To Live Freely: Part Three

Propositions and Predicates)

I have explained the necessity of adhering to physical truths in the field of aviation. In order to overcome the forces of gravity and air resistance, great minds had to search out the realities of the physical world and build machines that would act in accordance with them. Today I’d like to consider another example of this in the world of logic. This time we won’t just consider the usefulness of truth, though, but also the chaos of untruth.

There is a concept in mathematics called propositional and predicate logic. In this system, propositions are statements of truth, such as George is Abe’s father, Steven is George’s father, and Marcus is not Abe’s brother. These are simple facts that contain a single piece of valid information. Then there are predicates, which are rules for how these propositions can be combined to reveal entirely new truths. For example, we might have a predicate that if A is the father of B, and B is the father of C, then A is the grandfather of C. Given our initial propositions, we can derive that George is Abe’s Grandfather, a fact that wasn’t in the original set of information.

This might not seem that useful, but once we expand our set of propositions and predicates to thousands of items there are literally millions of implied facts that a computer can derive from, something that our brains simply don’t have the capacity to process. Our modern-day databases are built upon this system of logic, allowing a large dataset to have its parts combined in a multitude of ways, revealing hidden patterns and trends, secrets and truths that were hiding in plain sight.

Let’s build expand on our example of a family tree to see this process more clearly. Suppose we have the following propositions and predicates (feel free to skim over these):

Propositions:
#1 George is Abe's Father
#2 Susan is George's wife
#3 Penny is Susan's daughter
#4 Penny is Abe's sibling
#5 Helen is George's sister
#6 Gabe is Marcus's father
#7 Steven is Marcus's maternal grandfather
#8 Agnes is Helen's mother
#9 Steven is Agnes's husband
#10 Howard is Susan's father
#11 Jill is Susan's mother

Predicates:
#1 If A is the father/mother of B, and B is the father of C, A is the grandfather of C
#2 If A is the father/mother of B, and B is the mother of C, A is the grandmother of C
#3 If B is a parent of A, and C is the husband/wife of B, then C is also a parent of A
#4 If A is the father/mother of B, and C is the other parent of B, then A is the mother/father
#5 If A is the father of B, and C is the mother of C, then A is B's husband and B is A's wife
#6 If A is the child of B, and C is the child of B, then A and C are siblings
#7 If A is the maternal grandfather of B, and B's mother is C, then A is the father of B
#8 If A is the sister of B, and B is the parent of C, A is the aunt of C
#9 If A is the child of B, and B is the aunt/uncle of C, A is C's cousin
#10 A cousin is not a sibling
#11 A mother is not a father

Given this setup, we could piece together the following family tree:

This tree is a visual representation of all the separate facts we get by combining all of our initial information. We can ask our system any number of questions, even ones that go beyond the scope of the original data set, and it can derive answers for them. It will answer yes, no, or uncertain, and so long as our propositions and predicates are all correct, then we can know that any derived answer is also correct. This data is a source of truth because it is based on logically sound principles.

A Twist)

But what if all of our propositions and predicates are all correct…except one? What if among all the truth facts and rules we include just one falsehood? It might occur to you that this would tarnish our confidence in the system, because there would always be a possibility that the answer it gave to us was that one lie. But actually, the effect is far, far worse. It has been proven that introducing just one logical falsehood into a system such as this will make any possible lie seem true. It won’t just be one lie that comes out of the system, it will be all lies. That might seem improbable, but allow me to illustrate.

To the system up above I will introduce one logical falsehood. Given the previously established rules, it is impossible for this to be the case, but I am going to enter it as a fact even so:

Susan is Abe's father

This statement is completely contrary to the logic of Predicate #11, but we add it to our system regardless. This creates a logical contradiction, and now let us look at all the new falsehoods we are able to infer from it. By Predicate #4 we can infer that since since Susan is Abe’s father, then Abe’s other parent, George, must be his mother.

Of course, we previously had derived that Steven and Agnes were Abe’s paternal grandparents, because they are George’s parents. But now that we know that George is Abe’s mother, then they must also be his maternal grandparents. By the same token, Howard and Jill are now no longer only Abe’s maternal grandparents but also his paternal.

Of course, now that we know that Howard is Abe’s paternal grandfather we can combine that with the already-known fact that Agnes is his paternal grandmother, and we can now infer that they are married together, something we never knew before! And by the same token, Steven and Jill are now also married together. Thus all the grandparents are intermarried in some sort of free-love commune! This does have the unfortunate effect of making George and Susan, Abe’s parents, siblings to one another in addition to still being husband and wife! Furthermore, since Abe’s parents are also siblings, then his sister Penny is also his cousin because her mother is the sister of Abe’s father (and her father is the brother of Abe’s mother).

