Saturday, June 13, 2015

A slightly less terse semi-formal proof of the existence of G

Several Years Ago I published a semi-formal proof of the existence of G or God.  That little essay is here: (link).  It does not cover the Character of God - that is the question: "what is God like?"  It does, however, answer the question "Is there a God?"  It did it in such a way that it was a) new and b) preserved the intent of ancient proofs and c) was formalizable and sound in first order logic with identity and modal operators (necessity and contingency).  

The third requirement was the most interesting (and drove most of the answer to (a)).  Formalizability is a feature lacking in most demonstrations of God's existence.  The Ontological Proof is a stunning exception to this rule, essentially stating that because the definition of God is possession of every perfection, and existence is a necessary perfection, God exists by definition.  This proof however fell short in a few famous ways (in my mind, the proposition that Existence is a necessary perfection is questionable).    That is, the Ontological Argument has the feature that it is formalizable, but not the feature that it is formalizable using only provable statements from pure logic.  For instance, the notion of "perfection" is not a notion from first order logic and it is hard to see how it could be.   The requirement that my logical proof have only premises that were definitive of logic and science and only statements in it that were provable for those two disciplines was essential.  This was because the rhetorical form of the argument is roughly that since the existence of God is provable by pure logic, the only way to deny the existence of God is to deny the soundness of pure logic, that is, to be purely irrational.  One can't expect an irrational person to believe anything on the basis of reason, and thus they are exempted from this proof being interesting to them.  

However, the formalizability requirement made the argument terse and somewhat inapproachable to those without formal logic training.  The result is this reluctant essay in which I will try to present that same argument in a less formal way that my previous semi-formal presentation.  Unfortunately because part of the argument IS it's formalizability, it is impossible to do away with the symbolism of first order modal logic altogether.  There are plenty of good textbooks on the subject, but I have attempted to clearly state what each formal statement means in english and to try to present my derivations of propositions in english rather than formally in order to make the whole thing more readable.  This has had the unfortunate side-effect of making this whole subject much longer and less dense.   I personally prefer the density of information when it is purely information, and I regard what is contained in this as pure information, summarized by the statement "The existence of God is demonstrable in First Order Logic with Identity and Modality (I will refer to this as CORE LOGIC in what follows) which is itself fundamental to all science and human knowledge in general."  That is, that any rational person should accept that God exists is proved and provable.  I provide this information as a service to both God and Humankind in order that you all may know and love each other.  

The breadth of people who -should- be interested in this argument is extremely broad.  People who believe that mathematics expresses truths, that science discovers truths, that CORE LOGIC is valid, or that anything at all can be known by reason should be very interested in this proof.  That is, all of modern western academia should regard this proof as demanding their rational assent.  First Order Logic with Identity and Model extensions is -in fact- the language of science and reason.  

The proof starts by expounding on definitions.  In particular definitions of symbols used in CORE LOGIC so that it is extremely clear what is being proved.  

The most important definition and distinction for the proof is the definition of Necessity and Contingency.  That is, the unique operators of modal logic.   Necessity is, naively, the notion that some particular thing MUST be the case - can not possibly not be the case no matter what.  It is distinct from very high probabilities (there is a very high chance that the sun will come up tomorrow, but it is -possible- that a fast moving small black hole could collide with the earth tonight making sunrise impossible, for instance) in that some not-provably false statement could be true that would make the statement possibly false if it was true.    And here I don't just mean -physically- provably false.  I mean -logically provably false-.  That is, statements that break the first law of logic, namely that no pair of contradictory statements are true, the law of Non-Contradiction.  

