In a comment to a previous post, Franklin Mason writes:
You say:
X's conceivability renders it reasonable for me to believe that X is possible.
I reply by way of a fictitious story about me and God.
I am a mathematician. God gives me a list of 1000 mathematical propositions. He intones '500 are true and 500 false' and is thereafter silent. I easily understand them all, but a thorough search of the relevant literature provides no reason to think any of them true and no reason to think any of them false. [. . .]
I recall a certain post on Maverick Philosopher and wonder if, in this situation, it might provide a bit of insight. Bill has said that a proposition's conceivability renders belief in the possibility of that proposition's truth reasonable. (Well, he didn't say quite this, but what he said entails this.) Moreover, he says that conceivability of a thing means its conceivability without contradiction.
So, I say this to myself. It certainly does seem that each of the propositions God has given me is conceivable in Bill's sense. Thus it's reasonable of me to hold that each is possibly true. But every mathematical propositions is, if possibly true, necessarily true; and of course if a proposition is necessarily true, it's plain true. Thus it's reasonable of me to hold that each mathematical proposition in God's marvelous list is true.
Now, surely something has gone wrong here. It most certainly is not reasonable to hold that each of the 1000 is true, for I know that 500 are false. But what has gone wrong? I know of two possible answers.
1. I should not have been so quick to assume that the propositions in the list are conceivable. I know of no contradiction that would obtain if any were true, but this is no evidence that no contradiction would in fact obtain were any true. Indeed 500 of the 1000 are contradictory, or at least imply a contradiction. (1 assumes that false mathematical propositions are, or entail, contradictions. I have no idea whether this is true, if by contradiction we mean a proposition either of the form A &~A, or reducible to that form by purely logical manipulation.)
2. Although the propositions in the list are conceivable in the sense that they contain no contradictions, this is absolutely no evidence that any are possibly true. Rather for any proposition on the list we are simply in the dark about its possibility.
Moral if 1 is true: it ain't easy to know whether a proposition is conceivable. Indeed it may be just as hard as knowing whether a proposition is possibly true. Thus we're in no position to just assume that 'God exists' is conceivable.
Moral if 2 is true: conceivability is easy, but is no guide to possibility. Thus the conceivability of 'God exists' is no reason to assume that God is possible.
This is an interesting and welcome challenge. I wasn't talking about propositions and their possible truth, but about concrete individuals and their possible existence. But I'll accept the challenge any way. Franklin's argument is something like this.
a. Each of the propositions on the list is conceivable.
Therefore
b. Each proposition is reasonably thought to be possibly true.
But
c. If a math proposition is possibly true, then it is true.
So
d. Each proposition is reasonably thought to be true.
But
e. We have it on good authority that half the propositions are false.
Therefore
f. It is not the case that (d).
But it seems to me that Franklin is equivocating on 'conceivable.' In plain Anglo-Saxon, this just means thinkable. Now surely every proposition on God's list is thinkable. Franklin rigged it that way when he stipulated that "he easily understands them all." Take the proposition that there is a prime number between 3 and 5. That proposition is conceivable or thinkable inasmuch as we know exactly what it is that we have before our minds. But we cannot conceive it without contradiction. It looks as if Franklin is conflating these two senses of 'conceivable.'
But suppose he is not. Suppose by 'conceivable' he means conceivable without contradiction. Then (a) is false. For if each proposition is "easily understood," then any internal contradiction will be open to inspection, and the internal contradictoriness of the false propositions will be apparent.
But I think Franklin is on to something important. Let the proposition be that Zorn's Lemma is equivalent to the Axiom of Choice.
Suppose Jack understands what Zorn's Lemma states and what the Axiom of Choice states. He also understands what it is for two propositions to be logically equivalent. But Jack has never been taught that the two are equivalent, nor does he know any proof of their equivalence. But thinking through the equivalence as deeply as he can, he sees no contradiction. But our man also knows that his not finding a contradiction is no proof that there is no contradiction down the inferential road a piece.
