Predicate Logic Practice 2: The Four Quantifier Rules (Answers)

Difficulty: What the hell

Here are the answers to our second baby predicate-logic practice. This time the main point was to get used to the four quantifier rules—U.I., U.G., E.I., and E.G.—along with the irritating but necessary business of flagging.

If you made it through these without wanting to throw your book across the room, congratulations.

Exercise 1

1. (∀x)(Px ⊃ Qx)

2. (∃x)Px

∴ (∃x)Qx


3. Pa         E.I. 2 (a is a new flagged name)

4. Pa ⊃ Qa   U.I. 1

5. Qa         modus ponens 3, 4

6. (∃x)Qx   E.G. 5

Exercise 2

1. (∀x)(Px ⊃ Qx)

2. (∀x)(Qx ⊃ Rx)

3. (∃x)Px

∴ (∃x)Rx


4. Pa         E.I. 3 (a is a new flagged name)

5. Pa ⊃ Qa   U.I. 1

6. Qa         modus ponens 4, 5

7. Qa ⊃ Ra   U.I. 2

8. Ra         modus ponens 6, 7

9. (∃x)Rx   E.G. 8

Exercise 3

1. ¬(∃x)Px

2. (∃x)Qx

∴ (∃x)(Qx • ¬Px)


3. (∀x)¬Px     Q.N. 1

4. Qa           E.I. 2 (a is a new flagged name)

5. ¬Pa         U.I. 3

6. Qa • ¬Pa   conjunction 4, 5

7. (∃x)(Qx • ¬Px)   E.G. 6

Exercise 4

1. (∀x)(Px ⊃ Qx)

2. (∀x)(Qx ⊃ Rx)

∴ (∀x)(Px ⊃ Rx)


3. Let a be arbitrary.

4. Pa ⊃ Qa   U.I. 1

5. Qa ⊃ Ra   U.I. 2

6. Assume Pa

7. Qa         modus ponens 4, 6

8. Ra         modus ponens 5, 7

9. Pa ⊃ Ra   conditional proof 6–8

10. (∀x)(Px ⊃ Rx)   U.G. 9

That last one is the important one. It shows why flagging matters. You are only allowed to generalize from a because a was treated as arbitrary rather than as some special individual smuggled in from elsewhere.

Next time, we can either do a few more quantifier-rule exercises or move on to slightly nastier proofs that combine the quantifier rules with things like reductio and conditional proof. Predicate logic only gets more annoying from here, but at least now the annoyingness is beginning to take shape.

Predicate Logic Practice 2: The Four Quantifier Rules

Difficulty: What the hell

Today we move from baby predicate logic’s quantified-negation rules to the four quantifier rules themselves:

  • universal instantiation (U.I.)
  • universal generalization (U.G.)
  • existential instantiation (E.I.)
  • existential generalization (E.G.)

These are the rules that let you move between quantified statements and statements about particular individuals. And yes, this is also where flagging shows up and starts annoying people.

The basic idea

Predicate logic talks not just about whether P and Q are true, but about whether things are true of all individuals or of some individual. That is why we need quantifiers in the first place:

(∀x)Px = everything is P

(∃x)Px = something is P

The quantifier rules tell you when you are allowed to move from those quantified claims to singular claims like Pa, and when you are allowed to move back up again.

1. Universal Instantiation (U.I.)

U.I. is the easy one.

(∀x)Px

∴ Pa

If everything is P, then of course a is P. And likewise:

(∀x)(Px ⊃ Qx)

∴ Pa ⊃ Qa

Whatever is true of everything can be applied to any particular individual you like.

2. Existential Generalization (E.G.)

E.G. is also pretty easy.

Pa

∴ (∃x)Px

If a is P, then something is P. Likewise:

Pa • Qa

∴ (∃x)(Px • Qx)

If some particular individual has the relevant properties, then there exists at least one thing with those properties.

3. Existential Instantiation (E.I.)

This is where the mild irritation begins.

Suppose you know:

(∃x)Px

You are allowed to introduce a new name, say a, and write:

Pa

But only under a restriction: a has to be a new flagged name.