But we aren’t even really going yet! We still haven’t invoked the powers of NOT and ELIMINATION. First let’s consider the NOT. Predicate #10 stated that a cousin is NOT a sibling, and Predicate #11 that a mother is NOT a father. So, since we just proved that Penny is Abe’s cousin, then she is NOT his sibling. Of course, she also is his sibling, since Proposition #4 explicitly says so. Thus, she is his sibling, and she is not. These are both totally valid answers in the eyes of our data set. And Abe’s parents George is his father and Susan is his mother, but also, they are not. And his grandparents are his grandparents, but also, they are not.

And now that we’ve shown that we can prove that the exact same relationship can and cannot exist simultaneously, by ELIMATION we can also prove that every relationship can and cannot exist. So, from the initial data set we know that Abe has a sibling. But who is it? Well, we can go through each member of his family and prove that they are not that sibling. So, let’s do that for every family member except one, Steven, and now we know, by process of elimination, that Steven must be the one who is Abe’s sibling. And by the same process we can prove by process of elimination that it is Agnes, and Howard, and Gabe, and Helen, and George, and Susan, and Marcus. And by the same process they are all his father, and all his mother, and all his aunt, and all his uncle, and all his cousin, and all his grandfather, and all his grandmother.

I’m not going to try to show the family tree at this point, because it is simply all names connected to all other names in every possible way. But also…all names connected to none of the others. Every statement is true. Every statement is false.

Our data set was useful at one point. It was full of true statements, and it could be used to infer many other true statements. But now, after a single lie the entire thing has been corrupted. The only answer it has to provide are “yes, no, maybe, I don’t know…I guess it depends on how you look at it.” It has lost all confidence and isn’t useful for anything.

And sure, this is a rigorous and mathematical system, which is particularly prone to collapsing at the slightest instability. The system in our minds is far more nuanced, able to continue functioning with illogical assumptions and idiosyncrasies…but only to an extent. The same principle does apply to us to at least some degree. Adopt the wrong belief and suddenly every other concrete conviction starts to be undermined by it. People start going through logical acrobatics to try and make incompatible beliefs fit together, corrupting all that was once good and losing the certainty they once had. We cannot accept a lie without somewhat losing our grip on all truth.

To Live Freely: Part One

Axiomatic Truth)

There is a concept that has come up a few times in my previous series, including the last one. I have spoken to the matter in brief here and there, but now I want to consider it more fully. The concept is that living in the truth is the foundation for a full and happy life. Said another way, facing the facts as they really are is the only way to be truly free. Said a third way, only those who are willing to face the truth unflinchingly are ever truly alive.

This is a principle that is basic and fundamental to life. It is so foundational that sometimes it is difficult to really get a grasp on it. Axiomatic truths are, by definition, self-evident in their truthfulness, requiring no argument to prove them. That’s all well and good, but it means that if you then try to explain why an axiom is true you’re going to have a very hard time of it! Explanations tend to lead to circular logic, such as “living in truth is the foundation for a full life because…it just is!”

One way to come to full appreciation of these fundamental truths is to look at them in reverse. Fundamental truths are prerequisites for many other things in life, and by examining those things that are built upon foundational truth we obtain evidence that the underlying axiom really is true, for if it were not the things that we have observed could not be. We find that the fundamental truth is necessarily true, because it is necessary for it to be true for other observable things to be so.

Even so every good tree bringeth forth good fruit; but a corrupt tree bringeth forth evil fruit. A good tree cannot bring forth evil fruit, neither can a corrupt tree bring forth good fruit.... Wherefore by their fruits ye shall know them. -Matthew 7:17-18, 20

Jesus describes the same idea in these verses where he teaches that we may recognize that which was good by whether it brought about good or not. So if we want to know whether “a life founded upon the truth is joyful and free” is a true statement or not, then we merely have to look at those who live in harmony with this belief and see what sort of life they possess. I will begin my series today by doing exactly this. I will look at an example of people who are built upon this axiom, living their lives with the assumption that it is absolutely true. We may observe the reality of their lives, and infer whether they built upon a solid foundation or not.

