Time dilation and length contraction are horns on the same goat

I've been learning about special relativity and the Lorenz transformations lately (and I recommend this playlist), and I had an epiphany. The epiphany is that length contraction isn't really length contraction. Rather, it comes from the fact that different parts of a moving thing are in different times because of the relativity of simultaneity. Lemme explain.

Let's say we consider our own inertial frame of reference. For those not familiar with special relativity, an inertial reference frame is a reference frame that isn't experiecing any acceleration (i.e. change in speed or direction). If you were in a box that was moving at some fixed speed in one direction relative to, say, the earth, you wouldn't be able to tell that you were moving. If you had a small window to look out of, it might look like everything else was moving.

Special relativity says that the laws of nature and the speed of light are the same in all inertial reference frames. That means if you were standing on the earth, and you turned on a laser, the speed of light would be about 300k m/s. And if you were in a box moving at 30k miles per hour, and you turned on a laser, the speed of light would still be 300k m/s relative to you.

This is a little trippy because it's not like most other things. If you were on a bus, and you threw a ball 50 mph in the same direction the bus was moving, then the ball would move 50 mph relative to you. But if you were standing on the side of the road watching somebody on the bus throw the same ball at the same speed, the total speed of the ball relative to you would be the speed of the bus plus the speed at which the person on the bus threw the ball.

Not so with light. With light, whether you are on the bus shining a light forward, or you're on the ground watching somebody on a bus shine a light forward, you will measure the speed of the light as being about 300k m/s. It's this fact that gives rise to all the weirdness of special relativity, including time dilation, length contraction, and the relativity of simultaneity.

To illustrate this, imagine a light source sitting in the exact center of a box. When the light comes on, it hits both ends of the box at the same time.

But now imagine the box is moving relative to you. You're just standing there watching it go by, and the same thing happens. A light in the exact center of the box comes on. Since the light bulb is in the box, it's moving with the box, but that doesn't matter. Light will still move at the speed of light regardless of the movement of the source. So it will move just as fast to the left as it moves to the right. But since the box is moving, the light will hit each end of the box at a different time. Since the left side of the box is moving toward the source of the light, and the right side of the box is moving away from the source of the light, the light will hit the left side before it hits the right side.

This would be easier to show with an animation than with still pictures. Use your imagination.

Now, think about that for a minute. If you were in the box moving with the box, the light would hit both ends of the box at the same time. But if you're not moving with the box, and the box is moving to the right from your frame of reference, the light will hit the left side before it hits the right side. How is that possible?

Well, that's the relativity of simultaneity. Two events, like a light hitting a wall, can be simultaneous from one person's frame of reference, but they can happen at different times from another person's frame of reference.

Now, here's my epiphany with length contraction. Don't take this as gospel because I've watched a lot of YouTubers explain length contraction, and none of them have explained it like this. So maybe I've got it all wrong.

Anywho, let's say you are in the box, and the light hits each end of the box at 10 am. Let's say you actually place a clock at each end of the box, and they both read 10 am when the light hits each end.

But now imagine you're standing on the ground watching the box go by. From that point of view, the clock on the box in the back should read 10 am when the light hits the wall on the left.

We know from what we talked about earlier that the light hits the wall on the left before it hits the wall on the right from your point of view on the ground watching the box fly by. If the moment the light hits the wall on the left is 10 am, and it hasn't hit the wall on the right yet, that must mean the clock on the wall that's on the right is reading a time earlier than 10 am. Since the light hasn't hit the right end of the box yet, that means the right end of the box exists at an earlier time than the left end of the box.

Think about that for a minute. If the box is moving to the right, then as time goes by, it will have moved farther and farther to the right. So at an earlier time, it has moved less to the right than at a later time. If it's not 10 am on the right side of the box yet, then the right side of the box hasn't moved as far as it will have moved once it is 10 am on the right side of the box. If the left side of the box has moved to where it's supposed to be at 10 am, but the right side hasn't, then we should expect the right side of the box not to have moved as far as the left side of the box has moved. The result is that the box should appear to be shorter. The reason is because the person on the box measures the size of the box at 10 am on both ends, but the person on the ground measures the box at 10 am on the left side, but at some time earlier than 10 am on the right side.

The relativity of simultaneity is what explains length contraction. The length isn't actually different. It's the time that's different. Length contraction comes from the fact that the back end of the box has moved more than the front end of the box from the point of view of the person on the ground because the person on the ground is looking at both ends of the box at different times.

Does that make sense?

Notice that this weirdness is amplified the faster the box is moving. The faster the box moves, the bigger difference there is between when the light hits the back end and the front end of the box. The bigger that difference is, the earlier the front end of the box will be when the light hits the back end of the box, which means it will have moved less, and the box will seem shorter from the point of view of the person on the ground.

