Can Everything that is True Be Proven True?
Everything is either true or not true, but not everything that is true can be proven true, and not everything false can be proven false. Despite this however, we can, in some cases, calculate the probability that something is true or not true, even when we can’t know for sure.
The Technologies of Logic and Reason
The above statements can be shown with logic and reason by looking at concepts like:
- The Principle of bivalence which says, “every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.” In other words, every statement structured properly has a definite binary truth-value (two-value True or False logic).
- The laws of thought, specifically the law of identity, contraction, and excluded middles. These are “the laws” that logic and reason are based on, noted by Plato and Aristotle, which in simple terms also states that “everything” is either true or not true. This leads us to the rule-sets of logic.
- Constructivism (in the philosophy of mathematics) which shows that, even though everything is either true or false, there are some things we don’t know. So we have to consider 3-value logic: True, False, and Unknown.
- Since there are unknowns, from the human perspective there is probability. Thus, we have to also state certainty (using multi-value logic to make statements like likely true, or unlikely true, or 75% likely). This brings us to some important theorems for dealing with truth and probability.
- Gödel’s Incompleteness Theorems, which shows, using mathematical proofs, that there are things that are true that we can’t prove true (Gödel’s theorems are a set of equations that works out the logic behind a paradox like the sentence “this statement is a lie”).
- Bayes’ Theorem, which shows that we can compare conditional probabilities to find the “likelihood” something is true or false (even when we can’t know it for sure). We can use inductive logic to find likelihood.
- The Scientific Method, which shows that we can create a rock solid and usable theory based on a collection of related facts, testing, and a strong hypothesis. We don’t need to know F=ma directly, we just need to know it works every time when subjected to rigorous test. We can also use hypothesis and theory to great effect.
- The Rule-sets of Logic and Reason in general, which include all the above rules and show how we can deal with propositions (as semantic language or as mathematical equations). In general, the rule-sets of logic and reason (like the rules of a syllogism and deduction, and the rules for building a case with induction) can help us to decode truth-values, including certain truths and probabilities.
The above technologies of reasoning can help us understand many truths, and some extra heady logic that expands on concepts like those above can help us show many more. Still, the end result will be “not everything that is true can be proven true, and not everything false can be proven false.” We explain the logic behind this below.
Gödel’s Incompleteness Theorem – Numberphile. A video on Gödel explaining the gap between “truth and proof.” The reason to start with this video is that the other methods, the fundamental laws of truth, Bayes’ laws, the scientific method, and the rule-sets of logic and reason in general all point toward the idea that we can create a solid system for “knowing all their is to know.”… but Gödel’s theorems show us otherwise.
The Three Fundamental Laws of Thought
Before moving on, lets cover the basic logic of epistemology (what we can know).
(1) The Law of Identity—
Whatever is, is;
Every A is A.
(2) The Law of Contradiction—
Nothing can both be and not be;
Nothing can be A and not A.
(3) The Law of Excluded Middle—
Everything must either be or not be;
Everything is either A or not A.
From there one only needs to consider theories that deal with induction and probability, like Gödel’s theorems, Bayes’ theorem, and the Scientific method (which uses a mix of deduction and induction), to infer that there is some information we can know, some we know we can’t know, and some we can know with different degrees of certainty.
That can get complex, but for our purposes, we only need to accept the basics of the logic to show, “Everything is either true or not true (that there is objective truth and certain truth values), but not everything that is true can be proven true (that we can’t know for sure), not everything false can be proven false (again, we can’t know for sure), but we can prove some things are probable (despite what we can’t prove for certain, there is much we can).”
Known Knowns, Known Unknowns, Unknown Unknowns, and Formulating Good Theories
One major point here is that despite all we can know, there are always going to be things we don’t know for sure… and this can and should cast doubt on the world and turn us into skeptics (lovers of wisdom who search for truth, not sophists who are sure we’ve already found it).
Luckily, conversely, despite what we can’t know for sure, there are many things we can know with degrees of certainty (so again, as lovers of wisdom, we should take pride in our ability to know the objective truths we can know).
Thus, despite the uncertainty, we can still (for practical purposes) label many statements true or false based on current accepted knowledge, philosophical lines of reasoning, experiment and mathematical proofs, empirical evidence (experience-based evidence, rather than reason-based logic), and importantly by applying the scientific method.
To rephrase, the main goal then is to hunt for truth using the proper rule-sets, state certainty, make logical arguments, and show proofs… all while remaining skeptical (knowing better theories can come along and that our knowledge is always somewhat limited).