The Happiest of People)

I have mentioned before how the addiction-recovery groups I have attended are singularly focused on living in harmony with truth. Any addict working a twelve-step program can tell you that one of its most fundamental tenets is that we take a fearless inventory of our lives, facing all of the unpleasant and difficult truths in our character. Where most people attempt to cherry-pick their best qualities and define themselves by those, addicts in recovery open the door to all of their qualities. We do not care if the description of us is pleasant, only that it is true.

And what comes about by this strict adherence to seeing things as they really are? For an answer, let me offer an anecdote that occurred to me personally. I was speaking with an ecclesiastical leader about my efforts to overcome my addictions, and my time spent in my recovery group. As soon as he heard that I was part of a twelve-step program he said to me, “you know, I’ve never been a part of such a program, but I have been a witness to its meetings and its members, and those are the most humble, most sincere people I have ever met.”

The reason why the twelve-step program has grown at such incredible rates since its inception is entirely due to the quality of the men and women one meets when they walk through the door. People see men and women who have not only gained freedom from the most terrible of vices, but who also live with a clearness and a joyfulness that simply isn’t to be found anywhere else. Furthermore, the fact that that light has remained consistent throughout the decades and continues to burn brightly in every new generation of members is a testament to the fact that the happy way of life was not due to some pre-existing condition in the first AA members, but is cultured in its members from the principles that they live by. If people had not seen throughout the years that these people had uncovered a superior way of life by their principles, then no one would have stayed and joined the crew, and it would have been a long-extinct experiment.

It was the evidence of this joyful peace that also drew me into the ranks of the twelve step program. It might seem a counter-intuitive thing to say, but I quickly recognized that I had never seen a happier, more satisfied, and more productive people, than these addicts who sincerely identified their miseries and their flaws. One would have thought that bringing out those heavy truths would have crushed them, but so far as I could see those weights, once surrendered, were being taken away, so that they could live free and unfettered. They attested that one had to truly see their shackles before they could receive the key to undo them. Though I was not then converted to the notion of living my life strictly in harmony with the truth, I was persuaded enough by what I saw to give it a try. My result has been much the same as theirs.

But you don’t have to take my word for it. Even if you don’t consider yourself an addict, go and visit a few of the local meetings in your area. See for yourself what manner of men and women these are, and what sort of lives they lead. Granted, every group has its own culture and its own level of sincerity about the work, but attend a few different ones and you will quickly see that there is a clear correlation between those that genuinely face the hard truths and those that live joyful and free.

Our Own Reality- Proverbs 21:2

Every way of a man is right in his own eyes: but the Lord pondereth the hearts.

COMMENTARY

Every way of a man is right in his own eyes: but the Lord pondereth the hearts
Yesterday we considered the fact that those of us with obscured vision still believe that we see the truth perfectly. It is easy to make the assumption that if other people have wrong views, and our views are different from them, then our views must inherently be right. But that is a logical fallacy.
How then, when our perspective is genuinely clear and correct, do we even know that it is so? How are we to know when we see the truth that we even are doing so?
That is a whole other topic of study worth exploring. For now, though, I’ll settle with the second half of the verse I’ve quoted. We will know that we have finally the right perspective when it is not our own perspective anymore, but His.

The Way That Things Are- Gödel and the Incompleteness Theorem

Kurt Gödel was a logician and mathematician born in the early 20th century. He was a contemporary and friend of Albert Einstein’s, and as influential to the world of logic as Einstein was to the world of Physics.

Gödel’s most famous contribution were his Incompleteness Theorems, which proved that for any system there are truths which cannot be defined by the system itself. There are some things which are true, but which cannot be proven until you utilize outside sources.

Now this is a very crude example meant only as an illustration, but think of it this way: if we have a system called integer numbers (1, 2, 3, 4…) then there are patterns about those integers which do exist, but which we will never be able to discover with integers alone. We will have to add something else, perhaps fractions, in order to prove them. But now we have a new system, one of integers and fractions, and there are new patterns about this larger system which we will never be able to explain until we add something like irrational numbers, and so it goes on.

Now Gödel’s Incompleteness Theorem was related to logical and mathematical systems, but they raise a strong philosophical question as well. I have always been baffled that we humans are so arrogant as to believe that we could ever fully understand ourselves. It seems to me that we are much too close to the subject matter to ever find all of the objective truths locked within our souls. I find it far easier to accept that the only complete understanding of myself would have to come from a being that existed in a higher system than our own.

