Basics of Argument

So I believe these sentences:

"2 + 2 = 4"

"The Norman invasion occurred in 1066."

"California is a state in the United States."

"Smoking causes cancer."

"The Earth is spherical."  

And insofar as I believe them, I think they are true. I can't believe a claim and think it is false. To believe it is to take it to be true. So I do not believe that "Bubonic plague is caused by demon possession," although there have been some people who believed that in the 1300s. Furthermore, I do not believe, "Scorpios make better parents than Libras," "The American moon landing was faked," and "it is possible for unassisted humans to fly if they concentrate hard enough." I think all of those claims are false.  To believe a claim is to think that it is true, but the claim itself can be either true or false.  That is, people can, and often do have false beliefs, even though they don't seem that way to them.  

So, beliefs are subjective, or dependent upon being thought, whereas objective reality is what it is no matter what we think. Beliefs are mind-dependent--they must be in a mind to exist. And to believe a claim is to take it to be true, even though it might not actually be true. Some may still believe it. Belief is subjective because it is dependent upon an individual to possess it. Belief is mind-dependent. There are no beliefs without minds. We are employing belief in a broader sense than it is often used. We will include all claims that a person would agree to, whether they are trivial or momentous, among their set of beliefs. So you believe that February follows January. And you believe that the chair you are sitting on is not made of cheese. And you believe that when the sun is shining, it is day time.

Belief:  to believe a claim is to assent to it or to have an attitude toward it such that you think it is true.  Belief is subjective, mind-dependent, and may or may not be true.    

Truth vs. opinion.  Sometimes people assert the criticism, "Well, that's just your opinion." to undermine a speaker's claim.  We can make some important distinctions.  There are some claims that I make about my own preferences, like, "I don't like tomato soup." that apply to me, and when the claim is about me, it is true.  If the claim is made about or by someone else, it might be false.  Those are subjective preference claims, and it would be difficult to dispute them.  Sometimes, when a critic says that a claim is "just your opinion," they might be confusing an objective claim about the world for having the status of a subjective preference.  If the speaker says, "Donald Trump was the greatest president in U.S. history." they might be making the claim that they like Trump the best, like "Chocolate is my favorite flavor of ice cream."  But they might be asserting that there are properties that make presidents great, and Trump has more of them than anyone else.  This would be an assertion about the world, and it would be subject to evaluation as an objectively true or false claim.  Describing it as a "mere opinion," doesn't really address that claim or its truth. That is, to point out that a claim someone makes is "just their opinion," doesn't tell us anything important about it, and it doesn't give us any reason to think it is a false view.  It's a poor objection that doesn't help us figure out what's true or false.  So, the truth vs. opinion distinction, therefore, isn't useful for presenting and evaluating good arguments.  

We will adopt the Belief Principle (BP): Whenever a person considers any proposition, they must take one and only one of three attitudes toward it.  That person must believe the proposition, or disbelieve the proposition, or suspend judgment about the proposition. A person cannot at any time have more than one of these attitudes toward one proposition. So you either believe, disbelieve, or suspend judgment about the proposition, "The Earth is flat." You might be able to acknowledge why someone else might see it differently, you might even be able to recognized the evidence that would lead them to believe something else. But once we have clarified a claim, and you understand it you will be in a position of either believing it, disbelieving it, or suspending judgment about it. That is, you cannot both believe and disbelieve it at the same time.  To believe it is to think it is true, and to disbelieve it is to think that it is false.  It's not possible to believe both that, "Bachelors are unmarried," and "Bachelors can be married,"  for example.  

We can also see that people believe or disbelieve claims with different degrees of conviction.  You might both believe that "the woman who raised me is my mother," and "The Raiders will win the next Super Bowl," but you believe the first one with more conviction, more assurance, and more confidence than the last one.  You could, however, be wrong about both of them.  People can also change their minds over time such that they believe a proposition at one point and then later they come to disbelieve it.  I believed "Lance Armstrong is not cheating in the Tour de France," at one point, but evidence eventually led me to disbelieve it.  What we believe and how much conviction we believe it with varies over time and circumstance.    

We will adopt this Rational Belief Principle: In general, to be rational, our beliefs should be sensitive to the evidence and our conviction about them should be proportional to the strength of that evidence. So for a rational person, their evidence will reflect the three attitudes that they can take toward a proposition:

Believe: If a person’s evidence concerning a proposition supports that proposition, then it is rational for the person to believe the proposition.

