First order logic, set theory and God

[Belacqua]

No engineer builds a bridge for example using the 'law' of Causation.

All engineers use the law of causation when they build a bridge.

They know that if a lot of cars are on the bridge, it will cause the load on the bridge to increase. Without the law of causation, they could imagine that lots of cars on the bridge would not cause the weight to increase. They assume that the continued properties of the steel and concrete will cause the bridge to behave in certain ways. (These assumptions, not ideas about origins, are what the first cause arguments refer to.) 

Engineers take it so much for granted that they don't have to name it, but they know it applies.

[I_am_not_mafia]

No engineer builds a bridge for example using the 'law' of Causation.

All engineers use the law of causation when they build a bridge.

They know that if a lot of cars are on the bridge, it will cause the load on the bridge to increase. Without the law of causation, they could imagine that lots of cars on the bridge would not cause the weight to increase. They assume that the continued properties of the steel and concrete will cause the bridge to behave in certain ways. (These assumptions, not ideas about origins, are what the first cause arguments refer to.) 

Engineers take it so much for granted that they don't have to name it, but they know it applies.

That argument really is a case of if all you have is a hammer then everything looks like a nail. Philosophers generally only know how to talk about the world with the highest level of abstraction.

No engineering school would teach the law of causation. And if they did then they wouldn't turn out good engineers. Engineers need to work at a far more refined level than that. They wouldn't be considering stuff like if there are too many cars then the bridge collapses. They'd be working out how much stress a bridge can cope with under different environmental conditions. And from that they'd then be able to work out exactly how many vehicles it would take to cause it to collapse and the probability of it ever reaching that level.

And this is exactly what I am trying to express. It's all a framework to reason about reality and talking about first order logic and the law of causation is too high a level of abstraction to be meaningful or useful, especially when talking about and making predictions of the nature of reality.

First order logic is a language. A framework. It is not a part of reality itself. It doesn't matter if a logical proof is internally consistent if it does not properly map to the real world.

You use it for stuff like heavily constrained computer programs. Most of us have even given up using it for artificial intelligence because it does not adequately describe the real world which is noisy and continuous.

[GrandizerII]

No engineer builds a bridge for example using the 'law' of Causation.

All engineers use the law of causation when they build a bridge.

They know that if a lot of cars are on the bridge, it will cause the load on the bridge to increase. Without the law of causation, they could imagine that lots of cars on the bridge would not cause the weight to increase. They assume that the continued properties of the steel and concrete will cause the bridge to behave in certain ways. (These assumptions, not ideas about origins, are what the first cause arguments refer to.) 

Engineers take it so much for granted that they don't have to name it, but they know it applies.

You're referring to material and/or formal causation, it seems ... if we're talking Aristotelian language, that is. Or maybe final cause, I don't know. But it still wouldn't get us to the logical need for a supernatural "sustainer" of the universe. This universe may very well be due to -say- an absolute logical principle behind modal realism (all possibilities are necessarily actual). Since this universe is clearly possible, it necessarily is actual (if modal realism is [necessarily] true, that is). This is just one example of course, and perhaps it's something else happening that's nevertheless purely naturalistic, but there needn't be any "first cause" in the traditional sense (as far as I know).

[Belacqua]

But it still wouldn't get us to the logical need for a supernatural "sustainer" of the universe. 

That's the question at hand. I'm glad to see this formulation of it, as it is accurate about the First Cause argument. "Sustainer" not "beginner." 

I dunno. There are elaborate arguments as to why there has to be an end to the chain of causes in this sense. And, as I said earlier, whereas science is content with "that's just the way it is" after they get past a certain point, metaphysicians aren't. 

I have read some attempts to argue for an end to the causal chain. Why at some point there must be an uncaused thing that causes all the following causes. Sad to say, while I could kind of make out the reasoning, I am not clever enough either to show that the reasoning must be true or to show why it's false. 

Mostly I just make noise by whinging when people mis-describe the argument, since I can't manage any more than that. 

The most elaborate, careful, and (I suspect) easy to understand argument is in Edward Feser's book Scholastic Metaphysics. I read it. It made sense at the time. I can't say why it's wrong. But that isn't enough to overcome my skepticism.

[Pat Mustard]

from a transtemporal perspective

Although First Cause arguments come up on forums like this every few months, I despair of ever having people address the real argument.