What does that mean? It means you are not allowed to pretend that the existential claim is about some already familiar individual. You are just saying: “Fine, let’s call the thing whose existence is guaranteed by the premise ‘a’.” But that ‘a’ is temporary and special. It is standing in for an arbitrary witness to the existential claim.

So E.I. looks like this:

(∃x)Px

∴ Pa     (a is a new flagged name)

Why the flag? Because if you are sloppy here, you can prove ridiculous garbage. Existential claims give you some individual, not necessarily one you were already talking about.

4. Universal Generalization (U.G.)

U.G. goes the other direction:

Pa

∴ (∀x)Px

But again, there is a restriction. You are only allowed to do this when a is a properly flagged arbitrary individual.

In other words, you cannot just notice that Socrates is mortal and then conclude that everyone is mortal. That would be insane. You need a to stand for an arbitrary individual, not a special one. If you prove something about an arbitrary individual, then you are allowed to generalize it to everybody.

So U.G. looks like this:

Pa     (a is arbitrary / properly flagged)

∴ (∀x)Px

So what the hell is flagging?

Flagging is just a way of marking a name as special for proof purposes.

There are two main uses:

  • In E.I., the flagged name is a temporary witness to an existential claim.
  • In U.G., the flagged name must be arbitrary if you want to generalize from it.

So the short version is:

  • E.I. gives you a new flagged individual.
  • U.G. requires an arbitrary flagged individual.

If you blur those roles, predicate logic becomes a mess very quickly.

How Q.N. and C.Q.N. fit in

Q.N. and C.Q.N. often help you get a quantified statement into a form where you can actually use the four quantifier rules.

For example, from:

¬(∃x)Px

you may first use Q.N. to get:

(∀x)¬Px

and only then use U.I. to get:

¬Pa

Likewise, from:

¬(∀x)(Px ⊃ Qx)

you may use C.Q.N. first to get:

(∃x)(Px • ¬Qx)

and then use E.I. to work with:

Pa • ¬Qa

So Q.N. and C.Q.N. are often what let the rest of the machinery start moving.

What students tend to screw up

  • They instantiate an existential claim with a name that is not new.
  • They generalize from a name that was never arbitrary.
  • They forget that Q.N. and C.Q.N. often need to happen before the quantifier rules can be applied nicely.
  • They panic when they see a flagged name and start pushing symbols around like a raccoon in a trash can.

Exercises

Now derive the indicated conclusions. These are not one-step gimmies. Each of them should take at least several steps if you do them cleanly.

Exercise 1

1. (∀x)(Px ⊃ Qx)

2. (∃x)Px

∴ (∃x)Qx

Exercise 2

1. (∀x)(Px ⊃ Qx)

2. (∀x)(Qx ⊃ Rx)

3. (∃x)Px

∴ (∃x)Rx

Exercise 3

1. ¬(∃x)Px

2. (∃x)Qx

∴ (∃x)(Qx • ¬Px)

Exercise 4

1. (∀x)(Px ⊃ Qx)

2. (∀x)(Qx ⊃ Rx)

∴ (∀x)(Px ⊃ Rx)

That last one is where the idea of an arbitrary flagged individual really starts to matter.

Next time, I’ll probably post the answers. Until then, try not to let the quantifiers and flags make you spiral.

Predicate Logic Practice 1: Q.N. and C.Q.N. Rules (Answers)

Difficulty: What the hell

Here are the answers to our first baby predicate-logic practice. As you probably noticed, Exercises 1 and 2 are basically one-step applications of Q.N. That’s fine. They’re warm-ups. Exercise 3 can also be done as a one-step application of C.Q.N., but I want to make it a little more interesting by breaking it down into multiple steps.

Exercise 1

1. ¬(∀x)Px

∴ (∃x)¬Px


2. (∃x)¬Px         Q.N. 1

Exercise 2

1. ¬(∃x)Qx

∴ (∀x)¬Qx


2. (∀x)¬Qx         Q.N. 1

Exercise 3

1. ¬(∀x)(Px ⊃ Qx)

∴ (∃x)(Px • ¬Qx)


2. (∃x)¬(Px ⊃ Qx)         Q.N. 1

3. (∃x)¬(¬Px ∨ Qx)       conditional exchange 2

4. (∃x)(¬¬Px • ¬Qx)     De Morgan’s law 3

5. (∃x)(Px • ¬Qx)         double negation 4

Of course, Exercise 3 could also have been done in one step by applying C.Q.N. directly. But there’s something satisfying about watching the thing unfold instead of just smashing the correct rule on it like a button.

