I thought I'd put up a new thread for this topic, since the other one was getting a tad long. Also due to other commitments I just didn't have the time to allocate to it the way I wanted. Which led to some posts that weren't as readable as I wished. I've learned a bit in the thread though, and some of Brandon's patient criticisms were very helpful.
I'd ended with the idea that the problem is implicit but not explicit notions of infinity. That unstated but apparently implicit relationship to infinity is what generates the contradiction. It also isn't obvious from the pure definition that one can make the instrumental cause "the whole series." The isomorphism is really just to illustrate how we can say things that seem alright, but once we start applying it to infinite sets we find our language assumptions fail. Thus the appeal to highest integer which makes sense for finite sets but not infinite sets. Allow me to quote from Brandon's reply before expanding on the above.
I take it that, given an infinite set (simply speaking), we can conceive of one more member. But we don't have an infinite set (simply speaking) here; we have a series of causes mediating the causation of other causes, and this causal mediation is an important element. Or is that what you meant, i.e., the series of causes mediating the causation of other causes? But the issue here (as I said above) isn't whether you can have one more member (as far as I can see it makes no difference to the argument whether you can or can't) but whether you can have the whole series at all without something acting as the principal cause for it -- which is not the same question as whether an infinite set, or even an infinite series of instrumental causes, can have one more member. So I don't see what you mean by calling it a fundamental issue; it's not an issue at all. Or perhaps I'm just missing something.
Let me reply by starting with a series of analogies. Now the first few won't fit exactly, but will hopefully illustrate the problem.
Analogy 1
Let A be the set of integers such that it has three members, a1, a2, a3, and a1 < a2; a2 < a3
For any set A there will be an other set B with the same properties such that b3 > a3.
Thus we might have A = {1,2,3} and B = {2,3,4} for instance.
Now obviously Brandon will point out the difference between the mediation in the above (at least its formal representation) and that in the Aquinas example. That's because the mediation in Aquinas can be a set of other sets. (Which was what my initial example was attempting to capture) I just want the above example to illustrate that we can't make the following claim in any sensible fashion:
For a2, such that a2 is the highest integer, there exists no set B, as per the definitions above.
Now the reason for this is obvious. There is no highest integer since the set of integers is infinite. So let's expand our analogy slightly so as to capture the fact that Aquinas allows our mediating element, a2, to include sets.
Analogy 2
Let A be the set such that it has three members a1, a2, and a3. Each of those elements is either an integer or a set. If it is a set then it has the same properties as A. (i.e. three elements, each a set of type A or an integer) Each of these elements have the additional property that for all sets or integers making up a1, there is a lowest integer l1 and for all sets making up a2, there is a lowest integer l2 such that l1 < l2. Further for all sets or integers making up a2 there is a highest integer h2 and for all sets making up a3 there is a highest integer h3 such that h2 < h3. (you'll note I'm trying as best I can to avoid technical jargon relating to set theory - I could tighten this up if necessary with more careful terminology)
Let B be a set of type A with the additional property that for all integers in b3 there is a highest integer such that b3 > a3.
Thus for example we could have
A = {1, { 2, 3, 4}, 5} B = {2 , {3,4,5}, 6}
or
A = { { 1, 2, 3}, { 2, {3, 4, 5}, {4,5,6} }, 5} B = {2 , 3, 7}
Now I know that gets a tad more complex. And I'm trying to avoid jargon as much as possible. But basically it lets us have in A elements that are also sets of the same kind as A. That is, what mediates or what it mediates could also be a mediator. You'll see that this is fairly close to the formal structure of instrumental cause.
Now lets complexify it a tad more.
Analogy 3
Assume the definitions of Analogy 2.
There is some set D of type A such that d2 contains an infinite number of sets.
The problem is, that the formal presentation we gave entails that for a2 to make sense it's highest member must be lower than the highest member of a3. That is, the formal structure of our presentation won't allow it to work with infinite sets. But why won't it? Because it implicitly has the sense of "highest integer." Which was my initial criticism in the other post.
