By JD Snodgrass

This sentence is false.

Is the above sentence true or false? If it’s true, then what it says of itself — that it’s false — is true. In other words, it’s false. So it can’t be true. But can it be false? If it’s false, then what it says of itself — that it’s false — is false. In other words, it’s true. So it can’t be false. AHHHHHH!!!!!!!

This is the infamous Liar Paradox. The paradox has a storied history and continues to fascinate philosophers, logicians, and mathematicians alike. It originated in a list of logic puzzles written by the Greek philosopher Eubulides of Miletus in the fourth century BCE. Eubulides wrote, “A man says that he is lying. Is what he says true or false?” This eventually evolved into the famous Liar Paradox we all know and love, “This sentence is a lie,” or more to the point, “This sentence is false.” The problem with the Liar should be obvious. If it’s true, then it’s false. But if it’s false, then it’s true. So what do we do? How do we solve the paradox? Can it be solved? And why the hell should we care?

Regarding the last question, we need to know which statements are true and which ones aren’t. Without a formal theory of truth, we’re at a loss to say, for example, whether a given mathematical theory is true or false, as we don’t know what “true” and “false” really mean. The Liar Paradox opens holes in our theory of truth that must be closed in order to know that the statements we make about mathematics — or, in fact, anything — hold.

At this point, you might be tempted to say that the Liar is neither true nor false. I’m not going to address that solution here. Instead, I’ll refer you to the Stanford Encyclopedia of Philosophy entry on the Liar Paradox for an explanation of why that solution is ultimately problematic. In short, you actually end up creating another paradox.

So if we can’t say it’s neither true nor false, then what do we do? Numerous solutions have been devised over the years. They all fall into two camps: those that retain classical logic and those that reject it. I’m only going to address one solution in this post, a very recent one that works within the constraints of classical logic.

A few years ago, philosopher Kevin Scharp developed a new theory of truth that solves not just the Liar Paradox but all so-called alethic, or truth-based, paradoxes. In short, Scharp claims that truth is an inconsistent concept. Most of the time, we don’t notice the inconsistency. It’s not until we start examining certain kinds of self-referential sentences like the Liar that we run into problems. His solution is to replace our concept of truth with a pair of concepts he calls “ascending truth” and “descending truth.” According to Scharp, replacing truth with these new concepts dissolves the paradox.

Background

Before jumping into Scharp’s theory, we need to talk about an important concept in theories of truth. In the 1930s, Polish mathematician and philosopher Alfred Tarski developed a theory of truth for formal languages (think mathematics and logic). In it, he claims that for a theory of truth to be adequate, it must entail all possible T-sentences, or sentences of the form

“P” is true if and only if P.

Using logical notation, we can restate this as

T(P) ↔ P.

This double-arrow signifies a biconditional. It combines the following two statements into one:

If P then T(P), or using logical notation, P → T(P), and

If T(P) then P, or using logical notation, T(P) → P.

In the first statement, you can infer from a sentence P that it’s true that P, or T(P). Put another way, you can “capture” P with the truth predicate T. In the second statement, you can infer the opposite, and thereby “release” P from the truth predicate T. For Tarski, a theory of truth must entail both capture and release.

In more familiar terms, an adequate theory of truth in English should entail the sentence

“Snow is white” is true if and only if snow is white.

This should make intuitive sense: a sentence is true if and only if the proposition it expresses is true. Likewise, you can turn it around and say that snow is white if and only if the sentence “Snow is white” is true. Most philosophers, logicians, and mathematicians agree with Tarski that an adequate theory of truth must entail all possible T-sentences.

We can rewrite the Liar according to the T-schema. When we do, we get the following problematic T-sentence:

“This sentence is false” is true if and only if this sentence (the one in quotation marks) is false.

The issue is that we don’t want our theory of truth to entail this T-sentence because that would mean our theory is incoherent. But how do we ensure our theory doesn’t entail paradoxical T-sentences? Scharp thinks he has an answer.

