Monday, May 28, 2012

Introductory Logic, Part 1

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This is part 1 of my series on logic and critical thinking.
  1. Introductory Logic, Part 1—Introducing both logic in general (such as the difference between a deductive and inductive argument) and propositional logic in particular
  2. Introductory Logic, Part 2—More propositional logic
  3. Introductory Logic, Part 3—A defense of the material conditional

Bait

This post begins my series on logic and critical thinking. First a little bait for both atheists and theists. For atheists, here’s an argument from evil:
  1. If God exists, then gratuitous evil does not exist (we can define gratuitous evil as evil from which no greater good results from its existence).
  2. Gratuitous evil does exist.
  3. Therefore, God does not exist.
In logic, an argument uses a series of statements called premises (like lines 1 and 2 above) that attempts to support a conclusion (like line 3 above). Whether the premises in the above argument are true or not, one can use symbolic logic to prove that the conclusion follows from the premises, and this article will show how an atheist can do that. For theists, here’s a moral argument:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
Whether the premises are true or not, one can use symbolic logic to prove that the conclusion follows from the premises, and this article will show how a theist can do that.

And now that I have your attention...

In my previous blog entry I used symbolic logic to help analyze the “Could an omnipotent being create an immovable stone?” paradox, which helps illustrate the importance of logical and analytical thinking. Hopefully if it’s one thing theists and atheists can agree on it is the importance of logic as well as analytical and critical thinking. Part of what inspired this is seeing some atheists claim that a conclusion didn’t follow from premises when anyone with basic training in logic would’ve seen that the conclusion did indeed follow. To be fair, there are no doubt some theists out there who have likewise made embarrassing logical errors.

In this article I’ll explain the difference between deductive and inductive arguments as well as introducing some easy-to-learn symbolic logic. Why symbolic logic? Learning symbolic logic significantly helps one learn to think logically and for getting an intuitive feel for how an argument’s conclusion can follow logically from the premises. Symbolic logic is also great for introducing various rules of logic.

Deductive Arguments

A deductive argument tries to show that it’s logically impossible (i.e. self-contradictory, like a married bachelor) for the argument to have true premises and a false conclusion, and thus that the conclusion follows from the premises by the rules of logic. If it’s logically impossible for an argument to have true premises and a false conclusion the argument is deductively valid or valid. An example of a deductively valid argument:
  1. If it is raining, then my car is wet.
  2. It is raining.
  3. Therefore, my car is wet.
The above example uses a famous rule of logic called modus ponens which has this structure:
  1. If P, then Q
  2. P
  3. Therefore, Q.
Another famous rule of logic is called modus tollens where “not-Q” means “Q is false.”
  1. If P, then Q
  2. Not-Q
  3. Therefore, not-P.
An argument is deductively invalid or invalid if it is not deductively valid. An example of an invalid argument:
  1. If it is raining, then my car is wet.
  2. My car is wet.
  3. Therefore, it is raining.
In logic lingo, a deductively valid argument with all its premises being true is called a sound argument. And since a valid argument having true premises guarantees the truth of its conclusion, a sound deductive argument has a true conclusion.

Inductive Arguments

The symbolic logic I’ll deal with in this article will deal with deductive arguments, but it’s worth talking about another type of argument: the inductive argument. Whereas a deductive argument attempts to have premises that (if true) guarantee the truth of the conclusion, an inductive argument has premises that are intended to make probable the conclusion without guaranteeing it. There are two types of inductive arguments. One type is called an enumerative inductive argument and it’s basically the type of reasoning scientists use to establish physical laws, generalizing a truth about past experiences to future experiences. An example:
  • We’ve seen the law of gravity work in conditions C1.
  • We’ve seen the law of gravity work in conditions C2.
  • We’ve seen the law of gravity work in conditions C3.
  • We’ve seen the law of gravity work in conditions Cn.
  • Therefore, (probably) the law of gravity holds true in all physical situations.
Some enumerative induction arguments take on the following form:
  • All observed A’s are B’s.
  • x is an A.
  • Therefore, probably, x is a B.
Another example of enumerative inductive logic:
  • We’ve sampled a thousand animals of this species in this forest and three-fifths of them have black fur.
  • Therefore, probably three-fifths of the animals of this species in this forest have black fur.
Terminology warning: some people define “induction” to mean only the first sort of enumerative induction rather than including the “population sampling” sort. Here I’m using the broader definition of inductive reasoning as an argument with premises that are intended to make the conclusion more probable.

