(2) doesn’t require the graph to be finite. Infinite graphs also have the property that if you repeatedly follow in-edges, you must eventually reach (i) a node with no in-edges, or (ii) a cycle, or (iii) an infinite chain.
EDIT: Proof, since if we’re talking about epistemology I shouldn’t spout things without double checking them.
Let G be any directed graph with at most countably many nodes. Let P be the set of paths in G. At least one of the following must hold:
(i) Every path in P is finite and acyclic.
(ii) At least one path in P is cyclic.
(iii) At least one path in P is infinite.
Now we just have to show that (i) implies that there exists at least one node in G that has no in-edges. Since every path is finite and acyclic, every path has a (finite) length. Label the nodes of G with the length of the largest path that ends at that node. Pick any node N in G. Let n be its label. Strongly induct on n:
If n=0, we’re done: the maximum path length ending at this node is 0, so it has no in-edges. (A.k.a. it lacks justification.)
If n>0, then there is a non-empty path ending at N. Follow it back one edge to a node N’. N’ must be labeled at most n-1, because if its label was larger then N’s label would be larger too. By the inductive hypothesis, there exists a node in G with no in-edges.
Ah good point. OK yeah I believe that (2) doesn’t require the graph to be finite, and I also agree that it’s not tenable to believe all three of your statements.
If, hypothetically, we were to stop here, then you might look at our short dialog up to this point as, roughly, a path through a justification graph. But if we do stop, it seems that it will be because we reached some shared understanding, or ran out of energy, or moved on to other tasks. I guess that if we kept going, we would reach a node with no justifications, or a cycle, or an infinite chain as you say. Now:
A node with no justification would be quite a strange thing to experience. I would write something, and you would question me, and I would have literally nothing that I could say
A cycle would be quite a normal experience to go a few loops around—plenty of conversations go in loops for some finite time—but it would be strange for there to be absolutely no way out of the cycle. We would just go and go and go until we lost all energy, and neither of us would notice that we’re in a cycle?
An infinite chain would be perhaps the most “normal” of the three experiences. We would just have some length of conversation and then, what, give up? Since we have finite minds, there must be a finite program that generates the infinite graph, so wouldn’t we eventually notice that and say “huh, it looks like we are on a path with the following generator functions”. What then? Would we not go some place else in the justification graph other than the infinite chain we were previously on?
So it’s hard for me to imagine really experiencing any of the three possibilities you point out. Yet they would seem to be not just possible but actually guaranteed (in aggregate).
As you said, very often a justification-based conversation is looking to answer a question, and stops when it’s answered using knowledge and reasoning methods shared by the participants. For example, Alice wonders why a character in a movie did something, and then has a conversation with Bob about it. Bob shares some facts and character-motivations that Alice didn’t know, they figure out the character’s motivation together, and the conversation ends. This relied on a lot of shared knowledge (about the movie universe plus the real universe), but there’s no reason for them to question their shared knowledge. You get to shared ground, and then you stop.
If you insist on questioning everything, you are liable to get to nodes without justification:
“The lawn’s wet.” / “Why?” / “It rained last night.” / “Why’d that make it wet?” / “Because rain is when water falls from the sky.” / “But why’d that make it wet?” / “Because water is wet.” / “Why?” / “Water’s just wet, sweetie.”. A sequence of is-questions, bottoming out at a definition. (Well, close to a definition: the parent could talk about the chemical properties of liquid water, but that probably wouldn’t be helpful for anyone involved. And they might not know why water is wet.)
“Aren’t you going to eat your ice cream? It’s starting to melt.” / “It sure is!” / “But melted ice cream is awful.” / “No, it’s the best.” / “Gah!”. This conversation comes to an end when the participants realize that they have fundamentally different preferences. There isn’t really a justification for “I dislike melted ice cream”. (There’s an is-ought distinction here, though it’s about preferences rather than morality.)
Ultimately, all ought-question-chains end at a node without justification. Suffering is just bad, period.
And I think if you dig too deep, you’ll get to unjustified-ish nodes in is-question-chains too. For example, direct experience, or the belief that the past informs the future, or that reasoning works. You can question these things, but you’re liable to end up on shakier ground than the thing you’re trying to justify, and to enter a cycle. So, IDK, you can not count those flimsy edges and get a dead end, or count them and get a cycle, whichever you prefer?
We would just go and go and go until we lost all energy, and neither of us would notice that we’re in a cycle?
There’s an important shift here: you’re not wondering how the justification graph is shaped, but rather how we would navigate it. I am confident that the proof applies to the shape of the justification graph. I’m less confident you can apply it to our navigation of that graph.
