The Liar Paradox emerges upon analyzing the statement

⊥     This statement is not true.

Is ⊥ true or not? If it is, it isn’t; and if it isn’t, it is! Since ⊥ is either true or not, it follows that ⊥ both is and isn’t true. This is very unpleasant, to say the least.

I think the culprit is ultimately a certain way of thinking about truth. The paradox only arises if one thinks of truth as a really existing property that everything either has or lacks. Let’s call this the metaphysical framework (M) of truth. Now consider for example the statement

T     This statement is true.

According to framework M, T tells us something about the world (i.e. expresses a proposition), namely that T has the truth property. Another implication of M is that the Golden Gate Bridge, for example, being neither true nor false, lacks the truth property. Thus the Golden Gate Bridge is not true. This is not to say that the Golden Gate Bridge is false.

If one accepts M, then clearly ⊥ is either true or not. ⊥ also seems to express a proposition, namely that ⊥ lacks the truth property. It’s hard to escape paradox from there. But does one have to accept M? I don’t think so. In my post “Properties” (28 Dec 2025), I argue that properties don’t exist; and if properties don’t exist, M is false. In that post, I also present an alternative way of thinking about properties according to which property talk is just a convenient way of speaking. I will do something similar with respect to truth.

Let me introduce the linguistic framework (L) of truth. According to this view, truth talk is no more than a convenient–perhaps indispensable–way of speaking. To say that the proposition that the Earth is round is true, for example, is just another way of saying the Earth is round. Also, to say, for example, that the proposition that Santa Claus exists is false is to say that it’s not true, and both are just alternative ways of saying Santa Claus does not exist. A consequence of this view is that T tells us nothing about the world (unlike in framework M). This matches our intuitions: most people when pressed, I think, will say T seems vacuous. Another consequence is that it makes no sense to say that the Golden Gate Bridge is not true. On framework L, the truth predicate and its variations (“not true”, “false”, …) only apply to propositions.

If one accepts L, then clearly ⊥ does not express a proposition, and it therefore makes no sense to say ⊥ is not true. I can hear someone objecting at this point, “But it makes even less sense to say ⊥ is neither true nor not true!” The objector has either subtly shifted back into framework M or never really left. There’s no objection here if one really leaves M behind for L. Within L, non-propositions do not need to be either true or not true, because those predicates are ways of speaking that only make sense when applied to propositions. ⊥ doesn’t make sense, doesn’t express a proposition, and cannot be called “not true” except from within a faulty framework.

I’ve lived in Ohio all my life. Most people I’ve gotten to know, and I believe most people in the United States, are either apathetic or contemptuous towards philosophy. A lot of people seem to find it pointless or boring. So, what is the point? Obviously, I can’t answer this question for every topic in philosophy; however, I’ll give it a try with respect to the discipline as a whole.

Before we proceed, we must know what philosophy is. American philosopher Alvin Plantinga has defined it most simply and accurately to my knowledge: philosophy is “thinking hard” about something. We might throw in “about the so-called ‘big questions'” at the end for good measure, but Plantinga’s answer is sufficient. One might object that this definition is too broad, as it encompasses every other field of study. In my view, this is a feature and not a bug of the definition, since philosophy technically does include all other fields. Science, for example, is more or less “thinking hard” about the natural world, with the help of certain rules and restrictions. In its earlier stages, science used to be called “natural philosophy”.

Still, it’s useful to identify that part of philosophy which is distinct and separate from other fields, what one might call philosophy proper. Here are its three most important characteristics:

1. Philosophy (properly so-called) attempts to answer the “big questions”. Traditionally, philosophy is divided into four subdisciplines: logic (the study of reasoning), epistemology (the study of knowledge), metaphysics (the study of fundamental reality), and value theory (the study of value, moral and aesthetic).
2. Philosophy is the only discipline with an unrestricted universe of discourse. It can even include other disciplines.
3. Philosophy is the only discipline that can question its own assumptions.

Philosophy proper, hereafter “philosophy”, is what so many people find pointless or boring and what the rest of this article is about.

Now that I’ve explained what philosophy is, what’s the point of it? Why should we engage in it? First of all, it makes you think better. Thinking better helps you discover truth. Philosophy gives you conceptual and linguistic clarity and improves your reasoning skills. This carries over into every other intellectual endeavor in life from science to engineering to literature to conversations with your friends.

Second, unless you’re an exceedingly incurious, shallow person, philosophy is something you engage in already, whether you realize it or not. The only question is, will you be a good philosopher or a poor one? If you’ve ever thought about what you are, why you’re here, whether there’s a God, or what he’s like, if he exists, then you’re a philosopher–congratulations Plato.

Third, philosophy will change your life. Philosophy is the perfect tool with which to craft a well-grounded worldview, and your worldview influences how you live.

Fourth, philosophy is interesting. All truth is God’s truth, and it’s fun to discover the truths he’s brought about. Because philosophy is so interesting, studying it will make you a more interesting person.