Now, the question is, can we come up with a definition of mediation such that it avoids this reliance on highest integer? We'll modify Analogy 2 somewhat. (Changes in italics)
Analogy 4
Let A be the set such that it has three members a1, a2, and a3. Each of those elements is either an integer or a set. If it is a set then it has the same properties as A. (i.e. three elements, each a set of type A or an integer) Each of these elements have the additional property that for all sets or integers making up a1 and a2, for any element a'2 in a2 there exists an element a'1 such that a'1 < a'2. Likewise for the integers in the sets making up a2 and a3 there is an element a''2 in a2 and an element a''3 in a3 such that a''2 < a''3.
Now the question is, does this capture what we were going at with our instrumental cause? I think it does. But with the above form (as opposed to the form in Analogy 2) we don't have a problem. That is we can assert the following: a2 can have infinite number of sets making it up. That is because for any a''2 we can find an a''3 such that our formal presentation of mediation works.
Sorry for the complexity of the above. Hopefully the examples helped clarify it.
So it would seem to me, to continue the analogy a bit further, that efforts to posit a God as the First Cause or Unmoved Mover or Creator of the Universe are about as fruitful as defining God as the largest integer.
Do you agree?
Yes, I fully think so.
I could just be dense, but I don't find the examples very helpful at all. I still don't see the analogy to the actual causal relation; unless I'm missing something, the mediation involved in all your analogies isn't any stronger than formal precession and succession. But as I've said before, it's generally recognized by the people who make the argument against infinite regress that this is not the relation at issue. You seem to be trying to make it enough by changing the features of the elements in the succession, but that won't work unless it also makes the relation between elements a genuine relation of dependence that is able to be mediated. An instrumental cause is not a mediator in the sense of happening to be intermediate in a series (whatever the features of the elements in the succession); an instrumental cause, as I've said before, is intermediate in that it mediates causal dependence itself. Thus what makes a stick an instrumental cause is its being a mediating part of the causal dependence of its effects, without which they would not occur; that it is an instrumental cause entails that there is another cause to which it is instrumental, that every part of the stick involved in the causal activity is an instrumental cause, that if the stick is instrumental to another instrumental cause, these two causes form an instrumental cause with all the properties of an instrumental cause, that if it is instrumental cause to an instrumental cause B it is instrumental to all the causes of which cause B is an instrumental cause, that suppression of any causes to which it is instrumental would make it cease to be a cause at all, etc. I don't see any formal isomorphism of this type of relation in the relations you are proposing between sets or integers. But it's the point that I keep insisting is precisely at issue. Further, in your Analogy 4 you still don't seem to be making the important distinction between a causal series with infinite elements and an infinitely regressive causal series, only the latter of which entails there is no first cause (an infinitely regressive causal series is a causal series with infinite elements in causal relation and no first cause). An infinite series of causes does not of itself imply the nonexistence of a first cause; as I said in the other thread, there can be an infinite series of instrumental causes moved by a first cause (because that avoids the contradiction entailed by an infinite series of instrumental causes without a first cause). So it still looks to me like you're laying down red herrings. What relation between the elements in the set is formally analogous to the causal relation, and why?
If it were true, however, that an analogy with integers could be held up in every respect, you should be able to drop this elaborate analogy altogether and simply prove directly that it is logically impossible for anything to be a first cause, i.e., that if anything is caused an infinite causal regress is logically necessary. If you think the analogy with integers holds, you're committed to the existence of such an argument. This is, however, a very strong view: it entails that the occurrence of any effect depends on a series of infinite factors, such that if even one of them were otherwise, it would not occur; and it entails that this is a necessary truth about causation. If you're right there should be a feature in the notion of causation itself that yields this result, and you should be able to formulate it entirely in causal terms, without having to go through this analogy.
Brandon, I'd agree that is true for analogy 1, but I confess I don't see how the further analogies don't capture what you describe. That almost certainly is my fault. But I confess I'm racking my brain, and I just don't see how the analogies fail to be what you describe. It's not just sets of integers. But sets of sets and set and potentially an infinite regress of sets. It seems to me that Analogy 2 and Analogy 4 represent the two competing ways to read instrumental cause in terms of infinities.
