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. We can, however, in some cases, calculate the probability that something is true or not true, even when we can’t know for sure.
The above statements can be shown with logic and reason by looking at the law of thought, and specifically the law of excluded middles. This is “the law” that reasoning is based on, noted by Plato and Aristotle, which in simple terms states everything is true or not true. We can look at Gödel’s Incompleteness Theorems, which show there are things that are true that we can’t prove true (an equation that works out the logic behind a paradox like the sentence “this statement is a lie”), and by looking at 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). This can help us understand most truths, and some more heady logic that extends this thinking (see below) can help solve the rest.
Even though there are things we can’t know for sure, there are many things we can, and we can still (for practical purposes) label many statements true or false based on current accepted knowledge, experiment and mathematical proofs, through empirical evidence (experience based, rather than reason based logic), and importantly by applying the scientific method. This can give us useful information that can be considered “true enough” until a better theory comes along, or until any part of the theory is proved false. The key is asking well-phrased questions, knowing what technology we need to find the answers, and accepting the difference between theory and fact.
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, such is the nature of language and rhetoric.
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, 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, Principia Mathematica (PM) was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. 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.
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, 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. In science we aren’t looking to prove the truth, we are looking for theories that work 100% of the time. 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. One could argue that we don’t know that much for sure. Socrates said “I know one thing: that I know nothing,” these statements are some of the truer statements ever spoken.
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.