Next time, we’ll look at the four quantifier rules: universal instantiation, universal generalization, existential instantiation, and existential generalization—along with flagging, which is where predicate logic starts getting a little more annoying. I’m rusty on that stuff myself at the moment, so I’ll have to brush up before pretending to teach it. Until next time, then.

Predicate Logic Practice 1: Q.N. and C.Q.N. Rules

Difficulty: What the hell

Welcome to the beginning of baby predicate logic.

Predicate logic looks scarier than baby propositional logic because it adds quantifiers like “for all” and “there exists.” But today’s topic is actually manageable. We are just looking at two important families of equivalences: quantifier negation (Q.N.) and categorical quantifier negation (C.Q.N.).

The basic idea is simple: once you negate a quantified statement, the quantifier flips. But there is a bit more than the two bare-bones rules I gave earlier, so let’s do this more carefully.

Q.N. Rules

These are the four standard Q.N. equivalences:

1. ¬(∀x)Px ≡ (∃x)¬Px

2. ¬(∃x)Px ≡ (∀x)¬Px

3. ¬(∀x)¬Px ≡ (∃x)Px

4. ¬(∃x)¬Px ≡ (∀x)Px

In words, those amount to the following:

  • It is not the case that everything is P = something is not P.
  • It is not the case that something is P = everything is not P.
  • It is not the case that everything is not P = something is P.
  • It is not the case that something is not P = everything is P.

The last two are just the first two with the negation already built into the predicate.

For example, if Px means “x is pissed,” then:

¬(∀x)Px ≡ (∃x)¬Px

translates as:

It is not the case that everyone is pissed = someone is not pissed.

And:

¬(∃x)Px ≡ (∀x)¬Px

translates as:

It is not the case that someone is pissed = nobody is pissed.

C.Q.N. Rules

The C.Q.N. rules are just more specific quantified negation rules for the standard categorical forms. Lest you are intimidated by scary-looking symbols, we are using P’s and Q’s instead of Greek letters:

1. ¬(∀x)(Px ⊃ Qx) ≡ (∃x)(Px • ¬Qx)

2. ¬(∃x)(Px • Qx) ≡ (∀x)(Px ⊃ ¬Qx)

3. ¬(∀x)(Px ⊃ ¬Qx) ≡ (∃x)(Px • Qx)

4. ¬(∃x)(Px • ¬Qx) ≡ (∀x)(Px ⊃ Qx)

In ordinary English, those are:

  • It is not the case that all P are Q = some P are not Q.
  • It is not the case that some P are Q = no P are Q.
  • It is not the case that no P are Q = some P are Q.
  • It is not the case that some P are not Q = all P are Q.

What students tend to screw up

The most common mistake is to negate a quantified statement without flipping the quantifier. Don’t do that.

For instance, if you start with:

¬(∀x)Px

you are not allowed to conclude:

(∀x)¬Px

Why not? Because “not everyone is pissed” does not mean “everyone is not pissed.” It only means that at least one person is not pissed.

But if you start with:

¬(∃x)(Px • ¬Qx)

you are allowed to conclude:

(∀x)(Px ⊃ Qx)

because “it is not the case that some P are not Q” just means “all P are Q.”

A quick summary

For plain Q.N.:

¬(∀x)Px ≡ (∃x)¬Px

¬(∃x)Px ≡ (∀x)¬Px

¬(∀x)¬Px ≡ (∃x)Px

¬(∃x)¬Px ≡ (∀x)Px

For categorical forms:

¬(∀x)(Px ⊃ Qx) ≡ (∃x)(Px • ¬Qx)

¬(∃x)(Px • Qx) ≡ (∀x)(Px ⊃ ¬Qx)

¬(∀x)(Px ⊃ ¬Qx) ≡ (∃x)(Px • Qx)

¬(∃x)(Px • ¬Qx) ≡ (∀x)(Px ⊃ Qx)

Negate the statement, flip the quantifier, and then make sure the predicate or categorical form comes out right.