They should call it the relativity of length rather than length contraction. The person in the box measures each end at 10 am, but the person on the ground measures the left end at 10 am and the right end at some time earlier than 10 am. Of course 10 am and less than 10 am are actually happening at the same time for the person on the ground. I'm just talking about the person on the ground looking at the moving frame of reference. 10 am on the left side of the box happens before 10 am on the right side of the box. Let me know if I'm not explaining that clearly. I don't want anybody to get the wrong idea.

If you prefer, you can run the same thought experiment for when the light hits the right end of the box. The clock at the right end will read 10 am. But at the same time (from the ground point of view), the light has already hit the back end. From the grounded person's point of view, the time at which the light hit the left end of the box is in the past, and the clock at the back reads something later than 10 am. So the left end has moved farther than the right end, resulting in the box being shorter.

I don't know if this is right or not, but it makes sense to me. What do you think?

Being happy, more or less

Sometimes, you don't need anything more to make you happy. You need less.

What is philosophy, and how are science and philosophy related or distinguished?

If you take an introduction to philosophy class in college, the first issue they address is usually "What is philosophy?" This is an important question if you're interested in learning about philosophy, but it's also an important question if you're interested in other fields of study, like science, history, literature, etc. I've heard a lot of people argue about whether, for example, the multiverse is science or philosophy. When these arguments happen, it always comes down to what is meant by "science" and "philosophy."

The first thing I remember my intro to philosophy teacher saying was that philosophy was the love of wisdom. This defintion was based on the etymology of the word. "Philo" comes from the Greek word for love or friendliness toward. "Sophia" comes from the Greek word for wisdom. As a side note, I called my blog and my youtube channel PhiloChristos because I love Christ and I am a friend of Christ. I'm a Christian. Anywho, the etymology of a word is not the best way to determine its meaning because the meaning of words changes over time. The meaning of a word depends on how that word is used, and people change how they use words over time.

Let me first say what I wish was the meaning of philosophy. I wish philosophy was the use of reason to arrive at truths. The reason I don't think that's the best definition of philosophy, even though I wish that were the meaning, is because there's a whole field of philosophy and several schools of thought that deny even the existence of objective truth. They also deny the usual rules of logical inference. Philosophy, according to common use, appears to be broader than I would like it to be. Post modernism, continental philosophy, and much of eastern philosophy would not agree with my preferred definition of philosophy since they often reject reason, logic, and truth, but still publish in academic philosophical journals.

Alvin Plantinga gave a definition in one of his books (God, Freedom, and Evil if memory serves me right) that was pretty broad but mostly accurate. He said philosophy was just thinking really hard about stuff. That captures both my preferred definition as well as whatever post-moderninsts and eastern philosophers seem to think philosophy is. I'm afraid his definition might be too broad, though. I've spent a lot of time thinking really hard about conversations I had or decisions I made and what I wish I had said or done instead, but I wouldn't consider that philosophy.

Yesterday, somebody on YouTube commented that all philosophy is speculation. That comment is what gave rise to this post. I don't think that's true at all. When it comes to epistemology, which is undoubtedly the domaine of philosophy, there is a distinction between what we can know with certainty and what we can know with less than certainty. There are at least some things we can know with certainty. I know I exist, I know that 2+2=4, and I know that if two statements explicitly contradict each other, they can't both be true. I know these things with certainty. When I think about these things and what justifies my knowledge, I am engaged in philosophy. The question of whether the law of non-contradiction is true is a philosophical question with an answer we can be absolutely certain about. So philosophy can give us certainty, at least in some cases, which means not all philosophy is mere speculation.

In fact, it seems to me that philosophy is the only field of inquiry that can give us certainty. Science can't give us absolute certainty because it depends on observations, and it's at least possible that none of our perceptions correspond to an external world. But whether the external world exists or not, I am still certain of my own existence and of the law of non-contradiction. The person who made that youtube comment was so wrong, he was almost right again.

So what is philosophy? That's a hard question to answer. Even science used to fall under the broader category of philosophy. It was called "natural philosophy." We now consider science to be distinct from philosophy, which means our definition of philosophy has changed. I think it used to be the broadest way of referring to any field of inquiry where you're just trying to figure out what reality is like, which is why science was considered a branch of philosophy. Philosophers still use the findings of science, though. William Lane Craig often says that science gives us premises that are used in philosophical arguments with theological implications. Philosophers of time point to relativity to argue for what they think time is. Philosophers are happy to admit that they use science in their arguments.

Science uses philosophy, but for some reason, scientists are very reluctant to admit that they are using philosophy. However, science uses all the tools of reasoning that philosophers use. Science is a set of methods for discovering truths about the physical world. Where does science get the methods from? They can't get the methods from science or that would be circular reasoning. The methods come from philosophy. All the tools of reasoning that science uses come from philosophy. Science couldn't even get off the ground if not for philosophy. Without philosophy, one could not reason inductively or deductively, and science does both. Without philosophy, one could not draw any conclusions about the physical world merely by making observations, and that is the primary function of science.