Now, there are schools of thought that believe that -all of reality- is made up only of necessary facts - what I term Hard Determinism - the idea that everything that is the case must be the case and couldn't be any other way.  They deny this distinction between possibility and necessity.   Now I don't have a conclusive argument -against- Hard Determinism per se, but only considerations on why no person should accept Hard Determinism as factual.  The first, and most obvious, is that a Hard Determinist must regard every statement of a possibility as false.  That is, they can't even -consider the possibility- that something is true since that would be -for them- an extra-logical "possibility" for which they could not account in their core syntax.  They couldn't express the possibility in their language, that is.  But obviously, we all can all potentially express and understand possibilities, (in particular, that one) and I conclude that therefore the case for Hard Determinism can't be made in English, but some restricted language (I assume boolean logic with primitive arithmetic extensions might be the preferred language here).  But that language is not sufficiently expressive to be -our language- (we couldn't express fear or regret in that language) nor is it sufficiently powerful to express its own consistency  and therefore would be open to the basic attack of being inconsistent.  Simultaneously, Hard Determinism is a direct contradiction of Quantum Mechanics and is therefore anti-scientific (as I said, this doesn't make it false, just very hard to justify believing it).  That is, in short, Hard Determinism is very unlikely to be true and thus not part of my range of proof subjects.  Unfortunately I think this leaves Hegelian, Marxian and Spinozan historical determinists out of the conversation though I think their positions are very hard to defend in general anyway (though they are three of my favorite philosophers).  

Simultaneously, there are schools of thought that believe that there are no necessary truths.  This one is pretty simple to show is false.  If there are no necessary truths, then it is possible that there are no necessary truths and therefore possible that there are necessary truths.  If it is possible that there are necessary truths, it is possible that the possibility of the necessity of any of those truths is necessary.  If this were not the case, then it would be necessary that the truth in question was not possible, contradicting our previous supposition that it was possible, thereby contradicting the supposition that all truths are merely contingent.    (For those that don't know, a contingent truth is just a fact that could have been the case but is not or a fact that could is the case but might not have been).  

Thus are our core definitions established of Necessity and Contingency.  A necessary truth is one that could not fail to be the case no matter what.  A contingent truth is one that might not have been the case but is.  Now I've been loose on the question of Platonic Realism about any of this.  That is, whether or not real possibilities exist or whether they are just rhetorical or psychological objects of some kind, or whether or not numbers exist or whether or not Propositions exist.  These questions are beyond the scope of the current document and I hope you will forgive me for simply not addressing them except to say that I am in fact a Realist about platonic entities but that I intend for anything said here to not rely on the assumption of the existence of those platonic entities in any way so that the rhetorical position with regard to that question is effective either way and neutral with respect to either.  

With those preliminary distinctions and caveats in mind, we are ready to begin the proof itself.  

Proposition 1:  "Every set of contingent truths is also contingent."

Formally this is expressed : "Ax, y:  if !(x -> !y) & !(y -> !x), Pos(!x) & Pos(!y) <-> Pos(! x & !y)"

(where "Pos" is the contingency operator and stands -roughly-  for the english expression "Is possible" ).

Translating that formal statement into rough english: For all x and y, if x does not imply y and y does not imply x and if it is possible that x is false and possible that y is false, then it is possible that the group or set of x and y is false.    That is, formally it is affirming that the Contingency Mode is a distributive property.  So let's say you have two statements "The dog is in the yard" and "The dog is not in the yard" and both are contingent, it is therefore also contingent that "The dog is in the yard and the cat is in the yard" then it is possible that "The dog is in the yard and the cat is in the yard".    The requirement that x and y do not imply each other's falsity is there to catch cases like "the dog is in the yard and the dog is not in the yard" where the two together form a contradiction.  This is obviated by the negative case formation of the rule but is there just to make the expression clear for those who might think they've found a great exception (which they did!).

The proof of proposition 1 is pretty simple.   It is a proof by reductio ad absurdum - that is to show that the contrary supposition implies a contradiction.  Suppose the opposite, that there are two possibly false statements who's resultant set is not possibly false, that is, the set of them is necessary (since if something is not possibly false, it is necessarily true.)    Since necessity is Distributive (if it is necessary that a pair of statements is true, then it is necessary that each of them is true is a theorem of CORE LOGIC), it is easy to show that this supposition implies that is both necessary that both propositions are true and that it is possible that they are false, breaching our definition of possibility and necessity and resulting in a contradiction.  

