of both claims.
The argument for 3 is this. If the analytic/synthetic distinction makes sense, then the two concepts are exclusive and exhaustive among truths: a truth is synthetic just in case it’s not analytic. So, every truth is analytic or synthetic. But if 1 is true and the analytic/synthetic distinction makes sense, then it follows that every truth is a priori or a posteriori. But these phrases are short for a priori knowable and a posteriori knowable. Thus, if 1 is true and the analytic/synthetic distinction makes sense, then every truth is knowable.
The argument for 5 is the famous knowability paradox: If p is an unknown truth, then that p is an unknown truth is a truth that cannot be known (for if someone know that p is an unknown truth, then they would thereby know that p is a truth, and then it wouldn’t be an unknown truth, and no one can’t know what isn’t so).
One argument for 7 is an Ockham’s Razor argument: it is more plausible to think there is one being that knows all things than that the knowledge is scattered among many. A sketch of a deductive argument for 5 that skirts over some important technical issues is this: if you know a conjunction, you know all the conjuncts; let p be the conjunction of all truths; if every truth is known, then p is known; and someone who knows p knows all.
6 comments:
Hello! I've been thinking about analytic propositions as well, and came up with a fun argument based on a statement Bertrand Russel made in a debate on the Contingency Argument:
"But, to my mind, a 'necessary proposition' has got to be analytic. I don't see what else it can mean."
I'm going to uncreatively call this idea, that necessary propositions must be analytic, Russell's Idea (RI).
1. If RI is true, then all necessary propositions are analytic propositions.
2. If RI is true, then RI is true in every possible world.
3. If a proposition is true in every possible world, then it is a necessary proposition.
4. Therefore, if RI is true, then RI is an analytic proposition. (From 1-3)
5. If RI is an analytic proposition, then "necessary" has the same definition as "analytic."
6. "Necessary" does not have the same definition as "analytic."
7. Therefore, RU us not true. (From 4-6)
Typo! I meant:
7. Therefore, RI is not true. (From 4-6)
Fun argument. Here's a somewhat related arg that doesnt require anything as dubious as 1, for an 'omnicognizant' being (a being that entertains every proposition): (inspired by an Analysis paper by Chalmers)
Premise 1. Every instance of "p iff actually, p" is a priori knowable. (Justification: the validity of A(p iff @p) plausibly belongs to the logic of apriority.)
Premise 2. If one entertains a proposition, one (thereby) entertains its constituents.
Now suppose for reductio that there's a proposition, q, that's not in fact entertained. Then 'actually(Entertained(q))' is necessarily false. So "Entertained(q) iff actually(Entertained(q))" is unknowable (because any world where it's entertained will have a false right side and true left side). And since it's not knowable, it isn't a priori knowable, violating Premise 1.
(This gets us that every proposition is entertained. Then moves similar to those you suggest can take us to a single being who entertains all propositions.)
I think P1 is most questionable, but maybe it can be motivated by Williamsonian considerations (just as necessitism simplifies modal logic, maybe P1 is part of the simplest most elegant logic of apriority).
Brian:
That's really fun, too.
Everybody:
Here's a neater argument that if every truth is known, then there is an omniscient being.
1. Necessarily, there is a set of all knowers.
2. Suppose that every knower is ignorant of some truth.
3. By a plausible generalization of the Axiom of Choice to impure sets, from 1 and 2 it follows that there is a function f from knowers to truths such that for every knower x, x is ignorant of f(x).
4. Let I = { f(x) : x is a knower }.
5. Every set of truths has a conjunction.
6. Let p be a conjunction of I.
7. If you know a conjunction, you know the conjuncts.
8. So, every knower is ignorant of p.
So, if every knower is ignorant of some truth, there is some truth that every knower is ignorant of. By contraposition, if every truth is known, someone knows every truth.
Plausibly there is no set of all truths, and that fact makes it not so plausible that there is a conjunction of all truths. The above roundabout argument gets around this.
Do these types of arguments assume eternalism implicitly?
What if proposition p, about a supernova in a nearby galaxy, is unknowable by us today but knowable next year? Does this deny point
5. If every truth is knowable, then every truth is known.
William: I think the paradox is not that we can come to know this knowledge, but that by saying we don't know the proposition, we thus know we that is it a truth we don't know.
"if someone know(s) that p is an unknown truth, then they would thereby know that p is a truth"
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