TIP: Here it is vital to understand, all a theory is a probable truth, with many facts pointing at it, that hasn’t been proven wrong. If it works, then it is accepted as true until proven false. The thing that saves us from becoming dogmatic is skepticism!
The Key to Understanding
The key to “approaching knowing and understanding” is asking well-phrased questions, knowing what technology we need to find the answers, and accepting the difference between theory, fact, hypothesis, justified belief, opinion, and conjecture (by understanding the nature of epistemological truth).
The scientific method allows for what is shown true today to be labeled false tomorrow, and math- and logic-based theories back up the idea that this should be expected. Beyond this, truth as a concept is somewhat semantic, with much of what we call truth is actually a form of BS; but such is the nature of language and rhetoric.
TIP: Our ability to confirm and disprove a premise or inference differs depending on what types of truths we are talking about (for example whether we are talking about the formal or material in terms of the empirical, logical, or metaphysical) and how we frame our inquiry.
NOTE: United States Secretary of Defense Donald Rumsfeld gave a speech in 2002 about WMDs, in it he said, “as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don’t know we don’t know.” He was skewered for the riddle of a sentence… but, language aside, his core point is 100% correct (his point about epistemology, not the WMDs). This should remind us Language is complex, truth can seem paradoxical, and politicians aren’t always concerned with “the nature of truth” (they are also concerned with “rhetoric”). But this is another lesson, we can cite Bayes all day, but outside of some small circles “cuz’ (i.e. Bayes’ theorem) alone is not going to win the average argument…. although the same argument structured as a syllogism (which in this case uses conditional reasoning) might:
- Major premise: Most of the data points to A being true.
- Minor premise: If A is true then B is not true.
- Conclusion (or “inference”): Therefore it is likely B is not true based on the data.
TIP: Instead of trying to break out complex and heady logic to win arguments (for example instead of trying to go through each data point plugged into a Bayesian formula and explain the results), it is useful to transpose the results onto a simpler form of reasoning and use generalizations (while having the more heady proofs in your back pocket to cite). For example, I’m not going to go through Gödel’s theorem, the scientific method, or Bayesian theory line-by-line here, I’m going to simply tell you Gödel’s theorem proves to us that some truths can’t be known for sure, while the scientific method and Bayes’ theorem show us we don’t have to know something for certain for what we know to be useful and “true enough.”
Knowing and that Which We Can’t Know for Sure
There are some things that can’t, by their nature, be proven with today’s technology (technology meaning broadly all collective knowledge, tools, and computing power), this includes (but isn’t limited to) that which is metaphysical or ontological philosophy and that which is a current theory. Physics is one such realm where widely accepted theories remain unprovable in every sense (the goal is to find theories that work, improve them, and prove them false through rigorous testing to find theories with predictive power, not to prove absolute true).
Political theory is an example of another realm where we are sure to find ourselves unable to find absolute truth. There may be a “best system” (best model) for any given point in spacetime, all things considered. However, any solution will be at least partially subjective and include “unknowns” as well as room for argument and debate. Even when truth exists, some truths are dependent on other factors and are bound to be so complex that truth might be elusive, impractical, and even require “minimum mutilation” of one’s ideals or truths to find any common acceptable ground.
Although we can find half-truths, fuzzy logic, vagueness, and degrees of truth all around, and sometimes in practice we will have to sacrifice ideals and truths, we can still use reason and logic to prove the law of excluded middles.
Life is a complex matrix of gray areas, but the nature of truth is black and white regardless of our abilities to perceive and prove absolute specifics. Below we cover epistemological and statistical proofs by Bayes, Gödel, and others to better understand truth, but first let’s let Richard Feynman teach us about the Scientific method, because all our heady statistics theories aside, we will inevitably be falling back to on the scientific method to define useful truths and ideas for the world we live in.
Feynman on Scientific Method.
NOTE: A lot of the logic used to define truth is mathematics based, or proved with mathematics (although often using symbols to represent values like A, B, I, O, E, T, F, p, q, etc. more than using numbers), other proofs are purely semantical arguments of reason. With that said, we will just be giving a simple conceptual math-free overview to ensure that anyone can get the gist. Check the links and videos for explanations of the proofs.
TIP: Speaking of mathematics based proofs, Gödel, and logic… 1910’s Principia Mathematica (PM) (a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell) was an early attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven (a logic-and-proof-based Theory of Everything). However, in 1931, Gödel’s incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them. In other words, the only real ToE in logic mathematically speaking is Socrates’ [paraphrasing], “all I know is that I still don’t know for sure”.
