One thing can be proven with near certainty: do not stay up late the night before you must write an article on a subtle philosophical topic. I cannot prove that such a course will always be deleterious, but I can say with certainty that it sometimes is.
The truth of that sentence (where I moved from near certainty to certainty itself), like all sentences, follows a chain of argument. The conclusion is that a dissipated night will sometimes lead to an inability to write coherently. That conclusion is certain, but only with respect to the evidence that a particular observation proved that such a night resulted in at least one morning in which a blog author’s writing was strained.
All statements of knowledge, hence of probability, are conditional. That is, no statement is true without providing premises which support that statement. No statement (or event) has a probability without conditioning that statement on some evidence.
Thus, the truth of “Socrates is mortal” depends on what evidence supports or does not support that conclusion. If my evidence is that “All men are mortal and Socrates is a man,” then the statement is true. But if my evidence is “Some men are mortal and Socrates is a man,” the statements is no longer true but only probable.
It does not matter whether the evidence itself is true—and it can only be true with respect to other premises or evidence—it only matters that we accept or assume the evidence is true. This makes the fun with counterfactuals possible.
A counterfactual is a statement which is known (or assumed) to be false conditional on certain evidence. The counterfactual statement itself is assumed true as evidence in another chain of argument. For example, I might argue that “If Hitler did not invade Russia, Germany would have won World War II.” The counterfactual statement “Hitler did not invade Russia” is known to be false conditional on the premise that “Hitler invaded Russia.”
That last statement itself is true conditional on other evidence: books, memories, etc. And we accept the evidence (we assume this evidence true) of those books and memories based on yet other premises or evidence, which in turn are true based on earlier premises, leading in a chain all the way back to our most fundamental beliefs which are true conditional only on our intuitions (see the Day 1 post for more on this).
Thus, if I ask you what the probability a die will land a six, you cannot answer unless you first assign or assume true certain evidence. This is usually “I have a six-sided object, just one side of which is labeled six, which will be tossed, only one side showing when tossed.” Given that, the conclusion the “object will land with a six showing” is still not true, but only probably true, a probability we can quantify with the number 1/6.
You will often see probability treated as if it were unconditional, as if certain events had a fixed numerical value without respect to any evidence. This cannot be. For example, if I change a portion of my evidence to “just two sides of which are labeled six” the probability the conclusion in true changes to 2/6. Change the evidence and you change the probability. Just as when we changed the evidence of the mortality of men the truth of Socrates was mortal changed.
Nothing could be more obvious in logic—in logic we never ask whether an isolated statement is true or false without explicitly detailing or premises—but conditionality is often surprising in probably, perhaps because it is often thought that probability is a separate subject from logic. It is not. Probable statements are no different than logical statements, except that the former have quantifications (of being true) different than 1 or 0.
In advanced books on probability there will usually come a section called “Conditional Probability” which implies that all the probabilities that came before are “unconditional.” Again, this is not so, but there are certain quirky cases that require sophisticated mathematical techniques that are called, in a technical sense, “conditional probability.” These quirks are never met in real life, just in the arcane mathematical world of uncountable infinities.
But even some basic books treat probability as unconditional to some extent. As another proof that this cannot be, consider the probability of this sentence (or others like it), “A certain device takes a given number of states.” What is the probability that “This device takes state UUON”? You cannot give an explicit quantitative answer without first conditioning on more premises which relate to the states the device takes.
This is why it is better to write probability like this:
Pr( S | E )
which reads, “The Probability that S is true given the evidence (or premises) E” and where E might not be true with respect to other premises, but is assumed true for this calculation.
The bar, “|” just means given; it is often left out in elementary texts, but doing so leads to confusion and error.