Without philosophy, one could not design an experiment since it is philosophy that gives you the syllogism necessary to make an experiment relevant. Usually, experiments are designed in such a way as to test a hypothesis. A hypothesis is an educated guess about what the world is like. A hypothesis is testable if it allows you to make a prediction. Here is how the reasoning would work if you were designing an experiment to test a hypothesis:

1. If such and such hypothesis is true, then we should expect such and such to happen under such and such circumstances. <--This is the prediction.
2. Such and such did not happen under such and such circumstances. <--This is the result of the experiment.
3. Therefore, such and such theory is not true. <--We have falsified the hypothesis.

This reasoning, which science uses, is called modus tollens in philosophy. How do we know the reasoning is valid? Well, it isn't science that tells us that. It's the discipline of philosophy that tells us that. So philosophy underlies science, and it would be impossible to do science without philosophy.

And that's just one example. Many more could be given. I gave a deductive example, but here's an inductive example. Science also uses repeatability. The more repeatable an experiment is, the more sure we can be of the conclusion. Induction is when you extrapolate from specific examples to general conclusions. For example, if every time you heat water to 212ºF, it begins to boil, then you can draw the conclusion that 212ºF is the boiling point of water. That means if you heat water to 212ºF in the future, you can expect the same thing to happen. It will even happen if you're not watching. Extrapolations like this allow you to form generalizations which you can then use to predict what will happen in the future under similar circumstances. This principle has been stated in different ways:

• The future will resemble the past.
• Nature behaves the same way when we are not looking as it does when we are looking.
• What happens in the lab can tell us what happens in nature.

David Hume famously pointed out that the principle of induction itself cannot be demonstrated to be true since any experiment you could conjure up in an effort to prove the principle of induction would have to assume the truth of the principle in order to be valid. You couldn't say, "We know the principle of induction is true because every time we've used it in the past, it has yielded reliable results" because that would be circular reasoning. Since you can't use science to justify the principle of induction, where do we get the principle? We get it from philosophy. Philosophy, then, underlies science, and one cannot do science without certain philosophical presuppositions, like induction, logic, and the reliability of our sensory perceptions which are what enable us to make observations.

What is philosophy, then? I don't know for sure, but I usually recognize it when I see it. I do think that, strictly speaking, whenever you are doing science, you are doing philosophy, but at the same time, I understand what people mean when they distinguish science and philosophy. Whenever they are distinguished, science is always what we are able to conclude through making observations, taking measurements, etc. Philosophy, by contrast, is just whatever we are able to conclude through thinking and reasoning without necessarily having to go out into the world and make observations. I think that's what people mean when they make the distinction, but if you really wanted to be careful, you'd have to recognize that the distinction is not tidy. Science depends on philosophy, and philosophers often use the findings of science as a basis for what they are thinking about.

It may be that "philosophy" is just one of those words that doesn't have a precise definition. Since I've left the definition of philosophy kind of open-ended without nailing down a definition, how would you define philosophy, and how would you distinguish it from science? Or would you?

My new favorite science YouTuber - FloatHeadPhysics

I've been a big fan of science Youtubers for several years now. I even did a couple of blog posts on them in the past. PBS Spacetime was my favourite for a long time until they started running out of ideas. I discovered FloatHeadPhysics a few months ago and just decided today that Mahesh Shenoy is my new favourite science YouTuber.

I highly recommend any videos he has on special or general relativity. Mahesh is a superb communicator. He doesn't just explain the different phenomena; he strives to explain them in a way that makes them intuitive so they are not only easy to understand but they make sense as well. Before I watched his videos, there were some things I understood (or believed) intellectually but didn't understand (or know) in my heart. For example, gravity is sometimes explained by saying the earth is accelerating upward. I believed that because it's what the experts said, but it made no sense to me since the earth wasn't expanding. But I saw a video the other day where he explained it in a way that was intuitive and made sense. He made the light come on for me.

One of the first videos of his I saw was on the Twin Paradox. I had seen a ton of videos on this subject, and it seemed like multiple people gave different explanations. Most people resolved the paradox by pointing to the fact that the twin who left and came back had to experience acceleration when they changed direction, but Don Lincoln at FermiLab made two videos I found convincing arguing that acceleration had nothing to do with the solution. Then Sabine Hossenfelder (I think--my memory isn't perfect) made a video insisting that the solution involved acceleration, and Don Lincoln responded by saying, "We're basically saying the same thing," when they clearly were not. Well, I watched FloatHeadPhysic's video on the subject, and so far it was the best video of all the videos I've seen on the Twin Paradox.