For example, it's clear from earlier responses that everyone here is thinking of a temporal series of events leading back to a beginning in time. It doesn't seem to matter that this is not what the First Cause argument addresses. As far as I know the only temporal argument is the so-called Kalaam argument, and that is rejected by both Aristotelians and Thomists. 

It's hard to point out that the causal chain in question is essential, logical, and simultaneous. So the question is better posed as something like "what exists in order that people may exist?" or "what is it that holds people in existence?" The answer is then something like "the earth and its atmosphere." And again, "what exists in order that the earth and its atmosphere exist?" And pretty soon we're down to atoms and their parts, and then the fundamental laws of nature. That's where science stops, because it says that the fundamental laws of nature just are, with no further cause. 

As far as I can tell, it was alchemist, Bible critic, and physicist Isaac Newton who changed metaphysics so as to allow this sort of "it just is" argument -- a brute fact -- to end the causal chain. First Cause arguments wish to ask what holds these brute facts in existence as well. 

As for the rest of your post, perhaps it must be clarified that Hatcher's proof was meant to provide a cut-and-dried deduction of the existence of a minimal concept (ie. a "universal uncaused cause") from a set of self-evident and empirically grounded axioms. While the proof doesn't tackle the fuller characterizations of God, there is at least little logical doubt that something akin to this minimalist understanding of a "God" is entirely believable and plausible as a distinct entity that has an existence, though perhaps little relation to what religious folks these days think of as a personal God.

Likewise, many people object to First Cause arguments because such arguments don't prove that the God of the Bible has the characteristics which (they assume) he is supposed to have. But as you rightly point out, First Cause arguments only attempt to demonstrate a first cause, and anything more than that demands different arguments. 

I really don't know why people who have been posting on atheist sites for years haven't learned this yet.

Re Thomists, if they are against the chain of causality argument as you suggest, then they are against Aquinas, as that is his first cause argument, ie that everything must have a cause, therefore god (and don't ask about god's cause). This is because Aquinas restated Kalaam for a christian audience.

[Belacqua]

Re Thomists, if they are against the chain of causality argument as you suggest, then they are against Aquinas, as that is his first cause argument, ie that everything must have a cause, therefore god (and don't ask about god's cause).  This is because Aquinas restated Kalaam for a christian audience.

My reading would indicate something different. I checked the ultimate cause of all truth (Wikipedia) just now. Their summary of Kalam is this way:

1. Whatever begins to exist has a cause;
2. The universe began to exist;
Therefore:
3. The universe has a cause.

I may be wrong, but I see that as a temporal beginning. 

Aristotle thought the universe had no temporal beginning; it was eternal. Aquinas reasoned that neither science nor logic can show that the universe began. If we think that it had a temporal beginning point, as in a literal reading of Genesis, we have to take it on faith. 

But I'm pretty sure that Kalam goes for a temporal first cause, whereas Aquinas/Aristotle is talking about a non-temporal, essential series.

Here is how it was explained to me by a guy I chat with on line sometimes. He's at the University of Chicago, and is a leading expert on Aristotle:

Try: an essentially ordered series vs. a temporally ordered one.

e.g. Parents are temporally and necessarily prior to their children. Even should the parents die/vanish, the children remain (temporally ordered series)

Space-time is essentially and necessarily prior to carbon atoms. Should space-time vanish, so too, simultaneously, do any and allcarbon atoms vanish (i.e. there can be space-time without carbon atoms, but there can't be carbon atoms without space-time)

[Angrboda]

Since the first cause need not be a god, the proof is an example of ignoratio elenchi, at minimum.  It is not a proof of the existence of God.

There are other problems, but that is sufficient.  This is simply the basic first cause argument fleshed out with formalism.  Hardly worth even bringing up.

The point sailed over your head it seems. You'd have to provide some compelling and convincing arguments to support your position, instead of pointing out the fallacy that I'm allegedly guilty of.

No, I wouldn't have. The existence of a fallacy indicates that your conclusion doesn't follow from the premises. That's how logic works, dumbass. I could have pointed out other errors, but there was no need to do so having established the one. A point that apparently sailed over your head.

Are you really this stupid?

[polymath257]

Below is a copy-paste of my own thread that was posted in another forum. I'm reposting it here in hopes to spark an intelligent discourse on what I believe to be the most refined proof of God's existence.

Here it goes.

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Written in first order logic, Hatcher's proof of God is based on three axioms that he calls "empirically grounded" and an apriori assumption that "something exists."

