Justified knowledge (truth) requires evidence or proof. One problem that arises in epistemology is called the Regress Argument. The Regress Argument simply states that if any proposition is justified knowledge, it must be based upon another proposition which in turn must itself be justified knowledge or a justified true belief. This regression goes on infinitely with every piece of justified knowledge dependent upon another piece of justified knowledge.
Most traditional systems of logic have dealt with this problem through what is called Foundationalism. In Foundationalism, basic tenets are held to be irrefutably true, so that any proposition based upon those basic foundational beliefs must also be true as long as the system of logic is not violated.
Euclid’s system of geometry is an example of this. Euclidean geometry is a series of logical extrapolations based upon basic postulates that are held to be true by definition. The postulates are foundational. His fifth postulate is commonly restated as saying that two parallel lines never converge. As it turns out, this may not be true in all situations; it fails in some higher mathematical applications. This is and example of the problem with Foundationalism: once a basic postulate or proposition is called into question, then the rest of the derived beliefs are potentially invalidated. Foundationalism assumes that we know certain things to be undeniably true, but this type of absolutism can only be found in constructs and definitions. We define that 1+1=2, and because it is irrefutably true, it is foundational. But stating that two parallel lines never intersect assumes something about the fabric of space-time that might not be true.
Another response to the Regress Argument is called Infinitism. Infinitism states that there is no foundational knowledge and therefore the regression can go on infinitely. This view is often taken to a solipsistic extreme to state that, essentially, no proposition can ever be adequately justified as true since there is an infinite regression of dependency on other propositions. This view has practical implications: a nihilist might believe that justified knowledge cannot therefore exist and therefore action based upon justified knowledge doesn’t exist either. All truth is relative and each person’s truth is equal. Traces of this subjective philosophy can be seen in Existentialism and Nihilism.
Another response is Skepticism. Skepticism embraces the extreme of Infinitism in that it deals with the regress argument by stating that nothing can be justified beyond doubt. This is meant to encourage continued investigation and questioning and nullifies Foundationalism and other dogmatic systems. Everything should be vigorously questioned, and that which is left standing is most likely to be true. These principles are not absolutely true, but are likely to be true.
Coherentism attempts to solve the problem by arguing that evidence is, in fact, not dependent upon other evidence in a linear, chain-like fashion; instead, a belief is thought to be justified if it’s part of a coherent system of beliefs that support each other in a more complicated web of propositions. Unfortunately, this leads to circular argumentation where a proposition can support itself. Many things appear coherent to an observer which are not and the coherency may not break down because the observer isn’t aware of the right question to ask that creates a contradiction. It is definitely true that humans naturally tend to believe things that they find coherent with their other beliefs, but such coherency is an example of the confirmation bias.
Some modern philosophical systems dismiss with the Regress Argument altogether. For example, Critical Philosophy holds that beliefs do not need to be justified at all. This is a philosophical outgrowth of the Scientific Method and Skepticism because Critical Philosophy holds that any proposition should be criticized with rigorous attempts to disprove it and that those beliefs which best withstand criticism are those that require action. Pragmatism also dismisses the Regress Argument by stating that a belief is actionable at whatever level of justification is valid for a given individual. Different individuals may need different levels of justification to act. Unfortunately, this style of pragmatism, as endorsed by William James in The Sentiment of Rationality, creates a very subjective view of truth and therefore presents an ontological dilemma. It does not allow for a formal or objective method for determining which beliefs are justified.
This leads us back to Probabilistic Reasoning as a method for determining truth. Here, we can deal with the Regress Argument mathematically. Different statements of fact have different levels of evidence or justification. A regressive chain of statements that defines the various propositions upon which a statement of truth is based is useful if we can assign probabilities of our confidence in the truthfulness for each proposition. Probability theory then tells us the probability of our current proposition.
Probabilistic Reasoning does not believe in the absolute certainty or uncertainty of any proposition. P is never equal to 0 or 1. But many of the propositions that Foundationalism and other philosophical approaches hold as true principles will be assigned an exceedingly high probability of truth in a probabilistic reasoning system. For example, Foundationalism would state that we know it to be true that most humans are born with two eyes. This certainly seems like one of the truest propositions one could assert and reaches the standard of true belief by nearly any philosophical system of epistemology. But, alas, it might be untrue.