Another kind of inductive argument is called inference to the best explanation. This is where out of a pool of live options, a certain explanation is selected as the best and most rational one. When comparing competing theories, some factors include but are not limited to the following:
  • Explanatory power refers to making the data probable. A theory having more explanatory power than another means it makes the data more probable (e.g. given our background knowledge that doesn’t include data D, it’s more likely that we’d see data D on theory #1 than on theory #2).
  • Explanatory scope refers to how many things a theory explains. A theory has more explanatory scope than another if it explains more things.
  • Simplicity utilizes a number of factors, but among them is Ockham’s razor (also spelled Occam’s razor) that says to not multiply explanatory entities beyond necessity. All other relevant factors held constant, simpler explanations are preferred over more complex ones.
Inference to the best explanation is also called abductive reasoning or abduction. Some writers define the terms somewhat differently, defining induction to mean just the first case of enumerative induction and use the term “abduction” to distinguish between the more narrowly defined “induction.” A less ambiguous catch-all term for arguments that are not deductive is nondeductive. In this article, I’ll consider “inference to the best explanation” as a sort of inductive reasoning.

Inductive arguments are by nature invalid, but if the premises of an inductive argument make the conclusion highly probable it is said to be an inductively strong argument (another term I’ve seen used for this is nondeductively valid). The strength of an inductive argument comes in varying degrees, from making the conclusion highly probable to not at all probable (the latter case results in the argument being inductively weak). A strong inductive argument with true premises is called a cogent argument.

Symbolic Logic: Some Symbols

In this article I’ll talk about some symbolic with regards to deductive argument. Some symbols:

ConnectiveSymbolic
Logic
English
Meaning
Notes
∧ (conjunction)p ∧ qp and qThe p and q parts are called conjuncts.
∨ (disjunction)p ∨ qp or qThe p and q parts are called disjuncts.
→ (conditional)p → qIf p, then qThe p part is called the antecedent and the q part is called the consequent. Sometimes p → q is read as “p implies q.”
↔ (biconditional)p ↔ qp, if and only if qThis means the same thing as “p → q and q → p.”
¬ (negation)¬pNot-pThe negation of P is ¬P, and ¬P means “not-P” or “P is false.”


As suggested in the above table, the symbols →, ¬, ∨, and ∧ are called connectives. It’s a somewhat misleading name since ¬ doesn’t connect propositions even though the other three connectives do. Still, it’s a popular label used by a lot of logic textbooks. The type of symbolic logic I’ll discuss here uses letters to represent propositions. While the terminology varies among writers, I’ll call a single letter a simple statement and one more or more simple statements with one or more connectives is called a compound statement. For example, “¬P” and “A ∧ B” are compound statements.

Oddly enough, the symbols for the connectives are not as standard as one might think (though in my experience the disjunction operation being symbolized as ∨ is pretty standard). Below are some alternate forms:

OperationExampleAlternate Forms
conjunctionA ∧ BA & B, A • B, AB
implicationA → BA ⊃ B
material equivalenceA ↔ BA ≡ B
negation¬A~A, –A, A


In the system of logic I’m using (called propositional logic), simple statements are true or false. What makes a compound statement true? Here’s a summary of how the connectives work in making a compound statement true/false:

Type of
connective
EnglishSymbolic
Logic
When it’s true/false
Conjunctionp and qp ∧ qTrue if both are true; otherwise false
Disjunctionp or qp ∨ qFalse if both are false; otherwise true
ConditionalIf p, then qp → qFalse if p is true and q is false; otherwise true
Biconditionalp, if and only if qp ↔ qTrue if both have the same truth-value (i.e. both are true or both are false); otherwise false
NegationNot-p¬pTrue if p is false; false if p is true


We can also use truth tables like those below to express what makes a compound statement true/false, where T symbolizes true and F symbolizes false.

Truth Table of Connectives

pq   pq   pq   pq   ¬p
TTT   TTT   TTT   TTT   TF
TFF   TTF   TFF   TFF   FF
FFT   FTT   FTT   FFT     
FFF   FFF   FTF   FFF     


The disjunction (p ∨ q) being used here is an inclusive or, i.e. true if at least one of the disjuncts is true (otherwise false). This is in contrast to an exclusive or, which is true if and only if exactly one disjunct is true.

The type of conditional (p → q) being used here is called a material conditional. A material conditional is equivalent to “It is not the case that the antecedent (p) is true and the consequent (q) is false,” such that the only way for a material conditional to be false is for it to have a true antecedent with a false consequent, as the truth table for it illustrates. When you look at the truth table, a material conditional might seem like a pretty weak claim (in the sense that it doesn’t claim very much), since the antecedent and consequent don’t even have to be related to each other for a material conditional to be true. Thus, “If there is a married bachelor, then Minnesota is awesome” constitutes a true material conditional since it is not the case that we have a true antecedent (there is a married bachelor) with a false consequent (Minnesota is awesome). But it turns out that a material conditional is enough for modus ponens and modus tollens to be valid rules of inference, since in a true material conditional if the antecedent is true, then the consequent is true as well.