Replacing Truth

In his book Replacing Truth, Scharp claims that truth is an inconsistent concept and that this inconsistency is what causes problems like the Liar Paradox. Scharp claims that the T-schema T(P) ↔ P is really two separate concepts that we’ve improperly confused as one. He calls these two concepts “ascending truth” and “descending truth,” or AT and DT. They correspond to the capture and release concepts mentioned above: AT captures but doesn’t release, and DT releases but doesn’t capture. Put into logical form,

Ascending truth: P → A(P)

Descending truth: D(P) → P.

Scharp’s key insight was that these concepts cannot properly be combined into a biconditional. They are separate. He is denying the T-schema.

So how does this help with the Liar Paradox? Well, because truth is two concepts, the Liar sentence needs to be slightly modified. Let’s take a look at the Descending Liar, or D-Liar, a sentence P that says of itself that it’s not descending true: “This sentence is not D-true.” Is this sentence D-true or not D-true? If it’s D-true, then what it says of itself — that it’s not D-true — is D-true, and therefore it’s not D-true. Put another way, we’re just restating the release relationship D(P) → P; we’re moving from an assertion that a sentence is D-true to the sentence itself.

So far, the D-Liar is acting just like the traditional Liar. But, unlike the traditional Liar, our analysis ends here. You might be tempted to continue the analysis and say, “but if it’s not D-true, then what it says of itself — that it’s not D-true — is not D-true, and therefore it’s D-true.” Resist the temptation! Because of how DT works, you can’t make that move. Remember, DT only releases; it doesn’t capture. The move from P to D(P) doesn’t exist in our new scheme. That move is only possible for AT. So rather than being a paradox, the sentence is simply not D-true (it also works out as not D-true if you begin the entire analysis with the claim that it’s not D-true, as you can’t infer the sentence P from not-D(P)). It is, however, A-true, since the sentence P implies A(P). Separating truth into AT and DT prevents the paradox from arising in the first place. In the end, the Liar sentence is A-true but not D-true. Although not paradoxical, Scharp calls such sentences “unsafe.” Clever, right?

Scharp’s solution is incredibly powerful and convincing. However, it’s not without its problems. As philosopher David Ripley points out in his review of Replacing Truth, Scharp’s theory has some rather unpalatable limitations. First, we want a theory that allows us to use truth as a device of endorsement. We want to be able to assert that true things are true; we want to be able to assert both P and T(P). But Scharp’s theory makes this difficult. The strongest assertion we can make is that something is D-true, as the statement D(P) implies P itself. But, as Ripley points out by way of example, it’s impossible to endorse the contents of Scharp’s book without repeating all the sentences in it. If we say the book is D-true, then we’re actually disagreeing with it because it contains unsafe sentences like the Liar. But AT doesn’t work as a device of endorsement either. Claiming that P is A-true doesn’t give us P itself. The most we can say with AT is that if P is not A-true, then not-P. In other words, AT cannot serve to endorse. It can only serve to reject.

Second, Ripley claims that all this focus on truth is unwarranted. Scharp’s theory only helps us with alethic paradoxes and doesn’t apply to any other kind of paradox. Ripley points out that there’s one concept present in all paradoxes, and it’s not truth. Rather, it’s validity. Ripley thinks the concept of validity is the one worth pursuing as inconsistent.

Regardless of its limitations, Scharp’s theory is striking. And it upends a concept most of us never would have thought about as particularly problematic. It’s a fascinating addition to the literature on truth and a real sweet piece of brain candy.

JD Snodgrass is Philosopher-in-Chief of The Will to Power Hour. He is currently making great use of his philosophy degree as a professional contract negotiator. Who knows, maybe the whole philosophy thing will pan out and he can live out his days contributing to the literature on truth. But first, he’ll need to purchase a tweed jacket with leather elbow patches. He’ll also need to take up pipe smoking, although he could probably get away with one of those cool little plastic pipes that blows bubbles.

By JD Snodgrass

When Friedrich Nietzsche penned the phrase “God is dead” more than 130 years ago, he likely didn’t think it would rank among the most misappropriated and misunderstood quotes in history. He wrote it in a few places, perhaps most famously in his parable “The Madman,” which appears in his 1882 work, The Gay Science. The parable tells the story of a man who runs through a market crying, “I seek God! I seek God!” The market-goers don’t believe in God and ridicule the man. He responds:

“Wither is God?… I will tell you. We have killed him—you and I. All of us are his murderers… God is dead. God remains dead. And we have killed him. How shall we comfort ourselves, the murderers of all murderers? What was holiest and mightiest of all that the world has yet owned has bled to death under our knives: who will wipe this blood off us? What water is there for us to clean ourselves? What festivals of atonement, what sacred games shall we have to invent? Is not the greatness of this deed too great for us? Must we ourselves not become gods simply to appear worthy of it? There has never been a greater deed; and whoever is born after us—for the sake of this deed he will belong to a higher history than all history hereto.”