Another science YouTuber I discovered since my last post on this subject that I wanted to mention because it's worth drawing attention to is Space Mog with Maggie Liu. She also has some excellent content and explains things really well.

Oh, and there's also GeoGirl with Rachel Phillips. I don't remember if I mentioned her or not. I got really interested in geology in the 10th grade. I've been insterested in astronomy for as long as I can remember, and my high school offered an astronomy class that was one semester. To take astronomy, you had to also sign up for geology for the other semester. I wasn't that interested in geology until I took the class. It was awesomer than I expected. It's been a long time since I've really gotten into the topic, but Rachel's channel has re-awakened my interest. Her videos are outstanding because she goes into a lot of detail and uses powerpoint.

EDIT 11/26/2024: Somebody in the comments didn't agree with me when I said Don Lincoln and Sabina Hossenfelder were not saying the same thing about the resolution to the Twin Paradox, so here are the videos I was talking about so you can judge for yourself.

Don Lincoln/Fermilab's first video on the Twin Paradox

Don Lincoln/Fermilab's second video on the Twin Paradox

Sabina Hossenfelder's video on the Twin Paradox

Don Lincoln/Fermilab's response to Sabina Hossenfelder

Mahesh Shenoy/FloatHeadPhysics' video on the Twin Paradox

Mahesh Shenoy/FloatHeadPhysics' second video on the Twin Paradox (I just saw this one today. Mahesh agrees with Don that although acceleration is necessary to cause a change of reference, the solution to the paradox doesn't lie in the acceleration per se. According to Mahesh, the solution lies in the relativity of simultaneity. His explanation makes a lot of sense to me.)

It looks like I was a little skeptical when I saw Don's first video. I left a comment saying, "But from the point of view of the traveler, if they can always consider their own frame of reference stationary, then it isn't them who has existed in two reference frames. It's the person who supposedly didn't travel who existed in two reference frames."

I left another comment on his second video saying, "I've watched both of your videos, and I still don't get it. If Jim is station on earth, and Bob goes out, then comes back in again, then from Jim's frame of reference, he's in one frame of reference, and Bob is in two--outbound and inbound. But from Jim's frame of reference, wouldn't he be in one frame of reference and Jim in two since Jim moves away from him, then toward him?"

I left a comment on Don Lincoln's third video saying, "Maybe I'm stupid, but I don't agree with this at all. I think you and Sabine ARE saying something different. Sabine says that acceleration is what solve the paradox, not because acceleration is necessary to create two reference frames, but because of the acceleration itself. You deny it's the acceleration itself and say it's merely because of the two reference frames."

As a bonus, here's another video I saw seven months ago directly responding to Don Lincoln and explaining where he thinks Don went wrong. I left a comment on this video saying, "I remember trying to figure out the twin paradox, running into Don Lincoln's video after watching several others, and thinking, 'Finally! I understand!' But I guess I didn't understand. Of course not being a physicist myself, I'm still not sure. If the experts disagree, how can I adjudicate between them?"

For now, I think I'm going to go with Mahesh' explanation because his makes the most sense to me.

This is such an interesting topic to me! The whole reason I delved into the Twin Paradox several years ago was because of NaNoWriMo. NaNoWriMo is this movement where people commit to writing a 50k word novel within the month of November. I thought it would be fun to do, and I had a story cooking in my head for a long time. It was a time travel story. In the first chapter, I talked about how the main character came up with the idea for a time machine. I got bogged down talking about special relativity when I realized there was some stuff I didn't understand. So I went on YouTube and started watching Twin Paradox videos. I never did complete my novel because, as you can see, I hadn't found a solution to the Twin Paradox I was confident about enough to include it in my novel.

Does the Big Bang prove that the universe had a beginning?

TLDR: No, depending on what you mean by "prove."

Asking somebody to prove something is a way of raising the burden of proof to an unattainable standard. The idea behind our request is that we want the person to demonstrate certainty about their conclusion. That way if there's just an inkling of doubt, we can dismiss their case. In the real world, epistemological confidence (or strength of belief) comes in degrees. There are very few things we can be certain about, but certainty isn't necessary to function in life. We go on less than certainty about most things. Requiring certainty, then, is unreasonable in most cases.

For the purpose of this blog post, let's take it for granted that the big bang happened. I know there's a big stir on the internet about the James Webb Space Telescope calling the big bang into question. I don't think there's any merit to those doubts, but that's not what this post is about, so I won't go into it.

The question for today's post is whether the big bang, if it happened, proves that the universe had a beginning. If we're using "proof" as in establishes with certainty, then my answer is no. If the big bang were proof in that sense, it would mean that it would be impossible for the universe not to have a beginning as long as the big band theory is true. As long as it is possible for the big bang to be true without the universe having a beginning, then the big bang doesn't prove the universe had a beginning.