Disbelieve: If the person’s evidence goes against the proposition, then it is rational for the person to disbelieve the proposition.

Suspend Judgment: And if the person’s evidence is neutral, then it is rational for the person to suspend judgment concerning the proposition.

A person's evidence can lead them astray. They can believe things that are false and thus get the wrong conclusion, or they can make mistakes about what is implied by their evidence. We can see that given the evidence they have, what they believe makes sense, even though the truth is different than what they believe.  So it was reasonable for an illiterate 14th century peasant in Europe, for example, to believe that "Bubonic plague is caused by demon possession." even though we know that it is caused by the bacterial infection Yersinia pestis.  They were being reasonable given what they knew and the situation.  This position has a name:     

Fallibilism is the view that it is possible for it to be rational or reasonable for a person to believe a proposition even though it is false.  So under the right circumstances, many false claims could be quite reasonable to believe.  A member of an ancient bronze age tribe, living in the wilderness, might reasonably believe that the earth is flat.  A child raised in a cult with no access to the outside world or other information might come to believe, reasonably, that the cult leader is a divine being.  It is important to reflect, therefore, on what has to happen to a person and with the information that they have for a belief they have to cease being reasonable.  This is a surprising view; note that we have separated being rational from believing the truth to some extent.  If all of someone's evidence points to some conclusion's being true, then the rational attitude for them to take is to believe that conclusion, even if it turns out that it is false.  One reason to accept this view is that the implications of rejecting it are absurd.  Suppose we reject fallibilism and we assert that rational beliefs must be true.  That would mean that some person in history who was thoughtful, critical, and knowledgeable, who knew everything there was to know at the time and was operating within the best standards of evidence and rationality that they had in that era and who ended up believing something that was false was irrational.  Aristotle, one of the greatest minds in history, who knew as much or more than anyone in his time and place would be irrational in his belief that "the planets orbit around the Earth," was irrational.  There simply was no way that Aristotle, operating with the knowledge, the tools, and the resources that were available to him could have come to believe in a heliocentric universe.  And it seems absurd to conclude that Aristotle was irrational.  If Aristotle was not rational, then no ever was or could be rational.  We need to adopt a more useful, and practical notion of rationality.  

Furthermore, we should acknowledge that the better your evidence, the more sure you should be, and the weaker your evidence, the weaker your conviction about a claim should be.  

This view is the Proportional Strength of Belief Principle. It is rational to proportion the strength of one’s belief to the strength of one’s evidence. The stronger one’s evidence for a proposition is, the stronger one’s belief in it should be.

What is truth?  The truth is what is the case or what the actual state of affairs in the world is. The truth is the state of the real, mind-independent world.  We form beliefs about it on the basis of our information. The truth is objective—it does not depend upon people. Truth is mind-independent. Phillip K. Dick said, “A real thing is something that doesn’t go away when you stop believing in it.” Truth is the real world. And there are many beliefs that are not true. The truth remains what it is whether we form beliefs about it or not. When people believed that the earth was flat, in fact, it was not. The truth was that the earth was (and is) spherical. The Earth orbits the Sun. Evil demon possession does not cause bubonic plague. Furthermore, since it is mind-independent, there are truths that are not believed by anyone.  There are truths about the location, speed, and mass of a rock that is floating in space around the Sun that no one will ever confront or learn, for example.  Furthermore, since the truth is the real, mind-independent state of the world, claims like, "the truth is relative," or "there is no objective truth, only belief" are incoherent. Suppose Smith says, "there are no objective truths, there are only subjective beliefs." Paradoxically, his assertion presumes that what he is saying is false. He is asserting that his claim is actually, objectively true; it is objectively true that there are no objective truths. The real state of affairs in the world is that there are no real or true states of affairs. Relativism about truth is self-contradictory and incoherent. 