Exercises

Now derive the indicated conclusions. These are easy on purpose. Predicate logic is annoying enough already.

Exercise 1

1. ¬(∀x)Px

∴ (∃x)¬Px

Exercise 2

1. ¬(∃x)Qx

∴ (∀x)¬Qx

Exercise 3

1. ¬(∀x)(Px ⊃ Qx)

∴ (∃x)(Px • ¬Qx)

I’ll probably post the answers tomorrow. Have fun, and don’t let the quantifiers bully you.

Baby Logic Practice 3: Conditional Proofs (Answers)

Difficulty: What the hell

Here are the answers to Baby Logic Practice 3. As promised, each proof uses conditional proof, even in cases where there are quicker or more boring ways to get the conclusion.

Exercise 1

1. P → Q

∴ P → Q


2. Assume P

3. Q modus ponens 1, 2

4. P → Q conditional proof 2–3

Exercise 2

1. P → Q

2. Q → R

∴ P → R


3. Assume P

4. Q modus ponens 1, 3

5. R modus ponens 2, 4

6. P → R conditional proof 3–5

Exercise 3

1. P ∧ Q

∴ P → Q


2. Assume P

3. Q simplification 1

4. P → Q conditional proof 2–3

Exercise 4

1. P → (Q → R)

2. P

∴ Q → R


3. Assume Q

4. Q → R modus ponens 1, 2

5. R modus ponens 3, 4

6. Q → R conditional proof 3–5

Exercise 5

1. P → Q

2. R → S

3. P ∧ R

∴ P → S


4. Assume P

5. R simplification 3

6. S modus ponens 2, 5

7. P → S conditional proof 4–6

Exercise 6

1. P → Q

2. Q → R

3. R → S

∴ P → S


4. Assume P

5. Q modus ponens 1, 4

6. R modus ponens 2, 5

7. S modus ponens 3, 6

8. P → S conditional proof 4–7

Exercise 7

1. P → Q

2. P → R

∴ P → (Q ∧ R)


3. Assume P

4. Q modus ponens 1, 3

5. R modus ponens 2, 3

6. Q ∧ R conjunction 4, 5

7. P → (Q ∧ R) conditional proof 3–6

Exercise 8

1. P → Q

2. Q → (R ∨ S)

3. ¬R

∴ P → S


4. Assume P

5. Q modus ponens 1, 4

6. R ∨ S modus ponens 2, 5

7. S disjunctive syllogism 3, 6

8. P → S conditional proof 4–7

So there you have it—the answers to our third baby logic quiz. Conditional proof is simple in principle, but it gets nicer once assuming the antecedent starts to feel automatic.

Baby Logic Practice 2: Derive the Conclusions Using Reductio (Answers)

Difficulty: What the hell

Here are the answers to Baby Logic Practice 2. As promised, each proof uses reductio ad absurdum, even in cases where there are quicker ways to get the conclusion.