There are lots of models cooked up by cosmologists that are consistent with the big bang and that do not have a beginning. Most of these models are speculative and untestable. Most of them are probably wrong. But as long as they are possible, they show that it is possible for there to be a big bang without an absolute beginning. And that means the big bang does not show with certainty that the universe had a beginning.

But the fact that the beginning of the universe doesn't follow necessarily from the big bang doesn't mean the big bang isn't evidence for a beginning. Here, I'm using "evidence" to mean any artifact, data, or information that raises the probability of some conclusion. Since the beginning of the universe is more probable given the big bang than it would be without it, the big bang is evidence for the beginning of the universe.

Consider two scenarios - one in which the universe appears to be static and the other in which the universe appears to be expanding. Of these two scenarios, the expanding scenario at least makes it look like the universe had an origin as opposed to the static one which gives no indication. The origin would've been at or near the point in the past at which everything converged to a singularity. As you go back into the past, everything gets closer and closer together. There's a limit to how close things can get. Once they are all located at the same point, they cannot get any closer. If we look at the expanding universe and extrapolate back in time, it points to an absolute origin since everything is headed in the direction of infinite density.

Admittedly, there are horizons beyond which we can't look. One horizon is the epoch of recombination when subatomic particles first formed stable atoms. Before that, the universe was opaque. Recombination was the point at which the universe began to give off its first light. That light still exists as the Cosmic Microwave Background radiation (CMB). We can't see, using electromatic radiation, what the universe looked like before that no matter how powerful our telescopes become because no light was emitted prior to recombination.

We may be able to see earlier than that using gravitational waves, though. Gravitational wave astronomy is still new. I hope it advances to the point of being able to look at the universe earlier than the release of the CMB. There would be a lot we could learn.

Even if that succeeds, though, we run into another horizon. There is a point beyond which not even our best theories in physics can predict what we should expect the universe to be like. In general relativity, Einstein's field equation (and its various solutions) starts to yield nonsensical results when the curvature of spacetime approaches infinity. In quantum mechanics, the Heisenburg uncertainty principle would be violated if things were contricted to spaces smaller than the Planck length. Quantum mechanics and general relativity are two of our most successful theories in physics (and maybe in all of science), but if we try to use them to extrapolate back to a beginning, we run up against a wall.

It may be that we can push beyond this horizon if we can come up with a theory of quantum gravity that reconciles quantum mechanics and general relativity and allows us to describe the universe on scales smaller than the Planck length. But who knows if we will ever have such a theory? It may be that there isn't such a theory or it may be that such a theory is unknowable.

With all this fuzziness about what the universe was like beyond a certain point, it raises a degree of doubt that we can extrapolate from our observations about the universe now, using the known laws of physics, to an absolute beginning. That may always be the case.

I don't think this doubt should prevent us from inferring a beginning of the universe, though. Suppose the universe does not converge all the way to a singularity. Still, there's a limit to how far it can converge. At some point, it can't get any denser. Only one of two things can happen once you extrapolate to the limit of how dense the universe can get. Either whatever is left came into being, or it has always existed. If it has always existed, then why did it just begin to expand 13.8 billion years ago and not any sooner? After all, it would have had an infinite amount of time in which to do so. Whatever the reason or cause for why the universe began to expand when it did, that cause or reason would have always been there. I think the big bang points to a beginning of the universe even if we don't know what the universe was like beyond a certain point.

It may be that the beginning of the universe is the point at which our theories do make sense. The fact that we cannot push them beyond a certain point may be owing to the fact that the universe can't exist beyond that point, which in turn, means that's the beginning. Maybe the universe had some finite curvature and density at its beginning, avoiding all the problems of infinities, and avoiding the violation of any known laws.

This is all speculative, of course, but the speculation that the universe had a beginning is at least based on what we know. Based on what we know about the expansion of the universe and the laws of physics, it looks like the universe had a beginning. The fact that there are speculative models in which the universe didn't have a beginning doesn't change this fact. There's nothing in our evidence that makes it look like any of those speculations are true. There's nothing we can point to that would remotely suggest the universe had some finite density for infinite time before expanding.

So, while I don't think the big bang shows with any certainty that the universe had a beginning, I do think it shows with some positive probability that the universe had a beginning. It at least points to a beginning. It's the sort of thing we would expect if the universe had a beginning. It certaintly makes it look more like the universe had a beginning than it would if the universe was not expanding or contracting. So I think the big bang is evidence for the beginning of the universe even if it's not proof.