How to Prove That There are Statements That Can’t Be Proven True – Gödel’s Proof
In 1931, Kurt Gödel published two of the spookiest and most important results in mathematical logic: his self-named Incompleteness Theorems. In intuitive terms, what these two theorems prove is that “In any logical system, there will be statements that are true but which cannot be proven.” This is the conceptual takeaway, like Bayes theorem, this is proved with somewhat complex mathematics that goes outside the scope of this site. See the math behind Gödel’s Incompleteness Theorem is explained “simply” here.
For non-statisticians, the videos below will work just fine for an introduction to the concept of Gödel’s incompleteness theorem as it applies to truth.
Kurt Godel: The World’s Most Incredible Mind (Part 1 of 3). This video explains Gödel the man, how he is connected to greats like Einstein and Turing, and how his studying of systems, sets, and statistics led to some leaps forward in some interesting areas. This video mentions the different types of infinities (a related and interesting subject).
Godel’s Incompleteness Theorem (Doubting Math).
MYTH: Gödel’s incompleteness theorems disprove the idea that “everything that’s true can be proved”. Ironically enough, there’s a thorough and rigorous logical proof that there are lots of statements that are true, or are likely true, but can’t be proven true. Proving things false is actually much easier than looking for infallible absolute truths.
How to Prove That There are Statements That Can’t Be Proven True, Even Though They Are? – Gödel, Bayes, and the Scientific Method
Gödel’s incompleteness theorems (of which there are more than one) imply that there are theories that are true, even though we can’t prove them, but we can use Bayes’ theorem and the scientific method to find theories that are true enough, based on what we currently know, likely know, or can prove through testing.
A great example of the scientific method coming to our aid is special relativity and general relativity. We can’t prove these things true, but they work over and over again to make predictions, and thus are useful theories that are “true enough”. We didn’t prove them true; we have only proven them not to be false.
Employing Bayes’ theorem, a statistical theory of conditional probabilities, we can be pretty darn sure that a given theory is true (although we can’t prove it absolutely with statistics any more than we can with the scientific method).
Bayes’ Theorem for Everyone 01 – Introduction.
TIP: See our page on Bayesian thinking (which we explain without much math), or see An Intuitive (and Short) Explanation of Bayes’ Theorem from betterexplained.com.
How to Prove That There are Statements That Are Likely True, Even Though They Can’t Be Proven
Again, to prove something is likely true even though it can’t be proven we can employ the same tactics as above. We can’t measure light speed in a vacuum, but we have deduced light’s speed with math and have applied the results with repeated success. This hints we have likely found truth, although we can’t prove it.
In the famous Bayesian example, we can never be sure if a person has cancer without direct testing, but if we have enough data-points that point for or against cancer, we can compare these confidently and assume a diagnosis (with little chance of error).
A visual guide to Bayesian thinking.
How to Prove a Statement is False
If a statement contains any aspect that can be proven true but is proven false, then the whole statement is false. These fallacies are typically half-truths, but not always. I would consider alchemy and aether half-truth theories while I’d consider most rhetoric from politicians a form of semantic half-truths. These are just examples, but just like the scientific method insinuates, it is much easier to prove something false than true. This is why we use falsifiable statements as hypotheses in science. The method is one of deduction and reason.
The Scientific Method.
Complex Truths and Multiple-Valued Logic
Some truths are complex and have more than one value to compare (Many-valued logic). We can consider three values: true, false, and unknown (which still keeps with the law of excluded middles), or we can expand this concept to consider complex situations (like creating a constitution for a new nation who values liberty, justice, and truth).
This sort of logic helps to show us that there are truths we can’t know, and also begins to present us with complexity and contradictions that need to be overcome to find truth.
Some things are only true at a given point in spacetime and are contingent upon many factors. For instance, I can tell you where Jupiter is in the sky at any instant with the right equipment, but it is dependent on my reference frame at a point in spacetime. The truth is there, but it changes and is dependent on view point. We can aggregate viewpoints over space, or over time, but it is complex and on some level has aspects of subjectiveness in practice.
Although there is always the truth, finding “truth” that applies to all people, especially in complex social situations when considering ideals, will require the sacrificing of greater truths, and the accepting of lesser truths, to seek “the truest” outcome. This leaves us with complex factors to consider. Luckily, there are equations for this too. This logic requires a sort of mix of Bayesian thinking and moral reasoning. Thus, it can require philosophical reason as much as a logical mathematical approach, and may require that we break down complex truths down into more simple systems. See Modern Uses of Multiple-Valued Logic.
Many Valued Logics. Everything is either true or not true, but applying this to complex truths and belief systems requires a higher order of critical thinking.