Proposition 2:   "Every contingent proposition has a sufficient explanation for its truth"

Formally this expressed "A( P ): if  Con( P ) -> E(G): G -> P and G"

That is For every proposition, if that proposition is contingent then there exists some other proposition G such that G is true and G implies P.  This is equivalent to Lebniz's Law (The Principle of Sufficient Reason) in latin "ex nihil nihil fit" or "nothing comes from nothing".   Empirical science is -in fact- the expression of Leibniz's law in action in human society.  That is, science is the search for explanations and answers to the question "Why ?".  Anything -without- an answer to the question "Why?" is regarded as miraculous and suspect - the provenance of the spiritual if you will.  For example, the "Standard Model" in physics is the story of the big bang coupled with evolution to produce us humans here now alongside the laws of physics, chemistry and biology.  That is, all of it is just an answer to the question "Why?"   Now the further questions "Why are there laws of physics?" or "Why is there evolution?" or "Why was there a big bang?" are -out of line- for physicists.  This is not accidental I think, but in general if you want to make a physics professor happy, ask them why there is anything at all.  Now, given that we have restricted our audience further to "People who think science is relevant for producing knowledge", the existence of a BARE FACT - that is, one with no explanation for its existence at all, is absurd.  And for those people, I present the following proof:

Suppose that there is some proposition P such that no proposition G implies it.  But POSSIBLY, G is P - that is, G implies P because G = P.  Such truths we regard as NECESSARY TRUTHS, contradicting the supposition that P is contingent in our Proposition 2.  But since if P  and G are identical, if P exists, G exists and therefore our proposition would be fulfilled but by a NECESSARY proposition.  But suppose G is not P, then we regard G as P's explanation - why P?  G.     That is, in every case there exists some G such that G implies P, but in some cases P is not contingent but necessary.    That is, Leibniz's law is always true.  

Proposition 3:  "The set W of all contingent propositions exists"

Formally, EW: AP, Con( P ) -> P in W

Slightly less tersely: there is a set or group of things W such that if P is a contingent proposition, then P is a member of W.  Here I'm using the letter W to evoke the idea of World, but that is not the necessary meaning.  There are obviously MANY contingent propositions that are not members of this world (e.g. the statement that Santa Claus doesn't live in Manhattan is true (likely) but is not a fact about this world (since there is also no santa claus in this world).  That is - the W above is much more comprehensive than simply our universe or world, but rather includes ALL POSSIBILITIES WITHOUT RESTRICTION.  That is, anything that is possible and not necessary is a member of W.  I say this to encompass a variety of theories of possibilia, including classical, semantical theories, the possible worlds interpretation, platonic idealism, etc.  This is just to make sure that I'm not restricting the rhetorical reach of the argument by interpretation of modality.  

Now the rest is pretty simple and even obvious.  We know by proposition 3 that there is a set of all contingent truths.  We know by proposition 1 that if W is a set of only contingent propositions that W itself is contingent.  We know by proposition 2 that there exists some G such that G -> W.  Now suppose G = W, then W is necessary, contradicting proposition 1.  But suppose G is a MEMBER of W.  But then W is necessary, contradicting the supposing that W is the set of contingent propositions (since obviously W -> (… , W , …) where the "…" stand for arbitrary sets of true propositions).  Therefore we know that G is not W and G is not a MEMBER of W.  Since G is not W or a member of W we know that G is necessary (otherwise it would be a member of W).  

Hence we know the following:

Proposition 4: There exists some G such that G is necessary and G is the sufficient explanation of all contingent facts, W.  

That is, formally:

EG: Nec(G) & G -> W

(There exists an object G such that G implies the existence of the set of contingent truths).  