Michael, I should add, that Marion's approach to Anselm offers an interesting alternative to looking at first cause. However I'm not fully sure I really understand Marion. How I read him and how other friends read him seem to be quite different.
I was actually waiting for this thread to come to this.
I was waiting for somebody to bring in set theory and then look for isomorphisms to Aquinas' arguments. By the way Clark, I think you should have been even more rigorous, don't try and simplify it. The ironic thing about set theory is that it takes an amazing amount of rigour and technical jargon to talk about something so simple. I agree with you Clark, the only way to reason about sets of anything, especially infinite sets is to bring in set theory.
I also fully expected that the line of defense against this would be to deny that an isomorphism exists between numbers/set theory and what Aquinas is arguing. I think the only way to proceed is find agreement as to how Aquinas' argument is to be expressed using set theory and then follow the set definition(s) to its/their logical conclusion(s). You simply can't talk about any infinite/transfinite thing using normal language, it simply doesn't have the apparatus to do this.
Where can I find Marion's work on Anselm? I'm not familiar with that one (although, I must admit, I have three volumes of Marion unread on my nightstand...)
OK, Clark, but as I've said before, sets of sets to infinite regress isn't isomorphic to a series of instrumental causes to infinite regress. Where, for instance, is the dependence (of one set's being a set at all) on the set in which it is found? Where is the relation between the elements within the series such that if all prior elements were removed, the particular later elements would could not occur at all? Where is the distinction made between infinite series with first elements and infinite series without first elements (only the latter of which would be relevant)?
David Clark: I've already denied several times that there is or can be an isomorphism, so I suppose at the moment we're looking at whether Clark can set up an isomorphism. I don't think Clark and I actually disagree about the issues with infinity and transfinites on the mathematical side; he knows more than I do about it, but there's no real disagreement. What's at issue is whether one can map the causal dependence in terms of the relations of set theory without losing anything essential; I'm skeptical, and Clark's trying to show it can be done. But Clark has set a very hard task for himself given translation issues; to make an isomorphism in this context, we have to do a double translation, from the terms in which Aquinas puts it into the terms of the analogy, and back again. That's the real problem, and additional rigor isn't going to solve it; if the translation can't be made even with approximation, there's no real chance of its being made with precision. Were it just a matter of talking about infinity among sets and integers, greater rigor would suffice, because no translation would be involved; but it wouldn't do anything to handle the real problem here, which is to make the isomorphism genuinely isomorphic. It's accuracy, not precision, that's difficult to latch onto here. What's really at issue is not what happens when you build an argument in set theory, but whether this is a case adequately captured by relations among sets that Clark is using. Clark can be more rigorous if he wants, but at this stage it will just make his job harder, since what he has to show is not a strictly rigorous argument that there is no highest integer (which everyone concedes anyway) but that he is actually capturing what Aquinas says, at least to a reasonable approximation.
I don't know what Clark had in mind, but Marion discusses Anselm quite a bit in discussing Descartes in his Cartesian Questions.
Brandon, I have company and will have to wait to answer you and David later tonight.
Michael, my copy of Marion's paper on Anselm is in Flight of the Gods: Philosophical Perspectives on Negative Theology. Once again to add, I'm not sure my reading is right. To me Marion fails at what he is attempting to do. I'd touched on Marion and Anselm back in April.
Don't worry too much about responding to me. To be honest I can't say I really even know what this thread is about. I'll admit it, when things don't have a level of precision and accuracy, I am pretty stupid. I have never been clear about what the point of this thread is. At times I think I have an idea, so I make a post. When I come back later to see responses I am quite frankly baffled and so conclude that I did not grasp the point of it all. That's where I am right now.
Clark, take all the time you need.