Propositions about the truth:  A sentence is true if what it expresses matches or corresponds to what is the case in the world. So, "The earth is flat." is false, but "The earth is spherical." is true. And "The earth orbits the Sun," is true, and "The earth is at the center of the universe with the Sun and planets orbiting around it," is false. We might be tempted to say that when someone believes a false claim, these sentences were "true for them." To use "truth" in this way would conflict with the definition of truth above, and it is not intelligible. What these examples make clear is that belief and truth are distinct. When a person S believes some sentence, then S takes it to be true. S assents to that sentence; S believes that it is true. But whether or not the sentence is actually true is an independent matter from whether or not S believes it. There are countless cases where we have believed a sentence (taken it to be true), but we were wrong because it was in fact false--like the flat earth case above. And there have been countless cases where no one believes something, but it is true. There is a cure for cancer out there, we presume, but no one has discovered it yet. Believing does not make a claim true.  And notice that if one person S believes that the earth is flat, and another person R believes that the earth is round, they both take different things to be true. But only one of them is right, and the other one is mistaken. What they both believe cannot be true because what each one believes rules out, or is incompatible with what the other believes.  And what makes a belief actually true or false is the mind-independent state of the world itself.  The truth is out there, outside the head.   

Correspondence Principle (CP): A declarative sentence is true just in case it corresponds to the facts as they actually are. A declarative sentence is false just in case it fails to correspond to the facts as they actually are. So, "The Earth is spherical," is true because there is a planet in reality, the Earth, that is spherically shaped.

One Truth Value Principle.  Every proposition has exactly one truth value. It is either true or false, but not both. So, the proposition, "The Earth is the third planet in orbit around the Sun," is true. It cannot be both true and false at the same time in the same way. "Bubonic plague is caused by demon possession," is false, not true, even though there were people in the 1300s who believed that it was true. The proposition itself has only one truth value. No proposition can have more than one truth value.

Logical Consistency.  When two sentences make assertions about the world that are incompatible, we say they are logically inconsistent.  A set of sentences is consistent if and only if it is possible for all of them to be true (at the same time.) The set of sentences is inconsistent if it is not possible for all of them to be true at the same time.

When a sentence correctly describes the state of the world in some way, that means that the world is that way, not some other way. If it's true that Des Moines is in Iowa, then some other state of things such as Des Moines' being in Connecticut is false. One sentence's being true rules out other descriptions that don't match up with that description. So when we consider what one sentence says, there are some other sentences that are incompatible with it. If I live in Sacramento full time, then it is not true that I live in Paris. But if I live in Sacramento, then it is also true that I live in California. So living in Sacramento is not consistent with living in Paris, but living in Sacramento is consistent with living in California. And living in Sacramento is consistent with "70% of the Earth's surface is covered with water." The latter sentence in this case doesn't describe the world in a way that rules out or is incompatible with McCormick's living in Sacramento.

Additionally, when we consider a pair of sentences for consistency, it could be that neither one of them is true. Consistency and truth are different issues. So "aliens from Mutara all weigh more than 100 kilograms" and "Shreeek, an alien from Mutara, weighs 10 kilograms," are inconsistent, even though they are both fictions.

One way to test for consistency, is to imagine that one of the sentences is true and then ask, "does the state of affairs described by that sentence rule out or prevent the other sentence from being true?"

So these sets of sentences are consistent:

1) Arnold was the governor of California.

2) Arnold is a former movie star.

1) The earth orbits around the Sun.

2) Pluto orbits around the Sun.

1) If the door is locked, then the store is closed.

2) The door is locked.

3) The store is closed.

1) If you win the lottery then you will be rich.

2) Bill Gates is rich.

3) Bill Gates did not win the lottery.

1) Brad Pitt is taller than Vince Vaughn.

2) Vince Vaughn is taller than Tom Cruise.

But these sets of sentences are logically inconsistent:

1) Arnold was the governor of California.

2) Arnold was not the governor of California.

1) The earth is flat.

2) The earth is round.

1) God exists.

2) God does not exist.

1) If the door is locked, then the store is closed.

2) The door is locked.

3) The store is not closed.

1) If you win the lottery, then you will be rich.

2) Smith is not rich.

3) Smith won the lottery.

1) Brad Pitt is taller than Vince Vaughn

2) Vince Vaughn is taller than Brad Pitt

There is another very useful explanation of consistency and contradiction here.

Here are a couple of cases where we will employ concepts in a stricter fashion logically than common linguistic convention.  First, when we encounter a conditional sentence of the form "If P then Q" like "If you are in Reno, then you are in Nevada." we will not take it to imply that "If Q then P."  That is, if you are in Nevada, that doesn't imply that you are in Reno, because there are other non-Reno ways to be in Nevada.  So in general, "If P then Q," does not imply, not is it equivalent to "If Q then P."  Also, when we state that "Some As are Bs," we will not infer that to mean that "Not all As are Bs."  So, "Some CSUS students are female," and "All CSUS students are female," are logically consistent.  They both can be true.  "Some" doesn't imply "Not all."  These two points can deviate from some familiar linguistic practice, but it's useful when we're being logical and careful to tighten up our definitions and take care not to read in extra inferences that are not explicitly stated.  