Now this G is not equivalent to Aristotle's Prime Mover though I think the intent is very similar.   The existence of MOTION is not necessary (it might be that nothing ever moved).  So the existence of a prime mover is contingent on there being motion (and all motion being dependent on a single source).  Neither is it equivalent to the G of the Ontological Argument since there is no attribution of perfection to G (yet).  Nor the Cosmological argument (essentially, since there is a world, there must be a God) although this is closest in intent.  The difference here being that the existence of G is deeper than the existence of the cosmos (since there might not be a cosmos and that fact, too, must be sufficiently explained by G).  I know the word "deeper" is vague, I intend roughly that the cosmological argument presupposes a specific model of this universe and derives, I think correctly, the existence of a sufficient explanation for this universe.  But it may be that there are multiple universes or none, or that there is no correct model of this universe, etc.  The existence of the set of contingent facts is definitive and without restriction making some of the standard objections to the cosmological argument fail against this argument.  

Thus I believe myself to have accomplished my three goals of having a formalizable, new, yet preserving the intent of the ancient proofs of God's existence.  This third requirement is important because I do not wish to attribute to St. Anselm or St. Aquinas an irrational belief in God, but rather just a different language in which they expressed this similar idea.  Which is to say, that I believe this argument to be an answer to the objections to the previous proofs of God's existence.  

It is important to note that this G who's existence I have just proved is not characterized by the argument - that is G might be anything -for all we know-.  It could, of course, be the beardy white man of middle-age artwork, or maybe a field mouse or the God of Abraham, Isaac and Jacob, or Siva, or a kind of mathematical fact or something.   This will produce in the kind reader the desire to express the response to the argument presented that whatever I have proved the existence of, it is not God.  This will come from those religious folks who's faith have led them to a deeper understanding of God than I have described.  To them I can say only that the intent of this proof is to edify and encourage  - for those who's faith is weak to have stronger faith, and for those who have no faith to find it.  For those who's faith is already strong, this proof must be something you've known for a long time and have not thought it important enough to bring to the surface.  


For those who have not faith and who desire knowledge of what G is, I have the following considerations in other blog posts which you will find on the web in various places:


First, G CHOSE to build this world, these facts, the way you are, all of it, exactly the way it is and could easily have chosen otherwise.  That is, G is "all powerful" in that it -in fact- made this world, but could have made another.  This fact has a proof which I presented in my earlier work:

We define choice as follows:

Proposition 5: A fact P is "chosen" if and only if it is possible that P might not have been the case and there exists some N such that N implies P but not necessary N.

More formally:

X:{ AP: Con ( P ) & EN: N -> P}

That is N sufficiently explains P but P is not necessary.  Thus the sufficient explanation of P -could have- not been a sufficient explanation for P.  Now it should be easy to see that the set defined as X is -almost- equivalent to the set W above.  I say -almost- since in Proposition 5, N the explanans might be necessary since we are defining the concept of choice.  N is neutral with respect to modality.   We see here that in applications of the law of sufficient reason that the notion of implication of a contingent fact is unproblematic as long as its explanandum is contingent.  However, if we are talking about the set W of (4) above, the explanans G is necessary.  But if G is necessary and G implies W then W is necessary, contradicting the supposition that W is contingent.  Thus it is not the case that Explanation is always IMPLICATION.  This is why the negative form of Leibniz's law is preferred.  Here an example is in order:

Suppose a door closes as a result of a child siting there and deciding to close it.  Now, the child -might not- we think, have decided to close it.   Therefore, the decision in the child to close the door is the 

(a) insufficient but necessary part of the unnecessary but sufficient reason

 for the door closing (using Mackie's analysis of causation).  That is, something else could have closed the door, but didn't, and if the child had not decided to close the door, then this -most likely- would not have happened.  Simliarly, if the child's deciding to close the door is nowhere near an implication that the door is therefore closed.  LOTS of things could have gotten in the way of the decision to do X and X's being done.  Thus for the purposes of our proposition 2 above, the conditional -CAN- be implication but also -CAN- be this causation.  