It’s worth noting, however briefly, how English can be translated into symbolic logic.

EnglishSymbolic Logic
p is a sufficient condition for qp → q
q is a necessary condition for p
q, if p
p only if q


Notice that the “conversion” from English to the material conditional doesn’t necessarily work so well the other way (material conditional to English), e.g. p → q doesn’t necessarily talk about necessary or sufficient conditions; it just says it isn’t the case that p is true and q is false. On the other hand, if p is a sufficient condition for q, then it follows that p → q.

EnglishSymbolic Logic
p is a necessary and sufficient condition for qp ↔ q
p if and only if q
p just in case q


Odd as it may seem, in the philosophical literature “p just in case q” often means p ↔ q.

Order of Operations and Logical Terminology

Logic uses parentheses in a sort of “order of operations” (if you remember your algebra) to determine what to calculate first. For example, suppose we had this:

(A ∨ B) ∧ C

So if we were going for an “inside to outside” strategy of creating a truth table we’d first determine the truth-value (i.e. whether it’s true/false) of (A ∨ B) as opposed to checking whether B ∧ C is true. If we need more groupings inside other groupings, we first use brackets [] and then braces {} like so:

¬{[(A ∨ B) ∧ C] → D}

Another note: the order of operations is that negation is done to the immediate thing the ¬ is connected to. For example, this:

¬P ∨ C

Should be viewed as this:

(¬P) ∨ C

Rather than this:

¬(P ∨ C)

Two statements are said to be logically equivalent if it is logically impossible for one to be true when the other is false. So in propositional logic, two statements are logically equivalent or truth-functionally equivalent if and only if they yield identical truth-values in their truth-tables, and so p → q is logically equivalent to both ¬(p ∧ ¬q) and ¬p ∨ q. To make it easier to follow I’ve filled in the truth-tables in stages:

Stage 1

¬(p¬q)   ¬pq   pq  
 T  T    T T   T T
 T  F    T F   T F
 F  T    F T   F T
 F  F    F F   F F


Stage 2

¬(p¬q)   ¬pq   pq  
 T FT   FT T   TTT
 T TF   FT F   TFF
 F FT   TF T   FTT
 F TF   TF F   FTF


Stage 3

¬(p¬q)   ¬pq   pq  
 TFFT   FTTT   TTT
 TTTF   FTFF   TFF
 FFFT   TFTT   FTT
 FFTF   TFTF   FTF


Stage 4

¬(p¬q)   ¬pq   pq  
TTFFT   FTTT   TTT
FTTTF   FTFF   TFF
TFFFT   TFTT   FTT
TFFTF   TFTF   FTF


In every line of the truth tables, the truth-value is the same for p → q, ¬(p ∧ ¬q), and ¬p ∨ q; thus making all three statements logically equivalent.

Consider also these truth-tables:

Stage 1

p¬p   p¬p
T FT   T FT
F TF   F TF

Stage 2

p¬p   p¬p
TTFT   TFFT
FTTF   FFTF


It is said that a statement is tautologous if the form of the statement is sufficient to make it true, e.g. P ∨ ¬P, which is true for every truth-value assignment. A statement is self-contradictory if affirms both the truth and falsehood of the same thing, hence the law of noncontradiction which says “For any proposition p, it is impossible for p to be both true and false at the same time and in the same context.” In propositional logic, an example of a self-contradiction is P ∧¬P. We also have “truth-functional” synonyms for tautologies and contradictions, as well as other terms:
  • A statement is truth-functionally true if and only if every truth-value assignment makes it true.
    • Example: P ∨ ¬P
  • A statement is truth-functionally false if and only if every truth-value assignment makes it false.
    • Example: P ∧ ¬P
  • Two statements are truth-functionally equivalent if and only if every truth-value assignment gives them the same truth table.
    • Example: P → Q and ¬(P ∧ ¬Q)
  • Two statements are truth-functional contradictories if and only if every truth-value assignment gives them different truth-values.
    • Example: P and ¬P.
  • Two statements are truth-functionally consistent if and only if there is some truth value assignment that makes them both true.
    • Example: P ∨ Q and P ∨¬Q.
  • Two statements are truth-functionally inconsistent if and only if there is no truth-value assignment that maks them both true.
    • Examples: P ∧ Q and P ∧ ¬Q; also P ∧ ¬P and Q.
  • Two statements are truth-functional contraries if and only if there is no truth-value assignment that makes both of them true and there is some truth-value assignment that makes both of them false.
    • Example: P ∧ Q and ¬P ∧ Q.
  • Two statements are truth-functional subcontraries if and only if there is no truth-value assignment that makes both of them false and there is some assignment that makes both of them true.
    • Example: P ∨ Q and ¬P ∨Q.
Notice that if two statements are truth-functionally inconsistent, they are either contradictories or contraries. The words contrary and subcontrary are also used in a more general sense. For example, two propositions are contraries if they can’t both be true but they may both be false, e.g. “This ball is green all over” and “This ball is red all over” are contraries. Similarly, two propositions are subcontraries if they can’t both be false but they may both be true, e.g. “I went to the bathroom or I went to the kitchen” and “I didn’t go to the bathroom or I went to the kitchen” are subcontraries: they can’t both be false (I either went to the bathroom or I didn’t) though they can both be true (it’s possible I went to the kitchen). The term contradictory has a normal language definition too, whereby two statements are contradictory if and only if it is impossible for one to be true when the other is false, e.g. the statements “No cats have fur” and “There exists a cat with fur” are contradictories of each other.

You could prove the following is true via a truth table:
[(P → Q) ∧ P] → Q
But it would suck. It would be nice if you could instead do something like this instead:
  1. P → Q
  2. P

  1. Q 1, 2, [rule of logic used to arrive here]
Thankfully, logic has that. Natural deduction refers to types of proof methods that more closely match the way humans naturally think. There are many systems of natural deduction for a variety of logic systems, but for now I’ll concern myself with the system of natural deduction popularly used in propositional logic. For example:
  1. P → Q
  2. P

  1. Q 1, 2, modus ponens
Modus ponens is a famous rule of logic used in natural deduction.

Symbolic Logic: Some Rules of Inference

Some rules of inference I’ve already mentioned:

modus ponens
 
In English In Symbolic Logic
If p then q
p

Therefore, q
p → q
p

∴ q
modus tollens
 
In English In Symbolic Logic
If p then q
Not-q

Therefore, not-p
p → q
¬q

∴ ¬p


In the convention I’m using, the lower case letters p, q, r,...z are placeholders for both simple and compound statements. Thus, below is a valid instance of modus tollens.
  1. (A ∧ B) → C
  2. ¬C

  1. ¬(A ∧ B) 1, 2, modus tollens
It’s worth noting that the order of the premises doesn’t matter when using rules of inference. So below is also a valid use of modus tollens.
  1. ¬C
  2. (A ∧ B) → C

  1. ¬(A ∧ B) 1, 2, modus tollens
Some rules of inference can be used in more than one way. Examples include disjunctive syllogism and simplification.

Disjunctive Syllogism
 
In English In Symbolic Logic
p or q
Not-p

Therefore, q
p ∨ q
¬p

∴ q
p or q
Not-q

Therefore, p
p ∨ q
¬q

∴ p
simplification
 
In English In Symbolic Logic
p and q

Therefore, p
p ∧ q

∴ p
p and q

Therefore, q
p ∧ q

∴ q


Before moving forward, I’ll introduce a quick example of how to use some rules of inference. Suppose we wanted to get C from premises 1 and 2 below:
  1. A ∨ (B ∧ C)
  2. ¬A

  1. B ∧ C 1, 2, disjunctive syllogism
  2. C 3, simplification
Not too hard, right? After learning the above rules of inference, you might even have mentally “seen” that C followed from premises 1 and 2 above. Hopefully you are familiar enough with the symbols by now for me to remove the training wheels of english translation. Some more rules of inference:

conjunction
 
p
q

∴ p ∧ q
constructive dilemma
 
(p → q) ∧ (r → s)
p ∨ r

∴ q ∨ s


hypothetical syllogism
 
p → q
q → r

∴ p → r
addition
 
p

∴ p ∨ q
absorption
 
p → q

∴ p → (p ∧ q)


To illustrate, suppose we wanted to get (B ∨ D) ∧ E from premises 1-4 below:
  1. A
  2. A → Z
  3. Z → B
  4. C → D
  5. (B ∨ D) → E

  1. A → B 2, 3, hypothetical syllogism
  2. (A → B) ∧ (C → D) 4, 6, conjunction
  3. A ∨ C 1, addition
  4. B ∨ D 7, 8, constructive dilemma
  5. (B ∨ D) ∧ E 5, 9, absorption
Symbolic Logic: Some Equivalences