Here the madman fell silent and looked again at his listeners… “I have come too early,” he said then; “my time is not yet. This tremendous event is still on its way…”

Friedrich Nietzsche (1882), The Gay Science, § 125, translated by Walter Kaufmann

To fully understand what’s going on, you need some background on the quote’s context and Nietzsche’s broader project.

During Nietzsche’s time, Western society — certain portions of it anyway — was increasingly aware that it lacked any metaphysical foundation. In their unrelenting quest for truth, especially scientific truth, many had cast off their notions of God and transcendence, concluding they were groundless. God was no longer needed to explain how the world worked, and the very idea of God was increasingly untenable. But in ridding themselves of God, they failed to consider the horror that lay ahead. Once you pull the metaphysical rug out from society, you risk falling into widespread nihilism. What’s needed is not a replacement God, but an entirely new way of viewing life that negates the need for a transcendent realm.

The atheists of Nietzsche’s time had rid themselves of transcendence, but they hadn’t taken the next step of revaluing life. And the madman knew what lay ahead for them. When he said, “What festivals of atonement, what sacred games shall we have to invent,” he was warning them of the nihilism to come. Nietzsche’s task was to revalue life and avoid nihilism, and he was actually criticizing his atheist contemporaries for failing to do so — the madman knew he had come too soon, as the market-goers hadn’t yet realized the consequences of rejecting God. All this led Nietzsche to conceive the Übermensch, or Overman, who affirms life and its eternal recurrence through the will to power.

Rather than delve into an explanation of the Übermensch, eternal recurrence, and the will to power, I’ll just say this. Nietzsche seemed to think that some sort of aesthetic ideal was the answer to the problem of life — the problem of living in a world devoid of objective meaning. For Nietzsche, creation, especially the artistic kind, represents the highest exercise of the will to power and allows people to overcome nihilism. Creation is the most human activity, and a life of creation is a life well lived.

Although Nietzsche’s philosophy is problematic in certain ways, he expressed something important about human psychology: the need for meaning and purpose. And his solution to the problem of life contains real wisdom. Creativity not only brings joy, it allows you to build your very own apparatus of meaning and effectively cope with the absence of objective meaning. Creativity helps you avoid nihilism.

As I wrote in my previous post, I don’t believe in a deity. For me, the very concept of a deity is flawed and I don’t feel the need for one anyway. But that might leave you wondering where I find meaning and value. Ultimately, I find them in the same places as everyone else: family, friends, and the desire to have a positive impact on the world. And creativity helps me along the way. This podcast is a creative venture, and it allows me to explore the human intellect with my closest friends. There’s infinite meaning and value in that. Despite what right-wing commentators like Bill O’Reilly and comically bad movies like God’s Not Dead (2014) would have you believe, atheists aren’t a bunch of amoral misanthropic malcontents bent on destruction of all that is holy. We’re not all Richard Dawkins. Moreover, morality isn’t dependent on religion. In the end, I live my life by the words of the inimitable Frank Reynolds: “I don’t know how many years on this Earth I got left. I’m gonna get real weird with it.”

JD Snodgrass is the Executive Producer of The Will to Power Hour. He fancies himself a writer slash philosopher slash beer connoisseur. In reality, he is none of these and is lucky if he buttons his own shirt correctly. He was recently chewing gum when he yawned and lost command of the small, soft object, which landed on his desk in front of four coworkers. That was one of his better days.