The axioms are that:

P1. The principle of sufficient reason: All phenomena are either self-caused (i.e. A->A) or other-caused (B->A; B is not equal to A) but not both. Put another way, this principle says that the question "why?" is always meaningful. Everything happens for a reason.

P2. The potency principle: If A -> B then for all C element of B, A -> C. In other words if A is the cause of B then A is the cause of every part of B. There are several notions of causality in philosophy. Hatcher's notion of causality is total causality; i.e. it is not the straw that breaks the camel's back but the 1000 straws before it, the camel, gravity, and so forth, that give rise to the camel breaking its back.

P3. The principle of limitation: For all A, where A is an element of B, B -> A does not hold. This says a system (which Hatcher represents as a set) cannot be the cause of its own components. Hatcher justifies this by explaining any system has (1) form (the parts) and (2) function (the relationship between the parts). A car (the system) cannot be the cause of its own steering wheel (a part), because the car does not even logically exist until the steering wheel exists. Thus the car's existence cannot precede the steering wheel's existence.

Hatcher shows that the logical outcome of these 3 axioms together with the above noted assumption are the existence of a "unique, universal, uncaused cause."

Throughout this work, Hatcher strives to make his assumptions (his axioms) and modus operandi (first order logic) explicit. Unlike many proofs of God (beginning with the proof advanced by Aristotle) Hatcher's proof does not appeal to the absurdity of an infinite regression of causes. Hatcher argues that because his proof is formulated in first order logic one must invalidate one or more of his three empirically grounded axioms to refute it. At the same time he shows that doing so is difficult as it commits oneself to beliefs not commonly accepted in the scientific community such as the existence of non-causal systems (something not observed to date).

In the following I have provided a more complete formulation of Hatcher's proof.

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The following is a excerpt from pages 82 through 86 of "Love, Power, and Justice: The Dynamics of Authentic Morality" by William S. Hatcher.

Copyright 1998 by William S. Hatcher

Chapter 3, section 4

The Existence of God

Our purpose in undertaking this study is to establish the existence of God on a totally objective basis, as a necessary logical feature of the overall structure of reality itself.

+ ( R ) ;Reality = the totality of existence, everything there is.
+ (P);Phenomenon = some portion of ( R )  
+ Causality = relationship between (P) A and (P) B, which holds whenever A is a cause of B (symbolized A → B).

This means that A contains a sufficient reason for the existence of B. => everything B that exists must either be preceded by a cause A different from B (A → B and A ≠ B), or else contain within itself a sufficient reason for its existence (B → B).

<=>In the former case, we say that B is caused or other-caused and in the latter uncaused or self-caused.

* The principle that every existing (P) must either be caused or uncaused (and not both) is the principle of sufficient reason.

<=>Another basic relation between (P) is the relation of part to whole: we write A ∊ B whenever the entity A is a component of the system (composite phenomenon) B.

Notice that A may also be composite, but must be an entity (not just an arbitrary system) in order to be a component of another system B (whether the latter is an entity or not).

Two systems (whether entities or not) may also be related by one being a subsystem of the other. We write A ⊂ B whenever A is a subsystem of B. This means precisely that every component E ∊ A is also a component E ∊ B.
=> For example: a single leaf would be a component of a tree, but all the leaves together would constitute a subsystem of the tree.

If E is either a component or subsystem of B, then E is a part of B.

From the strictly logical point of view, the defining or characteristic feature of an entity A is that A can be a component of some system B, A ∊ B.

<=> entities are components while systems have components (they are composite phenomena). Moreover, some systems also are components.

Thus, with respect to composition, we have three distinct categories of phenomena:

- (P) may be noncomposite (have no components), in which case it is necessarily an entity.
- (P) may be a composite entity, in which case it both has components and is a component.
- (P) may be composite without being an entity, in which case it has components but can never be a component.

Causality and composition are related to each other by the obvious potency principle, which says that if A → B, then A must also be a cause of E, where E is any component or any subsystem of B.

<=> to be a cause of B is to be a cause of every part of B -- its components and its subsystems. This means that our notion of causality is that of complete cause (philosophy recognizes several different notions of "cause").

Finally, the existence of a whole system obviously cannot precede the existence of its components (rather, the constitution of a whole obviously supposes and depends upon the prior or simultaneous existence of its components).

We thus have the principle of limitation= every composite phenomenon A, A cannot be a cause of any of its components.

=> It follows immediately from these principles that no composite phenomenon can be self-caused, for suppose A → A where A is composite. Then, by the potency principle A → E, where E is any component of A. But this contradicts the limitation principle.