Rules of inference aren’t the only rules of logic. There are also logical equivalences which have the handy property of being replaceable anywhere within a line. In contrast, you can’t use rules of inference anywhere within a line. This for example would be a mistake:
  1. A → (B ∧ C)

  1. A → B 1, simplification Mistake!
Instead rules of inference have to be applied to “whole” lines rather than just anywhere “within” lines. Below are some equivalences:

equivalencename of equivalence
 
p ⇔ ¬¬pdouble negation
 
p → q ⇔ ¬q → ¬ptransposition (also called contraposition)
 
p → q ⇔ ¬p ∨ qmaterial implication
 
p ↔ q ⇔ (p → q) ∧ (q → p) biconditional equivalence
 
¬(p ∧ q) ⇔ ¬p ∨ ¬qDe Morgan’s laws
¬(p ∨ q) ⇔ ¬p ∧ ¬q
 
p ⇔ p ∧ p idempotence
p ⇔ p ∨ p


The contrapositive of a conditional p → q is ¬q → ¬p, e.g. the contrapositive of A → B is ¬B → ¬A, and the contrapositive of ¬C → (A ∧ B) is ¬(A ∧ B) → ¬¬C. Incidentally, the converse of a conditional p → q is q → p. For example, the converse of ¬C → (A ∧ B) is (A ∧ B) → ¬C.

As an example of how to use some equivalences, suppose we want to prove ¬H ∨ C from premises 1-3 below:
  1. A
  2. B → ¬A
  3. ¬(B ∨ B) ↔ (H → C)

  1. ¬¬A → ¬B 2, transposition
  2. A → ¬B 3, double negation
  3. ¬B 1, 4 modus ponens
  4. ¬B ∧ ¬B 5, indempotence
  5. ¬(B ∨ B) 6, De Morgan’s laws
  6. [¬(B ∨ B) → (H → C)] ∧ [(H → C) → ¬(B ∧ B)] 3, biconditional equivalence
  7. ¬(B ∨ B) → (H → C) 9, simplification
  8. H → C 7, 9, modus ponens
  9. ¬H ∨ C 10, material implication
Some more equivalences:

equivalencename of equivalence
 
p ∧ q ⇔ q ∧ pcommutation
p ∨ q ⇔ q ∨ p
 
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ rassociation
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r
 
p → (q → r) ⇔ (p ∧ q) → rexportation
 
p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r)distribution
p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r)


For example, suppose we wanted to get (E ∨ C) ∧ (E ∨ D) from premises 1 and 2:
  1. A → [B → (C ∧ D)]
  2. B ∧ A

  1. (A ∧ B) → (C ∧ D) 1, exportation
  2. A ∧ B 2, commutation
  3. C ∧ D 3, 4, modus ponens
  4. (C ∧ D) ∨ E 5, addition
  5. E ∨ (C ∧ D) 6, commutation
  6. (E ∨ C) ∧ (E ∨ D) 7, distribution
Hopefully, seeing these examples and rules of inference also gives you a feel for how a conclusion can follow logically from the premises. Logic is important if for no other reason than you don’t want to look foolish for calling an argument invalid when it’s really deductively valid. Also, if you want to construct a good deductive argument, remember the rules of logic and try to construct it in such a way that it’s deductively valid.

No, I Won’t Bait and Switch

If you’ve gotten this far, you’ve earned it! I’ll now show how to prove the validity of arguments I talked about at the beginning of this article. First, for the atheist:
  1. If God exists, then gratuitous evil does not exist.
  2. Gratuitous evil does exist.
  3. Therefore, God does not exist.
Let G be “God exists” and E be “Gratuitous evil exists.” Then using symbolic logic:
  1. G → ¬E
  2. E

  1. ¬¬E 2, double negation
  2. ¬G 1, 3, modus tollens
So if a theist is to deny the conclusion that God does not exist, he’ll have to reject a premise. A theist cannot accept the premises and deny the conclusion on pain of irrationality. But the sword of logic cuts both ways. Consider the theist argument below:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
The atheist cannot accept the premises and deny the conclusion on pain of irrationality. Using G to symbolize “God exists” and O as “Objective morality exists” we can prove the validity of the above moral argument as follows:
  1. ¬G → ¬M
  2. M

  1. M → G 1, transposition
  2. G 2, 3, modus ponens
An atheist cannot rationally deny that the conclusion follows logically from the premises on pain of irrationality.