I’ve heard three common criticisms of philosophy. The first boils down to there being a lack of agreement among philosophers. In science and mathematics, for example, one finds consensus on a wide variety of topics. In philosophy, by contrast, there is widespread agreement on almost nothing, and nearly every view imaginable–including those that seem bizarre or absurd to the regular person–is vigorously and expertly defended. This poses a big problem of not knowing who and what to read for those trying to get into the discipline on their own. That problem aside, while it would be nice for more people to agree, it’s not required for you to come to believe or know things.

The second criticism is that philsophy isn’t practical. I think people base this criticism on their being unable see specifically how to apply this or that philosophical insight in their life. For example, say you’ve become convinced through philosophical inquiry that properties don’t exist. What does this information do for you? Wanting to know specifically how each philosophical discovery impacts one’s life is understandable but mistaken. First, the improved thinking ability and better worldview you get from studying philosophy benefit you so broadly and fundamentally that it’s hard to identify specific benefits. However, that doesn’t make the benefits any less real. Think of a bustling economy. A minimal, moral, stable, and consistently enforced system of laws along with a conservative tax rate and unified culture would greatly improve its output over what it would otherwise be, but it would be futile to try to identify specific improvements. Second, as someone who’s been interested in philosophy for a decade, I can tell you that many discoveries which initially seem pointless later end up having amazing applications in the most unexpected areas. For example, the view that properties don’t exist turns out to have major implications for God’s greatness!

The third criticism is that philosophy is boring. To some extent, this is a matter of personal preference; and who am I to tell people what to like? However, I also submit that some questions are so fundamental that there comes a point where lack of interest turns into personal defect.

In conclusion, I think it’s valid to criticize philosophy for being hard to enter on one’s own both from a practical standpoint, given the volume and diversity of reading material, and from a motivational standpoint, given the hard-to-enumerate benefits. Philosophy is hard. However, if you study it with discipline, it will be worth your while.

A property is a way something could be. For example, if there is a brown dog named Fido, then it can be said that Fido has the property being brown and lacks the property being blue. This “having” relation between a thing and a property of the thing is sometimes called “exemplification”. Thus Fido exemplifies brownness.

Properties are a kind of universal. A universal is something that could be simultaneously multiply located, exemplified, or instantiated. For example, Donald Trump and I now both have the property being human. One and the same property–humanity–is simultaneously exemplified by billions of people.

Properties are abstract objects. Something is abstract if it is essentially causally impotent. This should be an obvious point. Properties can’t cause things. While it may be that a thing’s having a certain property could cause something, the property itself could not. What effect could, say, being blue have upon anything? All abstract objects are immaterial because any material thing could have impinged on another material thing.

There are two ways of thinking about properties. The first sees them as things that really exist, just as robustly as people and electrons. The second sees property talk as no more than a convenient way of speaking. For example, to say that Fido has the property being brown is just another way of saying Fido is brown. There is no property “brownness” that exists “out there” and stands in some mysterious exemplification relation with Fido.

I will now argue for the view that properties do not exist. If properties exist, then there is the property non-selfexemplification. This is the property something has when it does not exemplify itself. For example, the property being blue is not blue; thus it does not exemplify itself and has the property of non-selfexemplification. On the other hand, the property abstract is itself abstract and so lacks the property of non-selfexemplification. If such a property exists, then everything either has it or lacks it. What, then, about the property itself? Does non-selfexemplification exemplify itself or not? If it does, it doesn’t; and if it doesn’t, it does! This is very unpleasant, to say the least, and it seems to follow from the assumption that properties exist.

If we say instead that there really is no such thing as non-selfexemplification, then its associated problems vanish with it. If you have any doubts about this solution, consider the more down-to-earth example of a barber who shaves all and only those who don’t shave themselves. Does the barber shave himself? If he does, he doesn’t; and if he doesn’t, he does. Is this a problem? Not if we conclude no such barber exists.

Do we lose anything by getting rid of properties? I don’t think so. As abstract objects, they have no effect upon anything. Properties also don’t explain anything. Saying Fido exemplifies the property of brownness does nothing to explain why Fido is brown or why Fido is similar to other brown objects. A scientific explanation of Fido’s brown fur seems sufficient.

Something is subjective if it’s mind-dependent in the sense that it exists only in someone’s mind, like a mental state (figment, dream, preference, belief, …). Something is objective if it’s not subjective, that is, if it’s mind-independent in the same sense. Something is objectively true, for example, if it’s true not merely in someone’s mind but independently of what people think, wish, and so on.