David Clark, I waver in and out on this discussion in something like the same way you do; while I do have (I think) enough set theory in me to follow what Clark says, it takes me quite a while to process it, and I'm not always sure about whether I've processed it right -- which is why I keep trying to bring the discussion back to the points I know. :)
I think that's partially why I want to take it into set theory. It formalizes things so we're very clear what is being said. I suppose that's why I keep bringing up the formal sense of the definition. My inclination, as the analogies above demonstrate, is that there is a fundamental possible equivocation within the definition.
As for it's use. . . Well, it's philosophy. 99% of it isn't useful. But it can be interesting. Take here, even if what Aquinas says follows, it doesn't imply that the definition or way of thinking about causality is useful. Indeed I tend to be extremely skeptical of most conceptions of causality. I've been too corrupted by Hume and Nietzsche I suppose.
But I do find discussions of infinity interesting. I often find that they surprisingly come in handy months later too. I've never regretted any problem about infinities I've struggled to understand.
It's a different argument from Aquinas's, but I happened to run across an interesting clarification of (one of) Scotus's arguments against infinite regress (Clark summarized it in the comments to the previous post on this subject, August 12):
Duns Scotus on Natural Theology (scroll down to 14)
I'll have to think about it further, but it's an interesting clarification. If right, I think it would defeat Clark's argument that Scotus was begging the question.
Can I just say how much I'm coming to enjoy Scotus? I find him terribly difficult to read (and still try to forget the time I conflated his views of being with Eriugena). But even when I disagree I learn something from him. No wonder he held the place in philosophical education he did for so many centuries.
Anyway, that document was formatted kind of funny. So I'll quote the relevant section for everyone.
As to the inconsistency of a non-ending regress. Scotus does not offer a taxonomy of ordered natural kinds, and so we have to speculate about examples. Further, he is not committed at the outset to saying that all of nature belongs to a single such order. Rather, he relies upon some actual cases only. For if it is impossible that every regress of essentially ordered causes is infinite (unending), then a terminating one is possible. But that is possible only if a certain sort of thing actually exists, a First Being. And the sort of thing that actually exists will logically prevent there from being any non-terminating regresses of essentially ordered causes at all, because it will be the explanation of all contingent being. So the universal order in nature is a consequence, and outcome of the proof, not a premise of it.
Consider some cases. For asparin to help a headache, chemical reactions are required, and those require certain sorts of and arrangements of molecules (molecular natures: acetylsalicylic acid.) For that, certain molecular structure is required, along with molecular bonding; and for that, certain atomic organization is required, and, so, on. All the latter have to be actual and causing, “all at once”, ”all the way down” for the aspirin to work. The “all at once” can be physical and so, time-bound by the light constant and the medium, and there can even be quantum gaps between cause and effect; the nested causes must still be operating all together. Thus, Ockham’s doubts about whether “simultaneity” of all the causes is demonstrably satisfied, is obviated.
The part I don't quite follow is this. " And the sort of thing that actually exists will logically prevent there from being any non-terminating regresses of essentially ordered causes at all, because it will be the explanation of all contingent being."
The claim is that if there are at least some first beings that it follows that these are the explanation of all contingent being. I just don't see that. Further, it seems the very comment opens up the rejoinder of there being no first being at all. To abuse a saying, "it's turtles all the way down..."
I don't quite follow the second paragraph either. The issue of simultaneity seems a common one in philosophy. Yet it seems to me that this is an issue that process oriented philosophers have critiqued over and over again. Indeed I was just reading about a charge similar to this against Dewey earlier in this century. If we think in terms of process rather than absolute "nows" then this problem becomes alleviated.
Of course there is a very good chance I'm just misreading him. I've not read the full paper yet, just section 14.
I would have said it's not about first beings but about first causes; which is a different issue (obviously a first cause is a first being in a fairly straightforward sense, but it's the causality that makes it so). The part I was particularly thinking of occurs just after the part you quote:
One who says, “still, maybe such a line does not twist up to a first,” is committed to a contradiction. For he has to say that at every stage a sufficient condition is absent and one is never reached by stepwise regression; so one is always absent. And at the same time, he has to postulate the final effect, and, so, that there is a sufficient condition for it. That is explicitly contradictory.