It might be the case that humans all have three eyes but the presence of three eyes is so emotionally incongruent that we all suffer from some trick of the brain that convinces us that we only have two. We cannot confirm that our rather-certain view that most humans have two eyes is an ontological reality. Nevertheless, we and all our ancestors and all our descendants will likely live and function in a world where we accept that most humans have two eyes. Rather than make this proposition a statement of absolute fact (where P=1), Probabilistic Reasoning would merely assign the proposition very high probability. The number doesn’t actually matter that much. I think 99.99999999% is good enough but feel free to add or subtract a few nines.
We should therefore consider it a justified belief (truth) that most humans have two eyes and this will form a strong link in the chain of evidence for other propositions based upon this statement. But this formal system also allows us to consider propositions based on weaker statements of truth where Foundationalism and other strict logical systems of evidence will not.
The level of certainty sufficient to consider our knowledge justified (that is, where P ≥ j) is not dependent upon subjective assumptions or personal uses, as is the case with Pragmatism. Rather, it is based upon a definable cost:benefit analysis. What are the costs of being wrong about a certain belief versus the benefits of being right? This will determine the level of justification needed to accept a statement as justifiable truth. This is the sort of analysis pioneered by Blaise Pascal and considered in Pascal’s Wager. If someone hands you a gun, tells you that is unloaded, and asks you to pull the trigger and he will give you $100, you will need a high level of certainty that the gun is unloaded before you do so. If you hands you a water gun and says the same thing, you will need next to no certainty at all. This may seem like Pragmatism, but it is possible to formally weigh the true risks and benefits in a given situation and isn’t based just on how much you would like ot have a $100 (granted, though, that some people may make the wrong decisions based upon their greed).
The power of Probabilistic Reasoning is that it can utilize all sorts of potential evidence. There are numerous criteria of truth that have been defined and they are all of different value and quality. Majority rule is probably a better criteria of truth than emotions, but it is still of limited value. Rigorous consistency and coherence should be assigned more credit when assessing the value of evidence than emotions, but sometimes a particular problem won’t have rigorous consistency or coherent data available. Rather than ignore the question, Probabilistic Reasoning can still deal with the available evidence but the resulting belief will likely not be as certain and may not rise to the level of justified knowledge. What’s more, the chain of evidence can be updated and expanded upon as new data or new levels of confidence are discovered; this is the idea of Bayesian updating.
There was nothing wrong with Euclid’s fifth postulate that stated that two parallel lines never intersect; the only problem was that he held it as accepted and absolutely true. Had he assigned an evidence-based level of certainty to the proposition, he could have still created his derivative postulates and beliefs but it might not have taken us as long to develop non-euclidean geometry because those derivative beliefs themselves would not have been held with as much certainty. Subsequent generations would have felt more comfortable questioning things. Skepticism and Critical Philosophy would be more than norm. But Foundationalism promotes dogmatism and dogmatism is the enemy of science. The scientific method is designed to question critically even the most deeply-held beliefs. But, remember, the scientific method is a philosophical system that itself makes certain assumptions about epistemology and ontology.
Understanding the connection between philosophy and science is important because science is not above or removed from philosophy. Your philosophical perspectives will affect your science in your implementation of the scientific method. For example, if you philosophically believe in Foundationalism, and you’re taught from your youngest years the truth of Euclid’s fifth postulate, then your mathematical endeavors will be confined to a world where Euclid’s fifth postulate is absolutely correct. You will not be the one who discovers non-euclidean geometry. Or, if you believe in the constancy of the speed of light as a foundational principle that is inerrant and cannot be questioned, then your scientific endeavors will again be limited to a world where the speed of light is always constant. You will find it incongruous and dissonant to consider hypotheses where the speed of light is not constant. Your research into basic cosmology will be biased and limited.
The key point is that philosophical ideas like Foundationalism have real impact on science. Your ideas of epistemology and ontology, even though you might not even realize you have them, profoundly affect everything you do. Science is philosophy; once you truly understand this, then you also understand that science is in the same category as religion. Both can be utilized and abused in the same ways.