There are further fine-grained senses of conditionality that further refine the kinds of things that can -count- as sufficient explanations for the purpose of (2) above.  They are restricted by the same logical requirements that any sets of propositions would be restricted to (something can't both be and not-be a sufficient explanation).  The sense of explanation which we are interested for this particular purpose is one where the explanans is necessary and the explanandum is contingent.  This would seem to some to be initially contradictory but only because they are thinking of the strict conditional not the sufficient explanation requirement we asked for in 2 above.  

I propose the following for this particular case:

(b) a necessary part of a necessary and sufficient condition

Now the question is whether or not G's conditionality (b) above is more like (a) or more like ( c ) -pure- conditionality:

  ( c ) a necessary and sufficient condition

I say this because in the -spectrum- of kinds of conditionality I suggest that (b) being closer to (a) than ( c) in character (that is, by being severally complicated by necessity and sufficiency) makes the condition of G's conditionality on W closer to the child pushing the door than to the number 1 being required to exist before 1 + 1 could be 2.    That qualitative and quantitative closeness is sufficient to treat the action of G on W as a CHOICE rather than a mere condition.  Consider, does the -decision- to do X always result in X happening?  No, so it is not sufficient.  Is X's being chosen always a condition of X's happening?  No, so it is not necessary.  BUT if your choice to do something was necessitated (e.g. by brainwashing) it would still not imply that it would happen.  

The choice of God to create this world (out of all the other possibilities that might have been or might yet be) is thus very similar to the choice of a person to do something except that God might not have been able to fail (we don't know) and might have not done anything at all -at all- (that is, a person must always do something as long as they exist in some ways.  However, God might not have made anything contingent happen at all, and thus nothing -ever- would have happened, a choice us mere mortals do not have as far as I know).   It is also very dissimilar to that of a boulder rolling down a hill and hitting something in that the boulder was completely constrained by a large set of contingent facts.  The choice to create the world however, could not have been constrained by contingent facts in the sense of cause, but only by the existence of possibilities, that is, unrealized options.  But what is an unrealized option but the idea in a person's mind or perhaps a whole world?


We have seen from above that God Exists and God was Capable of Choosing this world from among all options and Did choose this world from among all options.  What we have yet to know is whether or not that was a GOOD THING.  A traditional attribute of God is God's Goodness.  While some people have worshiped as gods things that are not the creator of all things, among those that do worship the creator of all things, the common notion for all of them is that God is Good.  If the creator of all things were not Good, it would not be worthy of worship.  But the question is, is it good?

This part of the question is further complicated by the problem that an attribution of Goodness to God will require that Goodness be well defined as a predicate since, obviously, many people don't believe in Goodness per se at all.  Such people believe that Goodness is a culturally relative term, etc.   

I have found this attitude untenable in the face of the obvious evil that is in the world, and the great goodness often used to subdue it.  But is there a logical argument that would allow the introduction of the term "Good" to our purely logical vocabulary so as to remain within the confines of our requirement of formalizability?  

I suggest the following definitions:

If X exists it is good.
If Y causes X not to exist, Y is bad.
If Y causes X to exist, Y is good.  
If -on balance- Y causes more things to exist and for those things to continue to exist and produce things, etc., Y is, ON THE WHOLE, Good.
If -on balance- Y causes more things to cease to exist and for the things it causes to exist to cause other things to cease to exist, etc., Y is ON THE WHOLE Bad.

That is, existence and the spread of existence is good.  Nullification and the spread of nullification is bad.  That is, existence and goodness are co-extensive.   Salting the field so that nothing can ever grow is evil, but cutting the grass so that more things can grow is good. 

For choices, a Good Choice is something that causes, on the whole, more things to exist than it destroys.   A bad one does the opposite.  An intention is Good if it is the intention to make a good choice.  