Exercise 1

1. P → Q

2. ¬Q

∴ ¬P


3. Assume P

4. Q modus ponens 1, 3

5. ⊥ contradiction 2, 4

6. ¬P reductio 3–5

Exercise 2

1. P ∨ Q

2. ¬P

∴ Q


3. Assume ¬Q

4. ¬P ∧ ¬Q conjunction 2, 3

5. ¬(P ∨ Q) De Morgan’s Law, 4

6. ⊥ contradiction 1, 5

7. Q reductio 3–6

Exercise 3

1. (P ∧ Q) → R

2. P

3. ¬R

∴ ¬Q


4. Assume Q

5. P ∧ Q conjunction 2, 4

6. R modus ponens 1, 5

7. ⊥ contradiction 3, 6

8. ¬Q reductio 4–7

Exercise 4

1. P → Q

2. Q → R

3. ¬R

∴ ¬P


4. Assume P

5. Q modus ponens 1, 4

6. R modus ponens 2, 5

7. ⊥ contradiction 3, 6

8. ¬P reductio 4–7

Exercise 5

1. P → (Q ∨ R)

2. ¬Q

3. ¬R

∴ ¬P


4. Assume P

5. Q ∨ R modus ponens 1, 4

6. ¬Q ∧ ¬R conjunction 2, 3

7. ¬(Q ∨ R) De Morgan’s Law, 6

8. ⊥ contradiction 5, 7

9. ¬P reductio 4–8

Exercise 6

1. P ↔ Q

2. ¬Q

∴ ¬P


3. P → Q biconditional elimination, 1

4. Assume P

5. Q modus ponens 3, 4

6. ⊥ contradiction 2, 5

7. ¬P reductio 4–6

Exercise 7

1. P ∨ Q

2. P → R

3. Q → R

4. ¬R

∴ ¬P ∧ ¬Q


5. Assume P

6. R modus ponens 2, 5

7. ⊥ contradiction 4, 6

8. ¬P reductio 5–7

9. Assume Q

10. R modus ponens 3, 9

11. ⊥ contradiction 4, 10

12. ¬Q reductio 9–11

13. ¬P ∧ ¬Q conjunction 8, 12

Exercise 8

1. (P ∨ Q) → R

2. ¬R

∴ ¬P


3. Assume P

4. P ∨ Q addition 3

5. R modus ponens 1, 4

6. ⊥ contradiction 2, 5

7. ¬P reductio 3–6

So there you have it—the answers to our second baby logic quiz. Reductio can feel a little roundabout, but that’s part of what makes it useful: it forces you to see exactly where an assumption blows up.

Is the Fallacy Fallacy a Formal or Informal Fallacy?

Difficulty: What the heck

Is the fallacy fallacy a formal or informal fallacy?

Answer: it depends.

But first, briefly: the fallacy fallacy is the fallacy whereby the arguer concludes that an argument’s conclusion is false simply because the argument used to support it is fallacious.

For instance:

Debater 1: Trump is not evil, because you’re a cow.

Debater 2: Aha! That’s an ad hominem argument, and that’s fallacious. So you’re wrong that Trump isn’t evil. Therefore, Trump is evil.

Here, Debater 2 commits the fallacy fallacy by arguing that because Debater 1’s argument is a fallacious ad hominem abusive argument, Debater 1’s contention that Trump is not evil must therefore be false.

But the issue is this: just because someone’s reasoning is fallacious does not mean her conclusion is necessarily false. It does not mean it is true, either. It just means that the conclusion does not logically follow from that argument.

Think about it this way: if a kid were asked to do a multiplication problem—say, “2 × 2”—and, not knowing how to multiply, she used addition instead, she would still arrive at the right answer, “4,” even though her reasoning process would be wonky. It is the same way with logic and argumentation. A bad argument can accidentally land on a true conclusion.

However, the question I’ll have you ponder today is whether the fallacy fallacy is a formal or informal fallacy.

One way to interpret the fallacy fallacy is as relying on the following argument form:

Premise 1: If an argument is non-fallacious, then its conclusion is true.

Premise 2: This argument is fallacious.

Conclusion: Therefore, its conclusion is false.

If that is how the arguer is thinking, then sure: the fallacy fallacy can be understood as a formal fallacy. More specifically, it would have the form of denying the antecedent.

But here is another way to look at it. The arguer may instead be reasoning with a perfectly valid form—namely, modus ponens—while relying on a false premise:

Premise 1: If an argument is fallacious, then its conclusion is false.

Premise 2: This argument is fallacious.

Conclusion: Therefore, its conclusion is false.

If this is how Debater 2 is thinking, then the form itself is not the problem. The form is valid. The problem lies in Premise 1, which is false. In that case, the fallacy fallacy would be better understood as an informal fallacy.

So, is the fallacy fallacy a formal or informal fallacy?

Again: it depends on what, exactly, has gone wrong in the person’s reasoning.

And why does this distinction matter?

Because clarity matters. And because if we want to understand our own fallible thought processes, it helps to know whether the problem lies in the form of the reasoning or in the content of what we are assuming. But the psychology behind shitty logical reasoning is a topic for another day. Until next time, then.

Baby Logic: What the Heck are Sufficient and Necessary Conditions?