In fact, from these valid principles of causality and composition, we can logically deduce the existence of a unique, noncomposite, self-caused, universal cause G. This entity, whose existence we prove, is God (by logical definition). This God is not some abstract figment of our imagination but the actual, ultimate cause of all existing phenomena and entities, the origin of all being.

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[==== Since the proof is easy, we give it here in full. However, the reader who already accepts and understands the existence of a universal uncaused cause (i.e., God) can safely skip the details of the proof without diminishing his or her understanding of the subsequent sections of the course.

Let V=collection (universe) of all existing entities.

Since V is composite it cannot be self-caused (see above) and so must have a cause G (different from V itself).

Thus, G → V; G ≠ V

Moreover, every existing (P) A is either an entity, and thus a component of V, or else a system all of whose components are in V -- in which case A is a subsystem of V.

Thus, G is either a component or a subsystem of V. But, in either case, G → G by the potency principle.

Thus, G is self-caused and hence noncomposite (no composite can be self-caused as shown above).

Finally, since G → V and every (P) A is a part of V then by the potency principle, G = universal cause (the cause of every existing phenomenon, including itself).

Finally, we show that G = the only uncaused phenomenon, for suppose there is another such phenomenon G'.

Then G → G' (since G is a universal cause). But since G' is self-caused it cannot be other-caused by the principle of sufficient reason.
v
Thus, G = G' and the uniqueness of G is established.===]

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With that being said, I would be more than curious to see if anyone could spot a noticeable error in Hatcher's logical deduction

Which version of set theory are you using for your 'component' relation? In particular, what allows you to form the system V? How do you avoid Russell's paradox in your set theory? We *know* that naive set theory is inconsistent and no 'set' of the form V can exist in standard ZF set theory.

At the very least, you don't have a first order system when you allow the formation of arbitrary collections (that being a second order construction), *unless* you specifically give the rules of set formation.

This doesn't even address whether the axioms used  for the causality relation have anything at all to do with reality (and others have commented that issue).

[dr0n3]

A composite phenomenon, based on the principle of limitation, cannot be a self-caused phenomenon. That is one thing we should keep in mind.

Yes, but I am questioning the principle of self-limitation's validity as an axiom.

As for the rest of your post, perhaps it must be clarified that Hatcher's proof was meant to provide a cut-and-dried deduction of the existence of a minimal concept (ie. a "universal uncaused cause") from a set of self-evident and empirically grounded axioms. While the proof doesn't tackle the fuller characterizations of God, there is at least little logical doubt that something akin to this minimalist understanding of a "God" is entirely believable and plausible as a distinct entity that has an existence, though perhaps little relation to what religious folks these days think of as a personal God.

To field one more objection to the proof, it does not handle a scenario of cyclical time, perhaps from Big Bang/Crunch cycles.  Here you can have A causing B which turns around to cause A, with a sequence of causes regressing into the (repeated) past ad infinitum.

The validity of the principle is as clear as daylight - there is absolutely no way of rationally conceiving a counterargument to the notion that a composite can never be the cause of one of its own components. The logic is impeccable. The reason you're questioning it is because of the ambiguity of your previous example. Namely that, cells in a human body are continually dying out and being replaced by new cells through the process of mitosis. Yes, at first glance you might think it nullifies the principle in some way since one could consequently deduce that the whole (human body) has been the cause of one of its own parts (new cell).

Yet, upon careful analysis, the conclusion doesn't hold water.


First, we know that your example hinges on several time-parametered phenomena coming to being through a succession of stages( where a stage is the state of the phenomenon at a given instant of time), which of course logically presupposes a causal discrete-time system. Secondly, one must make a clear distinction that the new human body (the one that contains the new cell) is different than the (old) human body that manufactured the new cell.

In light of the above points, we can now suppose the old body H1 at time T1 and the new body H2 at time T2 where we understand that H2=H'+B, where H' is H1 however transformed in the process in producing the white cell B. Thus, the correct causality relation (logically and temporally) is then H1→B|B⊄H1 and not H2→B|B⊂H1.

As for your last argument, temporality is not necessary within the scope of Hatcher's proof. The notion of causality is first and foremost logical and relational.

Yes, but why call it god?  It's so divorced from both the common conception and most philosophical conceptions that the choice of term can only engender confusion.  This is an equivocation land mine just waiting for someone to trigger it.