What is an argument?  An argument is a set of premises or reasons that are given in support of a conclusion.  A Successful Argument for a conclusion (call it C) will be a set of premises or reasons (different than C) that are true and that when taken jointly would imply the conclusion C to a reasonable person who does not already believe C. A prosecuting attorney in a murder trial will attempt to give a successful argument that will convince the jury (who has assumed the defendant's innocence) that the defendant is guilty. So a reasonable person should accept the conclusion of a successful argument. Of course, people rarely hear a convincing argument and then abruptly change their minds. But reasonable people should be prepared to. If you hear an argument with premises that you believe are true, and you understand that the premises validly imply the conclusion, then you are rationally committed to accept the conclusion. Otherwise, you are being irrational. Disagreements about whether or not an argument is successful will be disagreements about whether or not all of the premises are true or whether or not the premises, if true, would imply the truth of the conclusion. This argument may not succeed in convincing you, but here is an example of an argument:

1. If there were no God, then the world would not be so well suited to our existence, our sustenance, and our survival.

2. But the world is well suited to our existence, our sustenance, and our survival.

3. Therefore, there is a God.

Whether or not an argument succeeds in convincing a particular person often depends upon her background information, the other beliefs she has, the extent to which she has critically scrutinized the premises of the argument and her underlying assumptions, and many other factors. So success varies from person to person. Determining when a person is being reasonable or unreasonable in accepting an argument or its premises can be a complicated and technical matter.

We will adopt the Rational Commitment Principle: 

If a reasonable person who does not already believe p:

a. understands and believes that all of the premises in the argument are true, and

b. understands and believes that the premises when taken jointly imply or indicate the truth of p,

then, that person is rationally committed to believing p. To fail to accept p as true in these circumstances would be patently irrational.

So if I believe:

1) If a person is in California, then they are in the United States, and

2) Arnold is in California right now.

then I am rationally committed to believe that:

3) Arnold is in the United States right now.

We will also adopt the Epistemic Culpability Principle:   If a person is epistemically culpable for believing claim, then they can be blamed or faulted rationally for believing or disbelieving some claim. The notion of culpability is borrowed from ethics. When your friend knowingly lies to you, we would fault her. We would say that she has done something wrong. She ought not to have done it. We use prescriptive language--"should" and "ought"--instead of merely descriptive terms about the facts. She is morally culpable for not telling you the truth. Someone is epistemically culpable when she has violated some duty or responsibility to be reasonable, rational, or thoughtful. There is a presumption that people ought to be reasonable or rational, so when they fail and they could have acknowledged the justified, reasonable conclusion, then we find fault in them. If someone has thought about the topic extensively, gathered evidence carefully, and correctly applied logical inferences, then we typically find him to be epistemically inculpable, or without blame, about the resulting belief.

If you found out that someone you know who otherwise seems reasonable and mentally healthy, and who has had a similar education to you, and who seems to know and believe many of the other things you believe, but they assert, and seem to believe, "The earth is flat," you'd probably find them to be epistemicallly culpable.  You'd say that it's irrational for them to believe that, they shouldn't believe it, they should believe that the earth is round, they are making and mistake and they should know better.  They should change this thing they believe in order to be more reasonable or rational.  

What is knowledge?  Classically, a person is thought to know a claim when three conditions are met:  a. they believe the claim, b. the claim is true, and c. they have justification for believing the claim.  If any of these conditions fails, then the person doesn't have knowledge.  They could have a true belief without justification, or a false belief with justification, or a false belief without justification, or justification without believing, and so on.  And none of these would be knowledge.  Only when it is true that the earth is round, I believe it, and I have justification for believing it does it become knowledge that the earth is round.  