A sentence with an infinite number of “if,if, if,if…”clauses cannot be made complete by adding more; so too, with a phrase inside brackets, inside brackets, repeating without end, never coming to an assertion. So supplying an infinite number of necessary conditions is not enough, by itself, to supply a sufficient condition. Thus supposing only the regression, each member necessary but none sufficient, contradicts the actuality of the effect, for which a sufficient condition is manifestly present. That is Scotus’ insight.
So turtles all the way down doesn't create a sufficient condition; it only approaches it. If that's so, your explanandum doesn't have an adequate explanans, and infinite regress will always be an incomplete explanation. (I suspect, although I'm not at all sure, that that's the point of the second paragraph: if the issue is necessary and sufficient conditions then the logical relation between explanation and effect will not involve any time interval, no matter what processes may be going on.)
Yes, but that part of the argument I'd actually addressed way back in the other thread. (I ought to put numbers on the comments so I can refer to them - perhaps in the next iteration of the software) Anyways, in one of the lines of the first post I actually brought up the "if, if, if..." approach. I think that the strongest approach, but it avoids the issue of whether that is actually how causality functions.
More later. My baby was up until six this morning screaming his head off with molars coming in. So I'm pretty thrashed and incapable of thinking too well.
"My baby was up until six this morning screaming his head off with molars coming in." Baby Tylenol + Baby Orajel makes a good combination for these kinds of situations.
Trust me, we tried. He was really having a night. He has about four teeth coming up at once. Anyway, I have about a half dozen things to write here. The next Nibley post, the next Tomasello post, the follow up to the science literacy, and then this post. Plus two others I've been wanting to finish.
As a fellow parent, you have my sympathies.
I misunderstood you then; I had understood you to be arguing that it assumes there must be a first cause, which the necessary and sufficient condition account doesn't.
Here's what I wrote in the other thread. Just so people don't have to waste time looking for it.
Just before I go to bed, I was reading some Scotus that might clarify things somewhat. (Always a dangerous situation though - sleepy and reading Scotus. Last time I did that. . .) Scotus' approach to this problem can be taken like the following. An infinite series ends up being like an infinite regress of ifs. So take the following.
Case 1: If A then B
Case 2: If ( if A ) then B
Case 3: If ( if ( if A ) ) then B
. . Case n: If ( if ( if ( ... [n ifs] ... ( if A ) )...) then B
Now if I have Scotus right, his objection is that it doesn't matter how many ifs you have. Even if you have an infinite number you still eventually need that A to be able to say B. And this is why I think Brandon is saying we are having a contradiction and not seeing how it is related to infinity. He sees it not a problem of infinite sets but a problem of causality. To be able to say if ... then ... you have to have something to put after the if.
The problem is that all this presupposes that the structure is an "if A then B." That is that assumption of an A is hidden in our belief about the nature of the "ifs." Thus it ends up being a sneaky way of merely saying that we have to have to have "if A then B." But supposedly, that was what we were setting out to prove. So it ends up begging the question.
I still don't know what this thread it about, but it's not valid logic, so I don't know what it is even supposed to mean. You can't do if(if a). "if" does not evaluate to a truth value so you can't ever have more than one. In programming languages this the difference between an expression and a statement.
There are two ways out. One is to say that in a sentence like if(if a), then the "if" outside of the parenthesis means something different than the "if" inside the parenthesis. The outside "if" would be part a metalanguage about predicate calculus. You would need to define this metalanguage and give the rules for producing well formed formula. In any case until that is done you would have no way of knowing if the argument is valid.
You could introduction functional notation so that you are evaluating something like if(g(a)). You could nest as many g(a)'s together as you wanted. However if(g(a)) is a statement about g, not about a. After all g could simply mean "always evaluate true for any x in the expression g(x)." At this point the infinite regress doesn't even come into play.
My conclusion would be that if this is what Scotus is arguing, then it's nonsense as there is no way to evaluate it.