Thus when society decides that truth must be one thing, but God’s word declares that the truth is something else…it frankly is not difficult to for me to side with the unknowable omniscient. Indeed, the fact that God is unknowable gives his argument greater weight to me, not less. If God was comprehensible to me, then there’s no way He’d be great enough to comprehend all of me.

Evolving Your Beliefs- Logic

We use the word “logic” quite a good deal, it is something we strive to live in harmony with. In its original form, “logic” is simply the study of how to make correct inferences from known truths. If A is true, then so is B.

Logic has many different branches of study, including propositional calculus, predicate logic, and modal logic. It has applications to mathematics, computational problems, and even philosophy. Many of our technologies today, such as the modern database, are based upon its principles.

Another way to explain logic is that it is the study of relationships between truths. Take for example the statements “Adam is the father of Seth” and “Seth is the father of Enos.” By these two truths we may logically infer that “Adam is the grandfather of Enos.” In fact, by being given only a smattering of relationship facts, logic can be used to recompile entire family trees, defining numerous relationships between every member.

And all of this works…until a lie is introduced to the system. It has been proven that a single lie can totally break down any logical system. By process of elimination, one can prove or disprove anything. You could simultaneously prove that Adam is the father of Seth, that he is the mother of Seth, that he is no one to Seth, and that Seth is actually his father. And you can also disprove all of those statements, too. In a word, everything becomes “relative.” Where before you could go to a system of truth and find verifiable fact, now all that remains is a shrug of the shoulders and a “maybe.”

This happens to us in our lives as well. Each of us is born with a very simple model of truth. We inherently accept principles of love, faith, and goodness. It is a small core of truth, but it is sufficient. As we go through life we discover new facts, accepting those that seem to fit with our already-establish model, and rejecting those that do not.

However somewhere along the way, each of us will make a mistake. It is very easy to do. Perhaps a trusted authority figure gave us a notion that we accepted without a second thought. So we added a falsehood, but we believed it to be a truth. We may not realize that anything is amiss for a while, but over time, that lie will corrupt our previous associations. We’ll start to notice logical contradictions in our beliefs, and finally we’ll know that our system has become untenable.

Sadly, many will throw the entire thing out at this point. The work of pruning out the lies from the truth seems impossible. They will claim that there never really were any truths to begin with. It can be a hard thing to let go of a misconception about God without letting go of God entirely.

But that is not the only option. Sometimes evolving our beliefs is a matter of going back to basics. We realize that we went astray, so we return to what few facts we really do know: that we are a child of God, that He loves us, that there is such a thing as “good.” It might be a much smaller belief system, but it will be true again. Then, with utmost care, we add back in only the parts that fit with this core.

Trial Before Blessing, Pleasure Before Anguish- Lorem Ipsum

No one rejects, dislikes, or avoids pleasure itself, because it is pleasure, but those who pursue pleasure irrationally may encounter consequences that are extremely painful. Nor again is there anyone who loves or pursues or desires to obtain pain of itself, because it is pain, but occasionally toil and pain can procure some great pleasure…
Who would fault a man who chooses pleasure that has no negative consequences, or who avoids any pain which produces no resultant pleasure? On the other hand, we denounce with righteous indignation men who are so beguiled and demoralized by the charms of pleasure of the moment, so blinded by desire, that they cannot foresee the pain and trouble that are bound to ensue.
The wise man therefore always holds in these matters to this principle of selection: he rejects pleasures to secure other greater pleasures, or else he endures pains to avoid worse.

COMMENTARY

Have you ever seen the Lorem Ipsum text? It’s a large collection of altered Latin that starts like this: “Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua…” You may have come across and not even realized it, you see it is commonly used as a placeholder or background text. If you look closely at a graphic with nonsense text written on a newspaper, then quite possibly this is what was used.
But it’s actually an excerpt from a work by Cicero, and one that is particularly profound. I have given an abbreviated form of the translation up above. Though this is not a work of scripture, I am convinced that there is a simple truth to it. There is nothing wrong in desiring pleasure and avoiding pain…yet only if one has the wisdom to recognize that actions of immediate pleasure sometimes are followed by a worse pain, and moments of immediate pain sometimes are followed by a better pleasure.
Thus one is right to avoid the pain of touching the burning stove, but one is also right to endure the pain of healthy exercise to enjoy a better physical condition. And yet, even knowing that eating too much will be bad for us, we still do it anyway. It is common knowledge today that smoking cigarettes is bad for us, but people still smoke. Our conscience warns against telling lies, but still we lie. Tomorrow let’s examine why this sort of illogic is baked into our very nature, and how God calls us to overcome it.