Justification:  A person S's justification for a claim p is the set of reasons, the other beliefs, or the matters that S takes to support or imply p. S can have justification for a claim and conclude that it is true, but it may still turn out to be false (Fallibilism.) But when S acquires a justification for p that S takes to be adequate to imply, prove, or indicate p, then S takes p to be true. It can also happen that S thinks that some piece of evidence is sufficient evidence to prove the truth of p, but it does not. That is, p may not be actually justified even though S thinks it is. One's justification can have varying levels of objective success in actually proving p, and this success should be considered separately from whether S takes her justification to be sufficient.  We think that the truth about the world is connected to justification because we think the Principle of Sufficient Reason is true: 

Principle of Sufficient Reason For every truth or thing that exists, there is a sufficient reason or explanation for its being true or existing, rather than otherwise. Truths have an explanation, existing things have a reason for their being. Events do not occur, things do not come to exist, and truths do not come to be ex nihilo or from nothing; there is always an explanation behind them.  Imagine that the Principle of Sufficient Reason were not true and that things came into and went out of existence without cause or reason, that there were no patterns or connections between causes and effects, that there were no explanations behind what exists and what happens.  if this were the world, then we could not understand it, we couldn't form any beliefs about what is true or why.  We could not hope to penetrate or form any accurate ideas about it with out thoughts.  But clearly this is not the case.  We are able to understand aspects of the world, what things exist, and what causes being them about.  It is the presumption in favor of the Principle of Sufficient Reason that keeps us moving forward in science, for example, to understand what is out there and why it happens.  

The Law of Non-Contradiction is the fundamental logical principle. Roughly, it is not possible for a thing to both possess and not possess a property at the same time in the same way. An assertion cannot be both true and not true at the same time. It corresponds to the One Truth Value Principle but it concerns reality, not beliefs.  The idea is that a thing in the world either has a property or it does not.  It is either one way or not that way, but it cannot both have a property and not have it at the same time in the same way.  Logically, this is often expressed as:

~(P + ~P)

Aristotle, the ancient Greek philosopher, said: "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect." And, "an affirmation is a statement affirming something of something, a negation is a statement denying something of something…It is clear that for every affirmation there is an opposite negation, and for every negation there is an opposite affirmation…Let us call an affirmation and a negation which are opposite a contradiction."

The affirmation and the negation cannot both be true of an object. To make an assertion of the form X is P, like "the ball is blue," is to claim that the ball has the property of blue and that it is false that the ball is non-blue. If we abandon the law of non-contradiction, then there is no meaningful difference between an assertion and its opposite. That is, our assertions cease to have meaning altogether. My claim that, "Today is Tuesday," doesn't say anything unless it denies some other state of affairs like, "Today is Wednesday." The law of non-contradiction is axiomatic to reason; that is, it is one of the most fundamental principles upon which reasoning and rationality are based. It cannot be argued for (it is the principle that makes arguments possible) nor can in be plausibly denied (to deny it is to already assume it.)

Logical Possibility is a concept that is built upon the law of non-contradiction. Things are logically possible if they do not imply a logical contradiction.  That is, anything that is not logically contradictory is logically possible.  More formally, any proposition whose opposite does not imply a contradiction is logically possible. Any description of a state of affairs that does not contain and implicit or explicit logical contradiction is logically possible. That is, if the sentence does not include a contradiction like, "Mike is a married bachelor." then the state of affairs that the sentence describes is logically possible. So it is logically possible that Mike is a bachelor. And it is logically possible that he is married. And it is logically possible that Mike (an unaided human) could fly. But it is not logically possible that 2 + 2 = 5, or that circles have sides, or that the Pythagorean Theorem is wrong, etc. Philosophers sometimes talk about logical possibilities in terms of possible worlds; there is a possible world where Arnold Schwarzenegger is the president of the United States. There is a possible world where you have super powers and can run faster than the speed of light. There are no colorless red balls, however, and no presidents who hold no political office, and no four sided triangles.

Logically possible:

Donuts rain from the sky.  

Dinosaurs have jobs in Wall Street finance.  

Superman flies faster than a speeding locomotive.  

Prof. McCormick is 20 feet tall.  

Logically impossible:

A cube with 4 sides.

A mother who is not a parent.  

A straight line that is not the shortest distance between two points. 

A number that is both even and odd.  

A female brother.  

Natural Possibility  We find regularities or patterns in the behavior of matter that we call Laws of Nature.  These are different from the Laws of Logic that define Logical Possibility.  The laws of nature, such as, "The speed of light is 186,282 miles per second in a vacuum," is a law that happens to be true in the physical world that we live in.  But this law is not logically necessary.  The speed of light might have been 186,283 miles per second," or even 50 miles per second.  That is, it would not have implied a logical contradiction had it turned out that this was the case.  