I think Clark just dropped the 'then' component of the phrases by accident. So more formally the idea would be something like (with 'A -> B' meaning 'If A then B'):
(A -> B)
((A -> I1) -> B)
(((A -> I2) -> I1 -> B)
...
and so forth, where each is a longer causal chain; the intermediaries (I1, I2, etc.) don't have any real role in the argument in themselves, so it would be easy to drop them without thinking.
We need to expand and modify Clark's original formulation a bit, though. The idea is that, because B exists, either B is uncaused or there is an actual sufficient total cause of B (a sufficient cause is a cause that is a sufficient condition, i.e., an A in the above; a total cause of B is all the causes needed to cause B). Now, the question is, How do we get a cause that's a sufficient condition in a regress of causes? It's easy to see how you would get a cause that's a necessary condition; by definition, every (particular) cause is a necessary condition of its (particular) effect. So for any series of causes, we could have a chain like this:
B -> I1 -> I2
And you can put in as many I's as you please. But all the causes together, being only necessary conditions, will only combine to create a necessary condition. But, since B exists, a necessary condition is not enough. Therefore there needs to be some cause that, either alone or in combination with the intermediate causes, is a sufficient condition for B. So we need an A, if I've laid it out correctly. To posit an infinite regress is to deny that there is an actual sufficient total cause of B.
Does that sound about right?
I was more thinking of the C language. If A evaluates to true if A is true. Then technically isn't a C construct but the { } symbols fulfill the role. Thus the then does the operation and the if does the evaluation. Mea culpa as I'm primarily a programmer.
Just to add, the reasoning you provide Brandon is pretty much why Leibniz has monads, as I recall. (It's been a few years since I last rigorously dealt with Leibniz, infinities and the nature of monads) He has an infinite regress but still a first point. So it's infinite in a different sort of way. His is more infinite in the sense of continuity.
But as I said, if we adopt Scotus' formulation, I fully agree with the contradiction. I'm just not at all convinced this is the way to think about causality. If we read Aquinas as Scouts does, then I likewise must confess to the contradiction. But I think this just reduces to the analogy 2 in the above post with me suggesting that analogy 4 is also a valid reading.
(More later when I'm more cogent)
Well, using C as the chosen language, if(A) is a statement not an expression, so it has no value, and can't be evaluated by a second if statement. Try compiling the following C program (scotus.c). The formatting will get all messed up but here it is:
#include int main(int argc, char* argv[]) { int i = 1; if(if(i)) printf("Yep, it's true!"); }
Also, since you want "if" statements to return values, you are no longer dealing with C or anything C like. You also are not in the realm of predicate calculus any longer, which was the point of my post.
Brandon
You example changes the syntax, but since the syntax maps directly to what Clark was using it has all of the same problems.
Also, I didn't realize, but since we are dealing with causes and not identities, you will have to add rules for logical inferences based on causation, which again makes it so that one is outside predicate calculus.
I don't think Aquinas is making the same argument, although two arguments seem to identify analogous problems with infinite regress.
I have plenty of time, so take as much time as you need.
David Clark,
I don't think I understand your point. The operation is ordinary implication, and would follow standard implication rules. It's ordinary propositional logic; we don't even have to kick it up to predicate calculus. There is a relevance restriction, i.e., we are only considering antecedents and consequents that are related to each other as cause and effect. But that doesn't change the rules of inference.
Brandon
You're right, propositional logic works just fine for reasoning about ordinary implication when dealing with individuals. However, this isn't implication in propositional logic either.
The statement "if a then b" has no truth value. "a" and "b" have truth values, which is why you can say "if a then b" because you can test for the truth value of a. You can't test for the truth value of "if a" or "if a then b", that is nonsense. Hence one cannot say if(if(a)) then b. You can't evaluate "if a", so there is no truth value to it, hence the statement is nonsense. This is what Clark was doing. Your modifications don't change this.
Brandon
I was out trimming my lemon tree when I thought maybe Brandon's syntax would work. I came back in an starting looking at it again. Unfortunately, it still doesn't work. In your post you said "A -> B" translates to "if A then B". So using your examples
(A -> B) translates to "if A then B". No problem
((A -> I1) -> B) translates to "if(if A then I1) then B", which isn't syntactically correct for propositional logic.