“huh, it looks like we are on a path with the following generator functions”
Not all infinite paths are so predictable / recognizable.
(2) doesn’t require the graph to be finite. Infinite graphs also have the property that if you repeatedly follow in-edges, you must eventually reach (i) a node with no in-edges, or (ii) a cycle, or (iii) an infinite chain.
EDIT: Proof, since if we’re talking about epistemology I shouldn’t spout things without double checking them.
Let G be any directed graph with at most countably many nodes. Let P be the set of paths in G. At least one of the following must hold:
(i) Every path in P is finite and acyclic. (ii) At least one path in P is cyclic. (iii) At least one path in P is infinite.
Now we just have to show that (i) implies that there exists at least one node in G that has no in-edges. Since every path is finite and acyclic, every path has a (finite) length. Label the nodes of G with the length of the largest path that ends at that node. Pick any node N in G. Let n be its label. Strongly induct on n:
If n=0, we’re done: the maximum path length ending at this node is 0, so it has no in-edges. (A.k.a. it lacks justification.)
If n>0, then there is a non-empty path ending at N. Follow it back one edge to a node N’. N’ must be labeled at most n-1, because if its label was larger then N’s label would be larger too. By the inductive hypothesis, there exists a node in G with no in-edges.
Ah good point. OK yeah I believe that (2) doesn’t require the graph to be finite, and I also agree that it’s not tenable to believe all three of your statements.
If, hypothetically, we were to stop here, then you might look at our short dialog up to this point as, roughly, a path through a justification graph. But if we do stop, it seems that it will be because we reached some shared understanding, or ran out of energy, or moved on to other tasks. I guess that if we kept going, we would reach a node with no justifications, or a cycle, or an infinite chain as you say. Now:
A node with no justification would be quite a strange thing to experience. I would write something, and you would question me, and I would have literally nothing that I could say
A cycle would be quite a normal experience to go a few loops around—plenty of conversations go in loops for some finite time—but it would be strange for there to be absolutely no way out of the cycle. We would just go and go and go until we lost all energy, and neither of us would notice that we’re in a cycle?
An infinite chain would be perhaps the most “normal” of the three experiences. We would just have some length of conversation and then, what, give up? Since we have finite minds, there must be a finite program that generates the infinite graph, so wouldn’t we eventually notice that and say “huh, it looks like we are on a path with the following generator functions”. What then? Would we not go some place else in the justification graph other than the infinite chain we were previously on?
So it’s hard for me to imagine really experiencing any of the three possibilities you point out. Yet they would seem to be not just possible but actually guaranteed (in aggregate).
Interested in what you make of this.
As you said, very often a justification-based conversation is looking to answer a question, and stops when it’s answered using knowledge and reasoning methods shared by the participants. For example, Alice wonders why a character in a movie did something, and then has a conversation with Bob about it. Bob shares some facts and character-motivations that Alice didn’t know, they figure out the character’s motivation together, and the conversation ends. This relied on a lot of shared knowledge (about the movie universe plus the real universe), but there’s no reason for them to question their shared knowledge. You get to shared ground, and then you stop.
If you insist on questioning everything, you are liable to get to nodes without justification:
“The lawn’s wet.” / “Why?” / “It rained last night.” / “Why’d that make it wet?” / “Because rain is when water falls from the sky.” / “But why’d that make it wet?” / “Because water is wet.” / “Why?” / “Water’s just wet, sweetie.”. A sequence of is-questions, bottoming out at a definition. (Well, close to a definition: the parent could talk about the chemical properties of liquid water, but that probably wouldn’t be helpful for anyone involved. And they might not know why water is wet.)
“Aren’t you going to eat your ice cream? It’s starting to melt.” / “It sure is!” / “But melted ice cream is awful.” / “No, it’s the best.” / “Gah!”. This conversation comes to an end when the participants realize that they have fundamentally different preferences. There isn’t really a justification for “I dislike melted ice cream”. (There’s an is-ought distinction here, though it’s about preferences rather than morality.)
Ultimately, all ought-question-chains end at a node without justification. Suffering is just bad, period.
And I think if you dig too deep, you’ll get to unjustified-ish nodes in is-question-chains too. For example, direct experience, or the belief that the past informs the future, or that reasoning works. You can question these things, but you’re liable to end up on shakier ground than the thing you’re trying to justify, and to enter a cycle. So, IDK, you can not count those flimsy edges and get a dead end, or count them and get a cycle, whichever you prefer?
There’s an important shift here: you’re not wondering how the justification graph is shaped, but rather how we would navigate it. I am confident that the proof applies to the shape of the justification graph. I’m less confident you can apply it to our navigation of that graph.
Not all infinite paths are so predictable / recognizable.