Difficulty: What the heck

In logic, words like “if,” “only if,” “if and only if,” and “unless” express different logical relationships. So if you conflate those terms and symbolize them incorrectly, you end up with confused garbage. Today’s post is on what “if,” “only if,” “if and only if,” and “unless” mean in logic.

“If”

This one is pretty straightforward. When we say “If P, then Q,” we mean that P is a sufficient condition for Q. That means that if P is true, that is enough for us to say that Q is true.

For instance, in the sentence “If Einstein is a parrot, then he is a bird,” Einstein’s being a parrot is sufficient for us to say that he is a bird. We symbolize a statement like this as follows:

P → Q

In this conditional, P is called the antecedent, and Q is called the consequent.

“Only if”

Now, that is logically equivalent to saying, “Einstein is a parrot only if he is a bird.” In ordinary language, whatever immediately follows “only if” is the necessary condition.

So the sentence “Einstein is a parrot only if he is a bird” means that being a bird is necessary for being a parrot. In other words, if it is true that Einstein is a parrot, then it must also be true that Einstein is a bird. So, once again, we symbolize the sentence like this:

P → Q

This is one reason people get confused: the phrase “only if” often makes them want to reverse the conditional. Don’t. The thing after “only if” gives you the necessary condition, not the antecedent.

Also note that this is equivalent to saying, “Only if Einstein is a bird is he a parrot.” That sounds more awkward, but the logical relationship is the same.

“If and only if”

Now consider the following sentence:

I will kick your ass if and only if you kick my ass.

This is what is called a biconditional. It means that each side is both necessary and sufficient for the other. In other words, the sentence can be broken down into two conditionals:

If you kick my ass, I will kick your ass.

If I kick your ass, you will kick my ass.

These can be symbolized as follows:

P → Q

Q → P

And those two together can be symbolized like this:

P ↔ Q

So “if and only if” means both directions hold. That is why it is stronger than plain old “if.”

“Unless”

And then there is “unless,” which seems like a pain in the ass to symbolize, but is not that hard once you get the hang of it. A useful rule of thumb is that “unless” can often be translated as “if not.”

So, “You will die unless you upload your brain to a computer” is logically equivalent to, “If you do not upload your brain to a computer, you will die.” This can be symbolized as:

¬B → D

where ‘¬’ means ‘not’, ‘B’ = ‘you upload your brain to a computer’, and ‘D’ = ‘you will die’.

So you can think of “unless” this way: “Unless B, D” means “If not B, then D.”

One last quick summary

Here is the baby-logic version:

  • If = sufficient condition
  • Only if = necessary condition
  • If and only if = necessary and sufficient condition
  • Unless = usually easiest to symbolize as “if not”

If you keep those straight, you will already be ahead of a whole lot of students who mangle conditionals into logical mush.

Apologies for the morbid tone today. Oh well. Until next time, then!

Is Philosophy Bullshit?

Difficulty: What the heck

Some philosophy is absolutely bullshit.

There is pseudo-profound nonsense. There is empty jargon. There is status-signaling dressed up as rigor. There are philosophers and graduate students who perform seriousness without being genuinely guided by the standards of inquiry they invoke. So if by “philosophy” you mean that sort of thing, then yes, philosophy can be bullshit.

But it does not follow that philosophy itself is bullshit any more than the existence of bad science shows that science itself is bullshit. The real question is whether philosophy, at its best, is a useless exercise in verbal fog—or whether it does something intellectually and practically significant.

My answer is: no, philosophy is not bullshit. At least, not when it is done properly.

To see why, we need to make one annoying but necessary move: we need to get clearer about what we mean by both “bullshit” and “philosophy.” Otherwise, people end up yelling past one another. One person means pseudo-profound academic fog. Another means logic, ethics, political theory, or the philosophy of language. One person means empty performance. Another means disciplined reflection on reasons, concepts, and arguments. Unless we sort that out, the question “Is philosophy bullshit?” is too muddy to answer well.