Subconscious and Implicit Reasoning

We are reasoning machines, but most of the reasoning we do isn't explicit and formal. Aristotle attempted to formalize our reasoning methods by recognizing laws of logic, including syllogisms that express laws of logical inference. Using these tools in a formal way allows us to recognize and avoid making mistakes in our reasoning. It also helps us understand what people are saying and thinking if we can formalize what they say.

Consider this conversation:

Sam I Am: I may have had Covid a month ago, but I'm not sure.

Doctor: Oh, if you had Covid, you would know it.

Sam I Am: In that case, I definitely had the flu.

This conversation didn't actually happen, so there's no need to nit pick about the fact that it's possible to have Covid and not know it. That's irrelevant to the point I'm trying to make.

Anywho, it might seem like I made some big leap in logic to conclude that I had the flu from the fact that if I had Covid, I would've known it. But there's some unstated premises in my reasoning as there almost always is in our day to day conversations. If I were to formalize my reasoning, it would look like this:

1. If you had Covid, you would know it.
2. I do not know it.
3. Therefore, I did not have Covid.

1. I either had the flu, or I had Covid.
2. I did not have Covid.
3. Therefore, I had the flu.

Formalizing my reasoning reveals that I used two different kinds of syllogisms. The first one uses the modus tollens syllogism, and the second one uses the disjunctive syllogism. Notice that the second premise in my second syllogisms says the same thing as the conclusion in my first syllogism. That's why the first syllogism comes first. I have to establish that conclusion before I can use it as a premise in my next argument. To simplify this sort of thing, we can combine all the premises and inferences into one argument, which eliminates repetition. In my case, it would look like this:

1. If I had Covid, I would know it.
2. I do not know that I have Covid
3. I either had the flu or I had Covid.
4. Therefore, I did not have Covid (this follows from 1 and 2 by modus tollens).
5. Therefore, I had the flu (this follows from 3 and 4 by disjunction).

Whenever we're talking to somebody, and they seem to make a leap of logic, or they come to some conclusion we disagree with, we tend to want to fill in the gaps where they didn't explicitly state all their permises. If we're charitable, we fill the gaps with whatever we think must be assumed in order to render their argument logically valid. Often, we don't even state the premise ourselves. If we disagree with their conclusion, the reason we give is always some denial of what we think the hidden premise was.

Now consider the following conversation:

Jim: I don't think I had Covid.

Bob: Why not?

Jim: Because if I had Covid, I would've known it.

Notice that if we tried to formalize this into a syllogism, there would be a missing premise.

1. If I had Covid, I would have known it.
2.
3. Therefore, I did not have Covid.

What is the hidden premise? It doesn't take too much creativity to recognize that Jim's reasoning assumes that he did not know he had Covid since that's the only way to render the reasoning valid. If you were trying to understand Jim, you would likely assume that's his hidden premise.

Jim could've said something like this:

Jim: I don't think I had Covid.

Bob: Why not?

Jim: Well, I didn't have any of the symptoms of Covid.

Bob: It's possible to have Covid without having symptoms.

There are a couple of things going on in this conversation. First, Jim is making an argument to justify his conclusion that he didn't have Covid. Second, Bob is objecting to Jim's argument by denying what he takes to be Jim's hidden premise.

At first glance, Jim's argument might seem unobjectionable, but even this line of reasoning contains a hidden premise, which can be exposed by trying to formalize the reasoning into a syllogism.

1. I did not have any of the symptoms of Covid.
2.
3. Therefore, I did not have Covid.

The best candidate for the hidden premise that jumps out at us is this: If Jim had Covid, he would have had symptoms of Covid. If that were the hidden premise, the conclusion would follow by modus tollens. Since that's the only premise Bob can immediately come up with, he fills in the gap and denies that premise. If the hidden premise is false, then Jim's argument is unsound even if it's logically valid.

Notice that Jim and Bob had this conversation without either of them ever stating the hidden premise in Jim's argument. This sort of thing happens every day. It happens in regular conversation, in heated arguments, and in civil debate and discussion.

There is a danger in filling in the gaps when somebody else delivers an argument with hidden premises. The danger is that you will fill the gaps with the wrong premise. If you do that, and the other person notices, they will think you are misrepresenting them. They may either chalk it up to an innocent misunderstanding, or they may think you're misrepresenting them deliberately. It's hard not to want to fill the gaps when the missing premise seems obvious, but we sometimes fill the gaps with whatever jumps out at us even if there are other options. We do this because we're hasty and sometimes uncharitable.

When I've had formal debates, I've tried two different tactics when responding to my opponent's argument. One tactic is to acknowledge that there's a hidden premise. I'll say something like, "Your argument assumes such and such because that's the only way to make the argument go through." Another tactic I use is to formalize their argument, leave one of the premises blank, and say that unless my opponent fills in the blank, his argument is invalid. I can only remember using that second tactic one time in a debate. I'm reluctant to use that in a formal debates because it requires more back and forth before getting to the point, and there are limited rounds. Usually, I make my best guess at the hidden premise. To avoid being accused of strawmanning their position, I make it known that I'm only guessing what their hidden premise is. Then I'll say something like, "If I've understood you correctly, here's the issue I have with your arugment. . ."