Three or More Value Logic. Sometimes we can’t know an answer; this is why, from a scientific angle we focus on asking the right questions.
TIP: Many-valued logic is a propositional calculus in which there are more than two truth values; it just so happens to be a type of logic that applies well to belief systems. See Gödel’s logics Gk and G∞ for examples of many-valued logic.
Proving a Statement True that is False, and Other Logic-Based Truths Like Contradictions
One of the more interesting things we can do with mathematics and logic is prove something that is false to be true or something that is true to be false, we can also find truth based on a false statement and vice versa, and we can seem to conclude that certain propositions are both true and false. Regardless of how complex things we get, the rule of excluded middles applies, and there is always a specific true or false answer (even if we can’t prove it).
Good examples include paradoxical statements such as: “This sentence is a lie.” This is known as the liar paradox and this is the paradox that Gödel’s First Incompleteness Theorem solves by translating the semantic puzzle into mathematics. The paradox is: if “this sentence is false” is true, then the sentence is false, but if the sentence states that it is false, and it is false, then it must be true, and so on.
This Statement is False.
The video below explains how to solve contradiction paradoxes like the Liar Paradox (a type of paradox that seems both true and false). We know the law of excluded middles tells us that everything is either true or false, so it is our logic (and not the universe) that needs to do more work. The key to solving this is properly wording the question or employing a specific type of propositional logic related to contradictions (demonstrated in the video).
Propositional Logic: Contradictions.
We won’t get into every logic-based proof for truth here, but you can see Three Ways to Prove “If A, then B” for some additional mind-bending logic related to direct proof, contrapositive proof, and proof by contradiction. Or see this Stack Exchange forum for more musings on proving true things false and false things true.
You don’t have to master logic to get the gist. Everything is either true or false, but that doesn’t mean we can prove it, and that doesn’t mean we should stop looking. The world is complex.
How to Prove a Statement is True
Most statements can be proved true or false with logic, but not all of them. This is reinforced by the following video.
Introduction to Logic.
On a deep philosophical level there isn’t much, if anything, we can prove 100% true from the inference we make down to the root (as it would require us to prove that what we observe is 100% true and subsequent reasoning is valid).
In science we aren’t looking to prove truth (we are looking to falsify; to prove something isn’t true). We are, in other words, looking for theories that work 100% of the time (not gospel truth).
This is as close as we get to truth.
When we employ math, logic, and reason, we can find those close enough truths we call mathematical proofs and scientific theories.
Descartes said “I think therefore I am,” but his proof was ontological. Socrates [sort of] said “I know one thing: that I know nothing,” these are some very wise and insightful statements (but they aren’t 100% by every measure).
We can make true statements about what we consider truth, but at the root we have to question rationalization, observation, and thus the inductive, deductive, and other reasoning we use to process the concepts we conceive.
Still, we don’t need to prove every aspect of philosophical certainty to have good and useful knowledge about the worlds of ideas and facts (and this is the point I’m trying to make here; the goal is, for practical proposes, to find truths that work, the ones we can’t prove false on a practical level no matter how hard we try; behind that lies the other truth, that everything is either true or not despite any degree of uncertainty we have).
Logic –The Structure of Reason (Great Ideas of Philosophy).
But For Practical Purposes, All Well Constructed Statements Worth Exploring Scientifically are Falsifiable
Whether it is FactMyth.com fact-checking statements or the scientific method testing a hypothesis, it typically isn’t proper form to ask a question that can’t be proven false.
We wouldn’t research a statement like “there are aliens,” because we know we don’t have the proof to back this up. Instead, we would research an answerable statement like “there are likely aliens.” We could employ the Fermi paradox and Drake equation and give a satisfactory answer.
The same goes for the scientific method; if we start with a hypothesis we can’t prove false, then we set ourselves up for a junk theory.
It isn’t that unanswerable questions are not worth exploring, Gödel ensures us that there is truth that we can’t know (and all the philosophers generally agree), it is that we work best when we use all the technology we have to answer the questions and fact-check statements and questions that can result in useful data. In this, we unlock more keys, and uncover deeper truths, and better questions can be asked.
If you have made it this far, you are in for a treat, below is one of the most mind-blowing metaphysical-mathematical-cosmological videos on YouTube. An introduction to Gödel, Escher, Bach (a Pulitzer Prize-winning book by Douglas Hofstadter that inspired many the philosophy/math nerd).
MIT Godel Escher Bach Lecture 1. This lecture series from MIT expands on the concepts on this page. If you found any of the above interesting, this lecture is a must watch. Check out the link for the full series.