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.