So the question is this: Does the coherent conceivability of the equivalence make it reasonable for Jack to affirm the equivalence? Well, if he slams into a contradiction, then he knows that the proposition in question is impossible. But as long as he meets with no contradiction, then it strikes me as reasonable for him to affim the possibility and thus the truth of the equivalence in question.
In other words: conceivability is no SURE guide to possibility, but it dooes provide some defeasible evidence for it. Conceivability does not entail possibility, but it is evidence for it. (Analogy: seeing a snake-shaped object on the trail at twilight is defeasible evidence that a snake in on the trail, but not entail or prove that there is a snake on the trail.)
If this be denied, then what could count as evidence for the possibility of any object or state of affairs that is not known to exist?
The topic is the possibility of God. My claim is that that the coherent conceivability, the conceivability without contradiction, of the divine nature is defeasible evidence for the possibility of God. I might proceed as follows. I try to tell a coherent story about the divine attributes and their compossibility as well as a story about the compossibility of the divine nature and various metaphysical truths that I need to uphold (e.g. Ex nihilo nihil fit), and to the extent that this account proves to be coherent and well-articulated, then I have defeasible evidence of the possibility of God.
In sum, to know whether or not a proposition is conceivable without contradiction is more or less easy depending on how complex the proposition is. And to the extent that the proposition is conceivable without contradiction, I have defeasible evidence of its possibility.
What say you, Franklin?
1. You said in your original post that by 'conceivability' is meant conceivability without contradiction. I followed you in this, as I made clear in my reply. Moreover, 'conceivability' has by now a long-established usage in philosophy, and means conceivability without contradiction. It does not mean mere thinkability. Both you and I have followed that usage.
When I said that I understood each of the mathematical propositions on the list, I did not mean that they were conceivable. I meant merely that they were thinkable, i.e. were possible contents of thoughts.
2. You say that any mathemtatical proposition on the list, if false, will wear its internal contradictoriness (what a jaw-breaker - I don't think I'll ever use that word in a paper that I have to read) on its sleave. But then you present a mathematical example in which you seem to wish to say that contradictoriness may be hidden even to a perceptive inquirer. The reason for my confusion should be apparent.
3. You assume that false mathematical propositions are contradictory. This makes mathematics into a branch of logic. This is a controversial assumption and should be marked. For what it's worth, I doubt that it's true.
Moreover, many hold that some substantive necessary falsehoods are not logically equivalent to any contradiction. (By 'substantive' I mean roughly 'non-logical'.) Take, for instance, the proposition that if Franklin exists, he arose out of the union of just this sperm and just this egg (I mean the ones from which he actually did arise). Many hold that this is a necessary truth but deny that that its negation is contradictory. Thus the mere fact that an investigation of a proposition of this form has revealed as of yet no contradiction is absolutely no evidence of its possibility. In such cases, purely conceptual inquiries are of no relevance.
In what class should we place 'God exists'? Is it purely logical? Of course not. Mathematical? Of course not. It is a substantive proposition that concerns the existence of a certain putative concrete individual. This makes me suspect that a purely conceptual inquiry is of little relevance to the question of God's possibility. Likely if 'God exists' is false, it is yet not contradictory.
4. Perhaps I should sharpen my claim. My story was intended in part to support the claim that mere conceivability by itself is no evidence of possibility. On reflection, I don't doubt that in certain cases inquiry into the conceivability of a proposition can produce evidence of its possibility. Example. If mathematical proposition p, a proposition that concerns, let us say, the set of primes, has been verified for very many cases and coheres well with a wealth of proven results, then even in the absence of a proof of p, we have nonetheless evidence of its truth.
But in my story I made clear that for none of the 1000 did we have any such evidence as this. The literature search turned up nothing.