It is interesting to me that in most debates, arguments, and discussions I've seen or been involved in, the disagreements seem to hinge on the hidden premises rather than the explicitly stated premises. It is also interesting to me to recognize that a lot of reasoning - even valid reasoning - happens without even exlpicitly thinking about all the premises we're invoking. We just jump from one fact or observation to our conclusion without explicitly thinking about the hidden premise or the logical rule of inference we used to draw the conclusion from the stated and hidden premise. In many, and possibly most, cases, we reason subconsciously. But formalizing our reasoning can help us be more clear. Formalizing our buddy's reasoning can help us understand him. Formalizing our reasoning can help us recognize and avoid mistakes in thinking. It can also improve our reading compreshension.

We should be thankful for Aristotle.

Defining knowledge as justified true belief with Gettier thrown in

Philosophers have been accused of using words in very unconventional ways, which is confusing for people who aren't philosophers. There's a reason they do it, though. They do it for the sake of clarity. Language, the way it is commonly used, can be ambiguous. Since philosophers are trying to address hard questions that require a lot of precision and careful thinking, they want to define their terms in precise ways so they can understand each other and communicate clearly. This requires them to give very precise definitions to the words they use. It doesn't matter whether the definitions they give are common or not. What matters is that the reader understands what they are saying. As long as you understand what somebody means by the words they are using, you can figure out what they are trying to communicate with those words.

Words and their meanings didn't just fall from the sky. A word doesn't mean something because the dictionary says so. It's the other way around. The dictionary is an attempt to capture what words already mean. What determines the meaning of words is just how they are used. Since any given word might be used in multiple different ways, there are some ways that are more common than others. Dictionaries attempt to put the most common uses first and the least common uses last.

In the case of the word, knowledge, the definition philosophers typically use may be unusually precise compared to the way most people would definite it, but it's not an arbitrary definition that's peculiar to philosophers. Rather, it's a definition that was arrived at in an attempt to capture the common use. Let me try to show that by asking some questions.

Would it make sense to claim that you know something if you didn't even think it was true? Probably not. Wouldn't it seem odd to say, "I don't think birds of a feather flock together, but I know they do"? I suspect that would seem odd to you. At a bare minimum, then, before you can know something, you have to at least believe it to be true. To "believe," just means to think something is true.

If I believe something, is that enough to claim that I know it? We have already established that belief is necessary for knowledge, but most of us would acknowledge that belief is not sufficient. Something more is needed. After all, I might believe something and be wrong about it. People believe all sorts of things that aren't true. You can't know something is true if it's not true. So at minimum, before you can know something, it first has to be true.

Now we've shown that to have knowledge, you need at least two ingredients. You need it to be true, and you need to believe that it's true. But are those two things enough?

Well, consider a situation in which somebody, for no reason at all, or for some erroneous reason, comes to believe that there is life on Jupiter. Up until now, no discovery of life on Jupiter has been made, and no chemistry or light on Jupiter has given us any reason to think there's life on Jupiter. But then suppose that years down the road, a probe sent into the atmosphere in Jupiter found that, in deed, there is mocrobial life floating in the atmosphere of Jupiter. If that were the case, then the person who believed there was life on Jupiter would have been right all along. He had a belief that turned out to be true. Would it be fair to claim that he knew that there was life on Jupiter?

I hope you said no. The way we commonly use the word, knowledge, seems to entail that he didn't have knowledge. Rather, he just made a lucky guess, and happened to believe it. What is missing? Well, if he just arbitrarily believed something he made up, the missing ingredient seems to be justification. If he had concluded that there was life on Jupiter because he performed some spectroscopy on the planet and discovered chemicals in the atmosphere that could only be created by living organisms, then he would have some justification for believing there was life on Jupiter. Or, if he had been part of the team that sent a probe to Jupiter, and he was privy to the data the probe sent back indicating that it found life on Jupiter, then he would by justified in beliving there was life on Jupiter. Then we might say he knows there's life on Jupiter.

Now, we have established that knowledge requires (1) belief, (2) truth, and (3) justification. That is why the typical definition philosophers use for knowledge is "justified true belief." That has served as a satisfactory definition for knowledge in most cases.

Being the persnickety people that good philosophers are, though, even this definition has been probed for its accuracy. Just as in the scientific method, we test hypothesies by trying to falsify them, so also in the case of philosophy, we test ideas by trying to think of counter-examples. A philosopher named Edmund Gettier came up with some counter-examples to the definition of knowledge as justified true belief. The issue he raised has become known as "The Gettier Problem" since there is no concensus on the resolution.