The laws of nature confine the behavior of matter in our world to a subset of the logically possible worlds. The laws of nature such as the universal law of gravitation, F = MA, and e=mc2 determine the range of what states of affairs are naturally possible. So it is not naturally possible for an unaided human body to fly—the musculature, bone structure, and other physiological traits prevent it. But it is naturally possible (we think) to cure cancer. The laws of nature, which are different from the laws of logic, could have been different without logical contradiction. All natural possibilities are a subset of logical possibilities. That is, anything that is naturally possible is also logically possible, but not everything that is logically possible is naturally possible. Being able to move objects with your thoughts alone through telekinesis is ruled out by physics and not naturally possible, but there is no logical contradiction in the scenario.

The periodic table could have been different, gravity could attract at a different rate, or force could be equal to something different than mass times acceleration. If one of those different sets of natural laws were in place, then the range of what events that is naturally possible would be different. The super powers of comic book heroes, for example, such as teleporting, flying, telekinesis, super speed, super strength, and so on would probably be violations of the laws of nature, but they are not logically impossible. Miracles such as walking on water or bringing the dead back to life are violations of the laws of nature, but they are not logically impossible. Roughly speaking, your rule for figuring out whether something is logically impossible vs. naturally impossible should be this: Ask yourself, "Does this sentence describe a scenario that creates an explicit logical contradiction like a square circle or a married bachelor, or does it merely describe a situation that runs contrary to the laws of physics as we find them in our natural world? If it is the former, then it is a logical contradiction, if it is the latter, then it is a natural impossibility.

Logically possible, but naturally impossible:

An unassisted human body running a 5 minute mile underwater.  

An elephant that can do a triple backflip.  

A bottle of wine that never runs out.  

A human body walking on top of water.

Raising a dead person from the dead with prayer.  

Anything that is naturally possible is also logicallly possible.  (But not everything that is logically possible is naturally possible, see above.)

Anything that is logically impossible is also naturally impossible.  

Possible vs. Probable

We should contrast events that are naturally impossible, or ruled out by natural law, with events that are statistically improbable. If we dropped a million dice onto a parking lot and all of them came up with a 1 on top, it would be statistically improbable, but it is not ruled out by the laws of physics. The odds of winning the lottery are millions to one, and if your friend said he had the winning ticket, you might say, "That's impossible!" But it violates no logical or natural laws. It's merely statistically improbable. We might also think about it this way: the laws of natural, such as the ones governing the periodic table of elements, constrain the regular behavior of matter. They also make some events improbable. Snow is Sacramento is rare because of the typical weather conditions in the winter. But the natural conditions can happen. On Feb. 5, 1976, two inches of snow accumulated in Sacramento. Such an event is clearly logically possible, and it is naturally possible. But it naturally improbable. Having it snow in Sacramento on 10 consecutive days this winter is exceedingly unlikely. We might even say that it is impossible because it is so improbable. But on the definitions we are using of natural and logically impossibility, this would be a misnomer. Every year, approximately 1 person in 500 million is eaten by sharks. So the odds of your being eaten by sharks, all other things being equal, are about 99.9999998%. It is not logically impossible; clearly there is no logical contradiction in the scenario the way there is with a four sided triangle. But it is naturally possible. There is nothing about the laws of nature that prevent such events from occurring. But is it likely? No. Is it probable? No. Is it reasonable to believe?

In probability theory, odds are depicted on a range from 0 to 1, with 1 being certain. The threshold where the odds begin to favor belief is .5. As evidence accumulates in favor of a claim and the odds increase towards 1, the claim becomes more and more reasonable to believe. The conviction that you have about the truth of a claim should be proportional to the quality and quantity of the evidence you have concerning it.

The mere logical possibility that a claim is true should not be enough to elevate it across the .5 threshold and make it reasonable. That is, possible does not imply probable. While is is possible that you will be eaten by sharks today, as we saw above, it is not probable. It is so unlikely, you should believe with a great deal of conviction that you will not be eaten by sharks. Here are more examples illustrating the point that possible does not equal probable:

It is possible that the Holocaust didn’t happen.

It is possible that wearing a raw steak hat wards off disease.

It is possible that even though you are taking birth control pills exactly as prescribed everyday you are pregnant.

It is possible that the government is watching everything you do and hiding it very well.