For sake of simplicity and because "god" is the more-or-less standard term for a "universal uncaused cause". Regardless, it's not as if nitpicking over terminology will render the substance of the original argument any less valid.

My point is though that first order logic is still nothing more than a way of describing reality.

Exactly, which makes it inherently linked to reality through its connection to truth. In that sense, one could just as confidently say that physical laws are descriptions of the dynamic behavior of matter in space-time.

I'm sorry, but what is your point again? 

Exactly. True and False are abstractions. They do not exist in nature.

Mathematics is abstract, yet inherently grounded in reality. Guess what? Same goes for True and False. Consequently, this renders your argument completely false.

Rubbish. It's extremely important.

What causes a hammer to exist for example?  When someone connects a handle to a head? When a wood is reshaped to create the handle? When metal is melted down and shaped into a head? When a tree is cut down? When ore is mined from the Earth? When a tree is grown? When a planet is formed? What is the single cause of a hammer? There isn't one. All we have is energy flowing through and reshaping matter in accordance of the laws of Thermodynamics. Your abstraction of True, False and causation miss all that because they are abstractions.

What you fail to grasp is that the law of causality is not descriptive but rather relational. A hammer's existence is not contingent upon whether or not the cause of its existence has to be known to us. Rather, the hammer's existence is already necessitated by an external cause preceding it, whatever that cause may be. Just as it is necessary for number 4, for instance, to occur between the numbers 3 and 5.

But just for argument's sake, the answer to your question would simply be - the causal link of all phenomenon that has brought the necessary and suffi­cient conditions for the realization of the hammer. That's it. Causality is merely established by the interdependence and relation between the cause and the effect.

 

The point sailed over your head it seems. You'd have to provide some compelling and convincing arguments to support your position, instead of pointing out the fallacy that I'm allegedly guilty of.

No, I wouldn't have.  The existence of a fallacy indicates that your conclusion doesn't follow from the premises.  That's how logic works, dumbass.  I could have pointed out other errors, but there was no need to do so having established the one.  A point that apparently sailed over your head.

Are you really this stupid?

In all honesty, the impression one gets by reading your posts is akin to a douche on a futile endeavor of throwing around "fallacy" this and "fallacy" that, and not even understanding what they're talking about. Perhaps you should understand that you don't get the decisive edge in an argument by simply brandishing retardedly your "Fallacy" wand at every chance you get and cramming your goddamn post with an impressive-sounding Latin term. Ironically enough, you're just as guilty of committing the fallacy of supposing you can defeat an argument simply by appealing to a named fallacy. You fucking dunce.

[I_am_not_mafia]

My point is though that first order logic is still nothing more than a way of describing reality.

Exactly, which makes it inherently linked to reality through its connection to truth. In that sense, one could just as confidently say that physical laws are descriptions of the dynamic behavior of matter in space-time.

I'm sorry, but what is your point again? 

My point is that first order logic doesn't describe reality very well.

There is an awful lot of fundamentally important stuff that it cannot describe at all. Making your so called proof worthless if you are trying to say something about the nature of reality. For example thermodynamics, complexity and chaos.

Exactly. True and False are abstractions. They do not exist in nature.

Mathematics is abstract, yet inherently grounded in reality. Guess what? Same goes for True and False. Consequently, this renders your argument completely false.

It is not grounded in reality any more than the English language.

Rubbish. It's extremely important.

What causes a hammer to exist for example?  When someone connects a handle to a head? When a wood is reshaped to create the handle? When metal is melted down and shaped into a head? When a tree is cut down? When ore is mined from the Earth? When a tree is grown? When a planet is formed? What is the single cause of a hammer? There isn't one. All we have is energy flowing through and reshaping matter in accordance of the laws of Thermodynamics. Your abstraction of True, False and causation miss all that because they are abstractions.

What you fail to grasp is that the law of causality is not descriptive but rather relational. A hammer's existence is not contingent upon whether or not the cause of its existence has to be known to us. Rather, the hammer's existence is already necessitated by an external cause preceding it, whatever that cause may be. Just as it is necessary for number 4, for instance, to occur between the numbers 3 and 5.

But just for argument's sake, the answer to your question would simply be - the causal link of all phenomenon that has brought the necessary and suffi­cient conditions for the realization of the hammer. That's it. Causality is merely established by the interdependence and relation between the cause and the effect.

Which completely misses the point I am making in that reality is continuous whereas your arguments of causality are discrete.

As I said, your description of reality is not sufficiently powerful enough to say anything worthwhile at all.