What then do we say about the 1000? Better to say nothing about their conceivability, their possibility, or their truth. Here's another argument:
a. We have no reason to suspect of any of the 1000 that it is false.
b. Thus we have no reason to suspect of any of the 1000 is inconceivable, for a reason to think a proposition inconceivable is also reason to think it false.
c. If a, then we also have no reason to suspect of any of the 1000 that its negation is false.
d. Thus we have no reason to suspect of any of the 1000 that its negation is inconceivable.
e. Consider now the principle that if we have no reason to think a proposition inconceivable, we thereby have reason to think it possible.
f. By b and d, this principle entails that for every proposition on the list of 1000, we have reason to think both it and its negation possible.
g. But again if a mathematical proposition is possible, it is true.
h. Thus by b and d, this principle entails that for every proposition on the list of 1000, we have reason to think that both it and its negation is true.
i. Moreover, as should be clear, the reason to think that it is true is equal in strength to the reaosn to think that its negation is true.
j. But this is absurd. If a principle gives us equal reason to hold of some proposition that both it and its negation is true, that principle must be defective. We are not post-moderns, and we do not ever feel the slightest inclination to affirm contradictions.
k. Given the story of the 1000 as I have told it, the principle articulated in e is the only principle that concerns the relation of the conceivable and the possible that is of relevance here.
j. Thus conceivability by itself is no guide to possibility.
5. How then do we come to know modal truths about things that we do not know to exist? I'm a bit of a defeatist about this. I think that we're largely in the dark here.
Ad 1. We have no real disagreement here.
Ad 2. Zorn's Lemma and the Axiom of Choice are provably equivalent. A proposition asserting that equivalence, however, cannot be seen to be true (by most of us anyway) simply by understanding it. I thought this example might strengthen your case.
Ad 3. It might be useful to distinguish narrowly logical and broadly logical contradictions. Some colors are sounds is not a NL-contradiction since Some Fs are Gs has both true and false substitution instances. But it is a BL-contradiction. Math props may well be like that.
We will agree that God exists is not a truth of logic, and that God does not exist is not a NL-contradiction. And I think we will agree that God exists, if true, is necessarily true. So its necessary truth, if it is true, is not grounded in its logical form.
I think we will also agree that an examination of our concept of God, no matter how protracted, cannot definitively establish that God is possible. But conceptual inquiry is of some relevance to the question of God's possibility. You say it is of "little relevance" which may be consistent with what I am saying.
Perhaps what you reject is my notion that conceivability without contradiction provides some defeasible evidence of possibility. Do you want to say that it provides no evidence whatsoever?
Ad 4. It now seems that this is indeed what you want to say. Conceivability without contradiction is no evidence at all of the possibility of the state of affairs that the proposition describes.
But surely it depends on the complexity of the proposition and its subject matter. I have never played golf and I never will. But the proposition, BV takes up golf, is conceivable without contradiction, and is surely some sort of evidence for the truth of It is possible that BV take up golf.
There are several fascinating questions here. If you are not a modal Spinozist, and grant that there are merely possible states of affairs, and also claim that we have some modal knowledge about them, how do we come by this modal knowledge if not by our power of conceiving? After all, we cannot interact causally with the merely possible.
Options: there are no modal disinctions in reality. There are modal distinctions in reality, but we have no modal knowledge. We have modal knowledge but we cannot account for it.
Lunch time, got to go.
2. Disallowing comments from a particular person, or deleting an offensive, off-topic, or otherwise substandard comment, has nothing to do with censorship. People who think otherwise confuse censorship with lack of sponsorship. I am under an obligation not to interfere with anyone's exercise of legitimate free speech rights. But I am not under any obligation to aid and abet anyone's exercise of free speech rights, legitimate or illegitimate.
3. The Comments area is not an open forum for anyone to say anything about any topic. As the name implies, it is primarily for commenting on the author(s)' posts. But to comment on them, one must have read them. And if I have spent three hours on a post, a reader will not understand it in thirty seconds. Secondarily, the Comments area is to facilitate civil discussion between and among commenters as long as the discussion remains on-topic.
4. Some undesirables: The skimmers, those who cannot read but only read-in. The sophists who, abusing argument, argue for the sake of argument. The ideologues, those who are out for power, not truth. The uncivil. The illogical. The politically correct. Worst of all, perhaps, are those who exemplify the anti-Socratic property: those who think they know what they don't know. If Socrates was famous for his learned ignorance, these types are marked by their ignorant unlearnededness.