It is possible that Christopher Marlowe wrote all of Shakespeare’s plays.

It is possible that having sex with a virgin cures HIV.

It is possible that eating the flesh of your enemies gives you power.

It is possible that birth defects are caused by wickedness from a past life.

It is possible that the Detroit Lions could win the Super bowl.

It is possible that fever is caused by demon possession.

It is possible that the earth rests on the back of a (invisible) turtle.

It is possible that lightening is thrown by an angry Zeus.

It is possible that the moon is made of green cheese.

It is possible that the moon landing in 1969 was faked on a secret Hollywood set by NASA.

It is possible that wishful thinking can help you win the lottery

It is possible that wearing your lucky underwear will help you win the basketball game

It is possible that Santa exists.

It is possible that there are still dinosaurs.

It is possible that if you concentrate you can levitate.

It is possible that tossing spilled salt over your shoulder improves luck.

It is possible that opening an umbrella indoors or breaking a mirror is bad luck.

It is possible that conceiving in the spring produces boy babies.

It is possible that swinging a wedding ring on a string in front of a pregnant woman's stomach will reveal the sex of the baby.

Some of these will seem outrageous to you, and you won't be tempted to believe them, or even argue that we should be agnostic about them. But some of these might strike you as being more plausible than others. Some of these might seem to have more evidence in their favor than others. And it might strike you that there have been or there are people who believe some of these things, and believe them with great conviction. And we are reluctant to conclude that some of those people are irrational or wrong.

The answer is that under some circumstances, some of these claims could be reasonable (but not true) for some people. If you have limited access to information, or the best sources that are available to you misrepresent the truth, and you have tried your best to gather and evaluate the evidence, then you might not be at epistemic fault for believing what you believe. Ptolemy, the ancient Greek astronomer, believed that the Sun orbited the Earth on the basis of the best information available in his day. He was justified, but his conclusion was mistaken. But most of these examples above will strike you as merely logically possible, but not even close to being reasonable, probable, or justified to believe.

A necessary truth is one whose opposite implies a contradiction. It is a proposition that must be true without exception. Sometimes, it's a proposition that is true in virtue of the meanings of the concepts involved. A = A is a necessary truth, as is "Bachelors are unmarried," and "Triangles have three sides." Given what the terms mean, the sentence cannot fail to be true, whereas some sentences, like contingent truths (see below) can either be true or false depending on what events unfolded.

Some necessary truths:

17 is a prime number.  

If McCormick is a bachelor, then he is unmarried. 

The whole is greater than the parts.  

If A is equal to B, and B is equal to C, then A is equal to C.  

A contingent truth is one that can be true or it can be false without logical contradiction. "The Washington monument is 555 feet tall," is true, but it could have been false if the stone mason carving the pinnacle block had opted to make it a foot taller. Contingent truths are assertions whose opposite would not have violated the law of non-contradiction. "The Washington Monument is not 555 tall," is not contradictory. "This triangle does not have three sides," is contradictory. 

Some contingent truths:

McCormick is 6' 1" tall.  

It rained in Sacramento on Jan. 14, 2023.  

There are 50 states in the United States.  

Carson City is the capital of Nevada.  

These all happen to be true, but history could have unfolded otherwise or they could have been false without any logical contradiction.

A priori truths are truths that can be known prior to or without experience. They are true by definition or in virtue of the meanings of the words involved. They are truths that we can know conceptually, just be analyzing the terms involved.  "A square is a four sided figure," is an a priori truth. We do not need to count the sides of objects that are squares in the world to know that it is correct. "Mammals have warm blood," is another a priori truth. A priori truths are truths that can be known without an appeal to experience. They are true by definition or in virtue of the meanings of the words involved. "A square is a four sided figure," is an a priori truth. We do not need to count the sides of objects that are squares in the world to know that it is correct. 

More examples:  

A bachelor is an unmarried, adult, human male.  

2 + 2 = 4

A line is the shortest distance between two points in a plane.  

A posteriori truths are truths that we discover and know on the basis of experience. "McCormick is 6' 1" tall," is a fact that can only be discovered by experience. It cannot be known by conceptual analysis the way "bachelors are unmarried" can. "Interest rates were at a 40 year low in July" is another example of an a posteriori truth.  "Obama was the 44th President," and "The high temperature in Sacramento on Jan. 1, 2021 was 58 degrees Fahrenheit," are other examples.