A counter-example to knowledge as justified true belief would be a scenario in which all three ingredients are present, but we still don't think the person has knowledge.

When I was in middle school, I had a friend named Chad who had a girlfriend named Wendy. One day, as a joke, I told Chad that Wendy said she was going to break up with him. Chad believed me because he didn't have any reason to think I'd lie about something like that. Seeing the look on his face, I felt bad and immediately told him it had been a joke. He went about his merry way. The next day, Chad came at me angry for lying to him. It turned out Wendy really did break up with him, and he was mad at me for claimimg that it had been a joke when it was actually true.

Notice that when I told Chad that Wendy was going to break up with him, he had all three ingredients for knowledge. (1) He believed Wendy was going to break up with him, (2) he was justified in believing Wendy was going to break up with him, and (3) it was true that Wendy was going to break up with him. Yet because I was joking, most of us would probably agree that Chad did not have knowledge. I wasn't reporting to Chad anything that I actually knew. It was just a coincidence that I happened to be right. This scenario, then, serves as a counter-example to knowledge as justified true belief. It appears that something else was missing. What was it?

Here, philosophers give different answers. Some answers involve tweaking or qualifying the criteria of "justification" in some way. What exactly is justification? Will any ole justification do? Other answers involve adding a fourth criteria. What other ingredient is required for knowledge?

Since Gettier problems in real life are very exceptional, I think that pragmatically speaking, we can just ignore them. The definition of knowledge as justified true belief is a good enough rough definition to cover most real life cases. But when it comes to arguing the nitty gritty details of the things weirdos like you and I like to talk about (including the topic of epistemology), it might be useful to tackle the Gettier problem.

While I haven't read a whole lot of literature on how other philosophers have tackled this problem, my own unrefined view is that whatever the justification for our belief is, it must be a proper justification. That is, the justification must actually bridge the gap between the belief and the reality. There has to be a connection between the two. There is no connection between the reality of Wendy breaking up with Chad, and Chad's belief that Wendy would break up with him since I just made it up. If Wendy had told me she was going to break up with him, and I had told Chad, then there would have been a bridge from the reality to the belief. In that case, Chad would have a proper justification for his belief that Wendy would break up with him. That would give him knowledge.

While I am happy to define knowledge as justified true belief without going into the Gettier problem, if I had to nail it down more precisely, I would just add the "proper" part along with the explanation. My more persnickety definition of knowledge would be "properly justified true belief."

How would you deal with the Gettier problem? What is the missing ingredient for knowledge that is left out by the usual definition?

Libertarian free will, Frankfurt cases, and the ability to do otherwise

Anybody who believes in free will thinks the will is free from something or free to do something. Libertarians thinks the will is free from absolutely all antecedent conditions, including one's own psychological states. That means if you have free will in the libertarian sense, then for any free act, you could have done otherwise even if everything in the universe prior to and up to the moment of choice had been exactly the same, and that includes all of your psychological states, including your beliefs, desires, preferences, biases, motives, etc.

Since the ability to do otherwise is so wrapped up in the notion of libertarian free will, many have taken to defining free will as the ability to do otherwise. A philosopher named Harry Frankfurt came up with some counter-examples to show that the ability to do otherwise is not necessary for libertarian free will. These are thought experiments designed to show that one can have libertarian freedom even if they lack the ability to do otherwise.

For example, imagine you're sitting at a table with a can of Coke on one side and a can of Dr. Pepper on the other, and you are given the option to drink one or the other. Imagine that unbeknowst to you there's a guy hiding behind the curtain watching you closely, and if he sees you reach for the Coke, he's going to jump out from behind the curtain and slap the Coke away, preventing you from drinking it. That never happens, though, because you choose to drink the Dr. Pepper instead of the Coke.

Thought experiments like this are meant to show that one can make a libertarian free choice without having the ability to have done otherwise. Even though you couldn't have chosen to drink the Coke, your choice to drink the Dr. Pepper was still a free choice.

One can nit pick about the particulars of the thought experiment (e.g. if you were free, then your choice wasn't just between Coke and Dr. Pepper, but between drinking Coke and not drinking Coke, etc.), but setting those quibbles aside, I think what Frankfurt thought experiments show is that the ability to do otherwise is not what is meant by libertarian free will. It shouldn't be part of the definition of libertarian free will.

However, Frankfurt cases almost never happen in real life. In real life, we make choices continuously every day without there being Frankfurt cases. In the absense of a Frankfurt case, if you have libertarian free will, then you do have the ability to do otherwise. The ability to do otherwise, then, is a consequence of libertarian free will in the real world. So it does make sense to talk about libertarian free will as entailing the ability to do otherwise, at least in the real world as opposed to imaginary scenarios.