Now, trying to define philosophy in a perfectly neat way is hard, just as trying to define science in a perfectly neat way is hard. Science includes physics, chemistry, and biology, but also things like geology, medicine, neuroscience, and perhaps at least some of the social sciences. Philosophy is similarly sprawling. It includes logic, epistemology, ethics, metaphysics, political philosophy, philosophy of language, philosophy of mind, and much else besides. So instead of hunting for some magical one-sentence definition, it is more useful to ask: what does philosophy do when it is working properly?

At its best, philosophy clarifies concepts, tests reasons, exposes hidden assumptions, and asks what follows from what. It tries to distinguish what merely sounds plausible from what actually makes sense. It trains us not just to have opinions, but to examine them. And that, I think, is already enough to show why philosophy is not simply bullshit.

To answer whether philosophy is bullshit, then, we also need at least a rough grip on what bullshit is.

Let’s clear away a few common misconceptions.

Bullshit is not the same thing as lying.

Bullshit is not the same thing as falsehood.

On Harry Frankfurt’s account, bullshit is, roughly, a product of someone who is indifferent to the truth but pretends otherwise, misrepresenting herself as someone who cares. So a student who bullshits on an essay to reach a minimum word count may do so not by lying or writing what is false, but by not caring at all whether what she writes is true. She merely wants to appear as if she gives a shit so that she can get a decent score.

On my account, bullshit is, broadly put, the empty performance of norm-guided speech or conduct without being genuinely guided by the norms that make the practice intelligible. A few of the norms that make philosophy intelligible, for example, are clear argumentation, logical reasoning, and openness to criticism or objections. A philosopher who merely performs those norms while refusing to be genuinely guided by them is no longer engaged in serious inquiry. At that point, what we are seeing is not actual philosophy, but bullshit.

Now, what many people seem to mean when they say philosophy is bullshit is that philosophy is useless. But that opens another can of worms, because we would then have to clarify what “useless” means.

If by “useless” they mean that philosophy does not reliably lead to a high-paying job, then sure: philosophy is not a guaranteed path to wealth. But it would be a leap to say that because many philosophy majors do not become rich, philosophy is therefore bullshit. Poor career outcomes, even when they exist, do not show that a discipline is intellectually empty.

And even on practical grounds, philosophy is not obviously useless. Because philosophical inquiry emphasizes reading, writing, argumentation, and logical reasoning, philosophy majors tend to perform very well on standardized tests like the GRE and LSAT, and many go on to law school and careers in law, policy, business, and education. The ancient story of Thales of Miletus makes the same point in a more amusing way. According to Aristotle, Thales, annoyed at those who accused philosophers of being useless, used his intelligence to invest in the olive industry and made a fortune. “[Thales] proved,” Aristotle writes, “that philosophers can easily be wealthy if they wish, but this is not what they are interested in.”

So what are philosophers interested in?

They are interested in exercising their minds the way athletes exercise their bodies. They do this not to guarantee victory in every situation, but to build habits of discipline, responsiveness, and control that matter when things get difficult.

And I do not mean only intellectually difficult, though philosophy is certainly helpful for that. I also mean difficult in the more painful sense: your life turns to shit, and you are barely holding on.

In the 1960s, when U.S. naval officer James Stockdale was captured and tortured by North Vietnamese forces, he applied the Stoic philosophy he had learned at Stanford to remain sane. When Marcus Aurelius led the Roman Empire, he drew on the same philosophical tradition to maintain equanimity while confronting war, plague, and political burden.

So I am not saying that philosophy matters only in classrooms or journals. At one of the worst moments of my life, a philosopher once gave me a simple argument that helped keep me alive.

Sometimes, clear thinking is not an academic luxury. Sometimes, it is what makes life bearable. And often, it is what makes life worthwhile.

So, is philosophy bullshit?

Sometimes, yes—when it degenerates into pseudo-profound fog, empty performance, or jargon without guidance. But philosophy itself is not bullshit when it does what it is supposed to do: clarify, test, examine, and help us think more clearly about what matters.

Try it seriously, and that much should become obvious.

What the Hell “Begging the Question” Really Means

Difficulty: What the hell

So it’s been around 18 years since my first girlfriend dumped me, but while I have long gotten over it, I have not come to terms with the stupidity of her argumentative strategy. Consider the first part of her argument:

“What your father does is wrong. Why? Because I feel really strongly about it.”