Is there objective truth? Rather than cite examples, I will first point out the self-defeating nature of believing that there isn’t. Consider the proposition (S) that there is no objective truth. Is that true or false, and is it objectively or subjectively so? Let’s look at the options. First, if S is objectively true, then there is objective truth after all, namely S, and S is therefore objectively false. Second, if S is subjectively true or false, that is, if it or its denial is just someone’s opinion, then it doesn’t preclude the existence of objective truth, just like my opinion that the sky is a certain color doesn’t preclude the sky’s being blue. If S is subjectively true or false in the sense that it or its denial is never more than an opinion in anyone’s mind, then it’s objectively true that S is subjective and S is therefore objectively false again. Third and finally, if S is objectively false, then, well, it’s objectively false—no problem.

To summarize, our options with regard to S are ultimately twofold: we must either affirm that S is objectively false, in which case there is objective truth (S‘s denial, for instance), or admit we can’t speak to the issue. When one considers this in conjunction with obvious examples of objective truths like

1. I exist,
2. 2 + 2 = 4,
3. If all men are mortal, and Socrates is a man, then he is mortal, and
4. It’s wrong to torture and rape a little girl for fun,

it’s hard to deny there are at least some objective truths. This is important because it means that there is an external reality with which we’re confronted. It is a certain way whether we think so or not, like it or not. What’s up for debate is not this reality’s presence but its scope and impact on our lives.

Will is the power to choose and act. Someone’s will is free if he isn’t causally determined to choose and act as he does. This kind of freedom is called “libertarian” (not to be confused with libertarianism the political philosophy!). Libertarian freedom is freedom proper, within philosophical discourse at any rate, and is to be contrasted with other kinds of freedom, improperly so called, such as voluntariness. A choice or an act is done voluntarily if the person who does it isn’t “dragged kicking and screaming”. To illustrate the difference between the two, imagine someone points a gun to your head and tells you to choose Cheerios for breakfast tomorrow or they’ll blow your brains out. You comply. Your choice of Cheerios is involuntary but still free in the libertarian sense, since you weren’t causally determined to choose Cheerios like a pre-programmed machine or billiard ball struck by a cue.

Do we have free will? The only proper sense of this question and the only version worth asking is, do we have libertarian free will? We must. This is a pragmatic “must”. If we don’t, then we might as well all become farmers (to quote one of my favorite philosophers). It would make life absurd and unlivable. Nothing we think, believe, say, or do would matter. We would cease to be persons, reduced to mere automata. We would cease to be rational: if you were determined by factors outside your control to believe, say, that the Earth is round, then how could that belief be rational? This would apply to all your beliefs, by the way, including any belief you may have that determinism is true, making determinism incapable of being rationally affirmed. But never mind all that. We must have free will because our existence depends on it. This doesn’t prove we have it, of course. But if in fact we don’t have it, who cares?

10 Jan 2025 update: added list item (12)

Modality is possibility/necessity, presumably called that because one can think of possibility and necessity as modes of truth. Something is necessary if it must be that way. The possible can be understood in terms of the necessary (and vice versa), so one might say that something is possible if it’s not necessarily not that way. Forgive the double negative, but that’s how it is. For example, it’s possible that I will be late for work tomorrow: the statement is not necessarily false.

It will be helpful to describe metaphysical modality in the context of two other kinds of modality: physical and logical. It’s incumbent on me to describe only necessity since, as I mentioned, possibility and necessity are definable in terms of each other. First, what is physical necessity? Something is physically necessary if it must be the case given the laws and boundary conditions of nature. For example, it’s physically necessary that a stone will fall if dropped. Something is logically necessary if it must be that way given the rules of logic. For example, it’s logically necessary that God either exists or he doesn’t.

What, then, is metaphysical necessity? Something is metaphysically necessary if it must be that way given our intuitions about what could have been real. Metaphysics is that branch of philosophy concerned with the nature of reality, whether physical or not. It’s best to look at examples. Here are twelve statements that, though logically possible, are necessarily false in the metaphysical sense:

1. 7 + 5 = 13
2. Jim is taller than himself
3. Prime Minister Thatcher is a prime number
4. The ball is both red and green all over at once
5. The chair created itself
6. Mary danced with the square circle
7. God committed adultery
8. There is something with a shape but no size
9. Joe traveled to a time before he was born and killed his grandfather
10. The desk on which I’m now typing is made of ice
11. Alex’s thirty-first birthday party preceded itself in time
12. It’s good to torture and rape a little girl for fun

None of these statements is contradictory, but all of them are absolutely incapable of corresponding to reality.

An argument is a set of premises leading to a conclusion. The strength of this “leading” can vary. Sometimes, that all the premises should be true but the conclusion false is logically absurd, and sometimes it’s only unlikely. We’ve all encountered arguments such that its premises are lovely, the connection between its premises and conclusion is very strong, and its conclusion is incompatible with other propositions we believe. When this happens, we’re forced to give up either some of the premises or these other beliefs. We should give up whatever we are less sure of. In my experience, arguments with the most unpleasant conclusions have premises such that one is far less sure of them than he is of the beliefs he would have to give up to accept the conclusion.