Now you could of course say (using ; as a line terminator): if A then I1; if I1 then B;. Of course in this situation the truthfulness of A guarantees the truthfulness of B.
David, I think you're complaining about the syntax of what I typed rather than the content.
If you prefer, use the following notation instead.
(a) → b
((a)) → b
((...((a))...)) → b
((...((...))..)) → b
With the latter never having an "a" and thus what Scotus rejects. Instead of saying "it can't be 'ifs' all the way down" we'll instead say, "it can't be braces all the way down."
With regards to Brandon's use of the inference operator though, you're incorrect. (A → B) → C means A signifies B which signifies C. However the braces clarify an ambiguity of signification. In other words we don't have (A → B) & (B → C) but rather we have the full if ( A → B ) then C.
While similar to the use of boolean algebra, it is syntacticly different. In logic classes one can say that "if a then b" is true. Indeed we do it all the time. It is because the claim that "if a then b" is either true or false. After all it may not be true that "if a then b" but rather "if a then c" only. This is different, I suspect than what you're used to. But in formal logic both "if ... then" and "if and only if" are unique truth functions just like "or" or "and."
Often in logic, to avoid connection with generic signification, we narrow it to the symbol ⊃ rather than the less specific →.
So what Brandon was writing is quite valid and meaningful. For a better overview, check out this nice Wiki on the subject.
Forgive me in advance if I'm misjudging your experience here - I'm dealing with a blind slate. It is true in pure propositional logic the inference operator doesn't function this way. And that's what you were taking Brandon's discussion to entail. We just have to be clear what we're talking about. Are we speaking propositional or logical calculus, are we speaking set theory, or are we speaking implication for propositions?
As I said though, one can rewrite the argument in any of these forms and still make the point. I prefer set theory since as I've asserted, analogy 2 and analogy 4 highlight a fundamental equivocation I see in Aquinas. But I'm too out of it to comment to that just yet.
"Are we speaking propositional or logical calculus, are we speaking set theory, or are we speaking implication for propositions?" I don't know, and that's the root of my confusion. I am comfortable with any of them provided that a consistent (and hopefully standard) syntax is followed.
"David, I think you're complaining about the syntax of what I typed rather than the content." Yes, I am, but I think I know what page everyone is on now. Why was it confusing, at least to me? (don't answer, the explanation follows)
What started it all was the initial statement where you say something like "if(if(if(a))) then b". My best guess at what you were saying using Brandon's more standard notation was "a -> -> -> b", which makes no sense. I assumed this because your (or Scotus') emphasis was on the fact that "it doesn't matter how many ifs you have." I assumed this meant many "ifs", with one "then" which the examples in the cases seemed to support.
Now when Brandon tried to explain what you were doing I got even more confused because I thought he was trying to map his more standard notation back into what I thought you were originally saying, which confused me to no end. In other words, I thought he was still trying to say something like "a -> -> -> b" (the Ix's should have clued me in that this was not the case). So Brandon you were right after all, you had the correct interpretation. My mistake, and you have my apologies.
However, the most recent example still confuses me. Following Brandon's more standard notation (I guess) you write "(a) -> b", "((a)) -> b", and "(((a))) -> b". However, these are all identical statements since parenthesis around single letters are optional. So I don't really know what the point of the statements are, unless of course parenthesis has some syntactical purpose other than grouping (in which case the notation isn't standard)? If the point of it is that parenthesis are optional in logic then I really *don't* get the point. I also don't get the point of no "a" being inside an infinite number of parenthesis not being valid. That would just be saying that you can't have material implication without an antecedent. But that's a definition, not something to be proved. I also don't see how this can be used to prove something else if that is the point. What am I missing?
As for where this goes from here, have fun, I still have no idea what any of this was all about. I jumped in because the syntax looked wacky and I thought someone was trying to pull a fast one. I guess someone could still be pulling a fast one, but I'll leave that to others to figure out.
I've closed comments in order to avoid spam since I don't check this older blog as much anymore.
and Theology
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Blogged by Clark Goble