By JD Snodgrass

Religion is great. It helps ground people and give them meaning in a chaotic, unfair world. It gives their life some semblance of order and direction. Buuut… I simply can’t take the idea of a deity seriously. I grew up in an orthodox Jewish community, and from a young age was inculcated with the one tradition that really separates Judaism from other religions: that of questioning absolutely everything, even God Himself, whom as I said I don’t believe exists. The importance of questioning even the Divine Will is illustrated by the story of the Oven of Akhnai. This story appears in the Talmud, which contains the legal and philosophical underpinnings of Rabbinic Judaism and even has a section on how to properly wipe your own ass (the Talmud is very comprehensive). It’s quite telling that “Talmud” is Hebrew for “learning” and that “rabbi” means “teacher.”

Back in the day, a bunch of sages were figuring out what to do with a clay oven that had become impure. One of them, Rabbi Eleazar, believed if it were reassembled with sand between the pieces, the oven would be pure again. The others disagreed, believing it could never be purified. So they did what Jews do best: they argued. Eleazer made his case and threw out every argument and justification he could think of. But the sages held firm in their conviction. Unable to persuade them with words, Eleazer resorted to miracles to show that halakha, or Jewish religious law, was on his side. He first called upon a carob tree to prove him right. And what do ya know, the darn thing promptly uprooted and flung itself a hundred cubits, or about 150 feet. Unconvinced, the others retorted, “Um, trees don’t have anything to do with ovens.”

After performing a few more miracles to no avail, Eleazer eventually called upon the Almighty Himself for support. Of course, God was all too happy to oblige given the vital importance of oven purity and said, “Yeah, um, Eleazer is right, you guys. You should listen to him. The oven is good to go.” Gloating, Eleazer told them all to shove it. But one of the other sages, the ever-obstinate Rabbi Yehoshua, shouted at God, “It is not in Heaven!” This phrase comes from Deuteronomy 30:12. By “it,” he meant the Torah. The idea is that the Law, having already been revealed to man, is no longer God’s preserve. Rather, it is for man and man alone to interpret. In other words, Rabbi Yehoshua had the chutzpah to tell God to mind His own damn business, as this was a purely rabbinic affair. GASP! And what did God do in the face of such willful recalcitrance? He laughed, “My children have defeated me! My children have defeated me!”

Critical thinking, independence, and incessant questioning are cherished values in Judaism to a degree not found in many other religions. Take modern mainstream Christianity. Faith is absolutely central to Christianity. Having faith in God and believing that Jesus Christ died for your sins is what it means to be a Christian. Questioning is encouraged only insofar as you are trying to understand God; you don’t tell God to butt out as Rabbi Yehoshua did. That kind of insubordination is considered blasphemy. But Judaism is different. Faith and belief are only part of the story. To be a Jew really means to question everything and seek the truth. That’s what matters. Faith is secondary. If your quest for truth leads you to clash with your Creator, so be it. The important thing is that you are constantly questioning and learning for yourself.

I spoke at length on this topic with a friend of mine, Steve McCloskey, who is pursuing his Master of Divinity at Princeton. He said the following:

Generally I agree with you. Judaism encourages more questioning of God than does Christianity. However, Christianity goes through periodic identity crises (e.g. the Protestant Reformation, the Great Schism, etc.) and has to re-think itself. In its current identity crisis, issues of biblical interpretation are at the forefront. For instance, how do you reconcile or respond “biblically” to current science and views about human sexuality?

Many contemporary theologians are seeing that Judaism, which is at the foundation of Christianity, has a kind of healthy dynamism built-in that could enrich Christianity and even help Christians understand what the writers of the New Testament, who mostly were Jews themselves, were saying.

I also asked Steve whether he knew of any stories in the Christian tradition like the Oven of Akhnai, stories in which man argued with God and won. He doesn’t know of any.

My own quest for answers has led me to the conclusion that God does not exist. If you spent your life thinking, questioning, learning, and seeking the truth, and you reached the opposite conclusion, then good for you. You exemplify everything great about the Jewish tradition. I might not believe in a deity, but that doesn’t mean I don’t have a religion of sorts. I have philosophy and that distinctly Jewish habit of questioning everything. The whole point of religion is to give order to the world and direction to your life. That’s what philosophy does for me. And isn’t theology just one branch of philosophy?

JD Snodgrass is the Executive Producer of The Will to Power Hour. He’s also Jewish. And an atheist. And he likes beer. He also likes his next-door neighbor, Lemar, who often gives him beer. Lemar is a great guy. You should get to know him. He might give you beer, too.