Summarizing Some Logic

Here are the connectives I’ve used:

Type of
connective
EnglishSymbolic
Logic
When it’s true/false
Conjunctionp and qp ∧ qTrue if both are true; otherwise false
Disjunctionp or qp ∨ qFalse if both are false; otherwise true
ConditionalIf p, then qp → qFalse if p is true and q is false; otherwise true
Biconditionalp, if and only if qp ↔ qTrue if both have the same truth-value (i.e. both are true or both are false); otherwise false
NegationNot-p¬pTrue if p is false; false if p is true


Some rules of inference:

modus ponens
 
In English In Symbolic Logic
If p then q
p

Therefore, q
p → q
p

∴ q
modus tollens
 
In English In Symbolic Logic
If p then q
Not-q

Therefore, not-p
p → q
¬q

∴ ¬p


Disjunctive Syllogism
 
In English In Symbolic Logic
p or q
Not-p

Therefore, q
p ∨ q
¬p

∴ q
p or q
Not-q

Therefore, p
p ∨ q
¬q

∴ p
simplification
 
In English In Symbolic Logic
p and q

Therefore, p
p ∧ q

∴ p
p and q

Therefore, q
p ∧ q

∴ q


conjunction
 
p
q

∴ p ∧ q
constructive dilemma
 
(p → q) ∧ (r → s)
p ∨ r

∴ q ∨ s


hypothetical syllogism
 
p → q
q → r

∴ p → r
addition
 
p

∴ p ∨ q
absorption
 
p → q

∴ p → (p ∧ q)


Some equivalences:

equivalencename of equivalence
 
p ⇔ ¬¬pdouble negation
 
p → q ⇔ ¬q → ¬ptransposition (also called contraposition)
 
p → q ⇔ ¬p ∨ qmaterial implication
 
p ↔ q ⇔ (p → q) ∧ (q → p) biconditional equivalence
 
¬(p ∧ q) ⇔ ¬p ∨ ¬qDe Morgan’s laws
¬(p ∨ q) ⇔ ¬p ∧ ¬q
 
p ⇔ p ∧ p idempotence
p ⇔ p ∨ p


And some more equivalences:

equivalencename of equivalence
 
p ∧ q ⇔ q ∧ pcommutation
p ∨ q ⇔ q ∨ p
 
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ rassociation
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r
 
p → (q → r) ⇔ (p ∧ q) → rexportation
 
p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r)distribution
p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r)


Bait for the Future

Lots more could be said about symbolic logic. As bait for my next entry (Introductory Logic, Part 2), consider this argument from evil:
  1. If God exists, then gratuitous evil does not exist.
  2. Gratuitous evil does exist.
  3. Therefore, God does not exist.
A theist cannot deny both premises on pain of irrationality (it is logically impossible that both premises are false) and I’ll show how one can prove it via symbolic logic in my next entry. Similarly, consider this moral argument:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
An atheist cannot deny both premises on pain of irrationality (it is logically impossible that both premises are false) and I’ll show how one can prove it with symbolic logic in my next entry on symbolic logic.

Sunday, May 20, 2012

Omnipotence, Creating an Immovable Stone, and YouTube

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I was thinking of having the title be “Omnipotence, Creating an Immovable Stone, YouTube, and Logic” (as we’ll see, logic plays an important but ironically overlooked role in the YouTube video being discussed) but the title was long enough already. This entry is inspired from a YouTube video I saw. Warning: the YouTube video contains strong language and violence (this article will itself contain some PG-13 level language, though I’ll limit myself) that is unsuitable for children, socially conservative non-maverick Christian old ladies, and bosses peering over your shoulder. If you decide not to see the YouTube video, I’ll just describe one part that I found annoying. The video:



If you didn’t see it, here’s the aforementioned part that annoyed me: At an atheist rally God shows up, violently killing atheists. One atheist and God (the Father) have the following conversation.


Atheist: If you’re all-powerful, could you make a rock so big that even you couldn’t lift it?

God: You idiot, of course I could!

Atheist: Aha! But then you wouldn’t be all-powerful!

God swears, begins to bleed, and turns translucent as if he is beginning to cease to exist.

Atheist: You see? Omnipotence is impossible!