To be beneficial is to cause, on average, more good than bad.

A choice is PERFECT if there is no other thing that could have been done that would have been more beneficial.

Now there is a very important one that is a bit harder to define and complicated to build up from smaller terms, so we'll jump ahead a bit - suffering.  Suffering is when a conscious being is the subject of a slow cessation of existence to the point of dread and death and pain.   The suffering of conscious beings is -worse- than the suffering of non-conscious beings for several logical and obvious reasons.  Firstly, a conscious being is capable of choosing and creating, something that non-conscious beings do not do (for whatever reasons).  As a result their potential for goodness is much greater than non-conscious beings (which can only do what they can do without choice).  For a conscious being can choose to improve the existence of others and thus multiply the overall potential for goodness whereas a non-conscious being can only serve as designated.  For instance, I am hoping to be doing a good thing by creating this essay to enrich your life so that you can also enrich the lives of others.  

Now since God, on the whole, has caused more existence than any other thing, it is the most beneficial thing in all of existence.  Could God have chosen to make a -better- world than this one?   Possibly.  But there is no reason to assume that God didn't do (or will not do) just that.  That is, this world as existing now is as close to infinite as we care to imagine, but even then there may be other worlds which are more or less close to perfect than ours, in terms of simply more existing things.    Now, I'm not sure if this is exactly right, but if God were to make the BEST OF ALL POSSIBLE DECISIONS - it would be to include everything that could potentially exist that -on the whole- caused the least suffering.  

Now, is this world, that world?  That is the problem of evil.  Could this world be the world that maximizes goodness and minimizes suffering, making it -the perfect choice-?

I'm hoping that that is the -strongest- formulation of the problem of evil possible.   The job in answering it is to decide exactly how the suffering to goodness ratio could be best optimized.  Consider that God could have made -every possible world- including those in which all people continually suffer at all times and those in which all people are free and happy and safe, etc., at all times.  I suggest that this world is -not- a world in which all people continually suffer (though I have heard something like this from some Buddhist friends).  I admit that I don't suffer much (I have a reasonable amount of pain and sorrow I think, but how would I know how much was normal?), but that many other people do -and have- suffered a lot throughout time.  

There are several debates and several possibilities here.  First is that God may have -in fact- made eternal life the reward for faith, in which case, no amount of pain leads to death, making all perceived suffering null and void for the faithful.  Second is that -just existing- the ability to feel anything at all, think, choose, be aware, is a precious gift.  And -percentage wise- the number of people who -on the whole for the duration of their lives- would rather have never existed is small, I think.  The price for the suffering in the world is all of the good that there is too - all of the happiness, all of the fish, all of the stars and galaxies, love, mystery, romance, and in some ways most importantly to me, freedom.   Third is that God -may be- in the process of creating every possible thing that can exist without being too much an admixture of evil and our observational context of -here and now- is simply unable to see that we are just a point in the great space of what is possible and thus something on the list of things to be done by God.  (The God of Spinoza, I think).  

Personally I think God only makes one world at a time - the group of coherent facts at any given time.  And this world changes as a result of God's desire to create as many good worlds as possible.  That is, I believe in the procession of time but only one "world".  I have great expectations for the future as a result.  For that reason I am very pleased to have been a part of this world and look forward to being part of those to come.  

In any case, I think that it is -very likely- that God took into consideration the possibility that all of us might like to exist and try it for a while even if we ultimately fail or fail in an untimely way.  We like being allowed to play -at all-.  

In short, I don't know that this is the best of all possible worlds, but it clearly might be and that is enough to answer the problem of evil as a logical problem.  As an emotional problem, in my opinion, the best thing to do is to be grateful for who and where you are at all times and to do your best to be as good as possible at all times, or, in the words of Our Savior, Love the Lord thy God with all thy heart and all thy soul and all thy mind and love thy neighbor as thyself.   If you are suffering, I can only say I'm very sorry and I'm trying to help in what ways I can.


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