I asked her why she feels “really strong about it.” She responded:

“Because it’s wrong.”

This circular reasoning—this is what philosophers call begging the question—went on for a good fifteen or thirty minutes.

Today, we’re looking at the informal fallacy of begging the question. You just read an obvious example. Now, we’re going to look at some subtler instances.

Christopher W. Tindale has us consider, in his Fallacies and Argument Appraisal, the following arguments:

Example 1:

A heavier-than-air craft could never fly because in order to lift up and travel over distance a machine would have to be lighter than the environs surrounding it.

Example 2:

God is the only perfect being and perfection includes all the virtues. So, we know that God is benevolent.

He then asks two critical questions to identify instances of begging the question:

Critical question 1:

Has the arguer avoided the obligation to provide independent support for a claim by restating it in similar terms?

Critical question 2:

Has an arguer avoided the obligation to provide independent support by assuming somewhere in the premises the very thing that has to be shown?

I’m going to add one more question to the list:

Critical question 3:

Do the premise(s) and conclusion depend on each other for justification?

These questions will help us articulate how Example 1, Example 2, and the ex-girlfriend example beg the question.

First, there is no real question when someone begs the question

Let’s disabuse ourselves of a potential source of misunderstanding. Nobody’s really asking a question when they commit the “begging the question” fallacy. Here, the word “question” is better understood as the “issue at hand.” If, for instance, a Bible thumper tells you,

“God exists because the Bible says so, and the Bible is the word of God”

it’s not that anyone is asking a question. It’s that the arguer presumes the issue at hand—namely, whether God exists—to be true, when that is exactly what she needs, but fails, to independently support.

In other words, the premise “the Bible is the word of God” hides the assumption that God exists. So duh, if she cheats like that, of course she can (fallaciously) reach the conclusion that God exists. She’s basically saying:

God exists because God exists.

Restating a claim in similar terms

Here’s critical question 1 again:

Has the arguer avoided the obligation to provide independent support for a claim by restating it in similar terms?

Now revisit Example 1:

A heavier-than-air craft could never fly because in order to lift up and travel over distance a machine would have to be lighter than the environs surrounding it.

This one’s sneaky. But yes—the arguer has avoided the obligation to provide independent support by restating the claim in different terms.

Briefly:

“Heavier-than-air craft” ≈ “machine” that is not “lighter than the surrounding air”

“Fly” ≈ “lift up and travel over distance”

These expressions are not strictly identical, but they function the same way in the argument.

So the arguer hasn’t explained anything. She’s just saying:

A heavier-than-air craft cannot fly because a heavier-than-air craft cannot fly.

Sneaking the conclusion into the premises

Reconsider this argument:

God is the only perfect being and perfection includes all the virtues. So, we know that God is benevolent.

Sounds good, right? Good my ass.

Rephrased, the argument goes like this:

Premise 1: God is the only perfect being.

Premise 2: A perfect being has all the virtues (including benevolence).

Conclusion: God is benevolent.

This looks like an argument, but it isn’t doing any real work. The conclusion is already built into the premises.

So while it doesn’t look like “G, therefore G,” that’s basically what’s going on under the hood.

Mutually dependent premise and conclusion

Now back to the ex-girlfriend argument:

I feel really strongly about what your father does. Therefore, what your father does is wrong.

What your father does is wrong. Therefore, I feel really strongly about it.

If this were an Escher drawing, it’d be a masterpiece.

Recall critical question 3:

Do the premise(s) and conclusion depend on each other for justification?

Here, the answer is yes.

Each claim is being used to justify the other. That’s classic circular reasoning.

You might be tempted to ask whether circular reasoning and begging the question are really the same thing. Fair question—but we don’t need to settle that here. For now, it’s enough to see how these arguments go wrong.

A psychological note on begging the question

In A System of Logic, John Stuart Mill observes that people are unlikely to commit this fallacy in their own private reasoning, but they do commit it in dialogue.

My guess is this: people know their reasons are weak, but instead of admitting it, they just restate their conclusion in slightly different words and hope no one notices.

Don’t do that.