Later in the video God eats a hundred dollar bill to fully materialize and before shooting the atheist dead he says, “You forgot one thing: logic is for pussies.”
The “logic is for pussies” remark was annoyingly ironic. To emphasize the irony I’ll use a bit of symbolic logic in this post:
  • ◊ symbolizes the possibility operator, e.g. ◊P means proposition P is possibly true, i.e. P is true in some possible world.
  • → symbolizes implication, i.e. “P → Q” represents “If P, then Q.”
  • ¬ represents the not operator, e.g. ¬P means “not-P” or “P is false.”
A handy english-to-symbolic-logic table:

EnglishSymbolic Logic
P is possible◊P
If P, then QP → Q
not-P¬P

Symbolic logic is handy for showing that a conclusion follows from the premises by certain rules of logic. For example, the following rule of logic is called hypothetical syllogism:
  1. P → Q
  2. Q → R
  3. Therefore, P → R
If you wanted to show that a proposition P is false, premises like these would be great:
  1. P → Q
  2. Q → ¬P
  3. P → ¬P 1, 2, hypothetical syllogism
Someone who grants premises 1 and 2 would have to reject P as false (a little more symbolic logic would be needed to rigorously conclude that, but you get the idea).

Now let’s consider the following argument against omnipotence.

In this case, “omnipotence” is defined as the ability to do anything logically possible. If an omnipotent being exists, it is logically possible for him to create an immovable stone. Yet if there is an immovable stone, then an omnipotent being does not exist (for an omnipotent being could move anything). Therefore, an omnipotent being does not exist.

Now watch what happens when we use some symbolic logic. Using the following letters to represent statements:

O: An Omnipotent being exists.
S: An omnipotent being creates an immovable Stone.

Then we take the following argument...
  1. If an omnipotent being exists, then it’s possible for an omnipotent being to create an immovable stone.
  2. But if an immovable stone exists, then an omnipotent being doesn’t exist.
  3. Therefore, an omnipotent being doesn’t exist.
…and translate it into symbolic logic:
  1. O → ◊S
  2. S → ¬O
  3. Therefore, ¬O
The argument is invalid, i.e. the conclusion doesn’t follow from the premises. Why? Here’s one reason: just because an omnipotent being could give up his omnipotence (as by creating an immovable stone) it doesn’t follow that the omnipotent being has in fact done so. Inferring O → ¬O from premises 1 and 2 breaks the rules of logic. The “Could an omnipotent being create an immovable stone” argument is hopelessly invalid; so says logic.

But maybe some atheists think logic is for pussies.

Wednesday, May 16, 2012

The Leibnizian Cosmological Argument for God (Page 5)

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LCA 2

Yet another argument:
  1. There is an explanation for why there is something rather than nothing.
  2. If there is an explanation for why something exists, that explanation is God.
  3. Therefore, the explanation for why something exists rather than nothing is God (from 1 and 2).
One could reject premise 2 by being a Platonist, but otherwise we can accept premise 2 for the same sort of reason as we can affirm “If the contingent universe does have an external cause for its existence, that cause is God” if we accept that there are no necessarily existing things with the possible exception of God (apart from God and maybe abstract objects, there doesn’t appear to be any other plausible candidates for necessarily existing things). If God exists he exists necessarily, and if God exists something exists. This would explain why something exists rather than nothing. God also explains why the contingent universe exists and why the physical universe exists. Once again, if the atheist insists that God can’t be used an explanation, we can tone it down and just say that an eternal, transcendent, metaphysically necessary personal entity is the best explanation for why there is something rather than nothing.

One advantage this version of the LCA has is “Why is there something rather than nothing?” more perspicuously requires an explanation. By my lights, the best atheist response to “Why is there something rather than nothing?” (since I wouldn’t be a Platonist if I were an atheist) is “it just exists inexplicably.” All else held constant though, we’re better off with a worldview that gives us an explanation for why there is something rather than nothing as opposed to a worldview that gives us no such explanation.

Conclusion

I’ll summarize and highlight some key points for the series.

Explaining Why the Physical Universe Exists

The first version of the Leibnizian cosmological argument (LCA), what I labeled LCA 1A, goes like this, where premise 1 is a version of the principle of sufficient reason (PSR).
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The universe exists.
  3. If the universe does have an explanation for its existence, that explanation is God.
  4. Therefore, the universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation for the existence of the universe is God (from 3 and 4).
Premise 3 is plausibly true as a result of “If God does not exist, then the universe has no explanation of its existence” being likely true, since a transcendent personal cause seems to be the only viable explanation for the universe’s existence. With a physical external cause of the universe’s existence being impossible, the universe’s external cause would have to be ethereal and no less strange than God creating the universe. By far the best option for the atheist is to say that universe exists eternally, inexplicably, with no external cause of its existence. “If God does not exist, then the universe has no explanation of its existence” is much more plausible than its denial. But “If God does not exist, then the universe has no explanation of its existence” is logically equivalent to “If the universe has an explanation of its existence, then God exists,” which is almost synonymous with premise 3. Still, if the atheist insists that God cannot be legitimately used as an explanation (for whatever reason) in spite of the strong justification for premise 3, there is the following toned down argument that I labeled LCA 1B:
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The universe exists.
  3. If the universe does have an explanation for its existence, that explanation is a transcendent personal cause.
  4. Therefore, the universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation for the existence of the universe is a transcendent personal cause (from 3 and 4).
The physical universe having a transcendent personal cause still makes atheism less plausible.

By my lights the weakest part of the LCA is the PSR, such that it is the premise I would reject if I were an atheist. The argument crucially hinges on the universe having an explanation of its existence, but there is good reason to accept that there is an explanation for the existence of the universe. The justification offered:
  1. The translucent ball in the woods illustration. Increasing the size of the ball doesn’t remove the need for an explanation. This helps to illustrate that it is rational to accept that there are explanations for the existence of things, at least when we have no reason to think that an explanation doesn’t exist (if most of the universe were merely a translucent ball, it seems we’d still need an explanation for the ball’s existence).
  2. The nature of rational inquiry. It’s the nature of rational inquiry to look for explanations for why things exist. We seek explanations for the existence of humans, of planets, of stars, and of galaxies. Avoiding all that and saying, “It all just exists inexplicably” would cripple science. And if we are rational to accept that there are explanations for the existence of planets, stars, and galaxies, why not also accept that there is an explanation for the existence of the physical universe? Simply not liking the only viable explanation for the universe’s existence isn’t a good enough reason. The rational thing to do is to accept that there are explanations for the existence of things if we don’t have good reason to believe otherwise, especially if we have an explanation readily available and no evidence for the explanation being false (e.g. believing that the cosmic microwave background radiation just exists inexplicably is less rational than accepting that the big bang theory explains it).
  3. If the shoe were on the other foot… If we had an explanation for the existence of the physical universe that devastated theism (imagine that the best explanation for the physical universe’s existence required that God does not exist) and it was the only viable explanation, and a theist gave a “Maybe there’s just no explanation” response, atheists would argue that the response is just an attempt to avoid a serious intellectual problem for theism, and that the rational thing to do is to accept that there is a an explanation for the existence of the physical universe if we don’t have good reason to believe otherwise. And such atheists would be right! But if that is true, rationality dictates that we be consistent and acknowledge that the best (and apparently the only viable) explanation for the existence of the universe being a transcendent personal cause is a serious intellectual problem for atheism, and that we should accept that there is an explanation for the existence of the universe if we don’t have good reason to think otherwise.
One could say that the universe is eternal, but it’s clear that something being eternal doesn’t necessarily remove the need for an explanation. For example, we can conceive of a three-dimensional hologram projection that exists eternally and is also eternally sustained by a hologram projector. With that in mind, if there is some thing X that meets the following conditions...
  1. X is eternal but contingent (it could have failed to exist).
  2. There is an explanation for why X exists.
  3. It is the only explanation of X’s existence that is a live option.
  4. There is no reason to believe that this explanation for X’s existence is false.
…then it seems we should accept that explanation for X if we have no good reason not to. It’s also good to remember the three reasons above for believing that the rational thing to do is to accept that the universe has an explanation of its existence if we have no good reason to think otherwise. In the “shoe on the other foot” case for example, suppose we had an explanation for the physical universe’s existence that was devastating to theism and the explanation met conditions 1 through 4 (it explains why the universe exists, there is no other viable explanation etc.). Couldn’t the atheist justifiably accept this explanation as evidence against theism? If so, then we should recognize that the only viable explanation for the physical universe’s existence being a transcendent personal cause poses a serious intellectual problem for atheism.

Even if the theist grants that maybe not every eternal contingent thing has an explanation of its existence, all things considered it seems the theist is on good grounds to say that at least the universe has an explanation of its existence, in which case the theist can use this version of the LCA that I labelled LCA 1C:
  1. If the universe exists it has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The universe exists.
  3. If the universe does have an explanation for its existence, that explanation is a transcendent personal cause.
  4. Therefore, the universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation for the existence of the universe is a transcendent personal cause (from 3 and 4).
This still yields a conclusion that, if true, makes atheism considerably less plausible.

The Argument from Contingency

One form of an argument from contingency (what was called LCA 3) goes as follows:
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The contingent universe exists.
  3. If the contingent universe has an explanation for its existence, that explanation is God.
  4. Therefore, the contingent universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the external cause of the contingent universe is God (from 3 and 4).
To summarize the justification for premise 3: the contingent universe does not exist necessarily, i.e. there is a possible world where no contingent thing exists. Thus the contingent universe does not exist by the necessity of its own nature and (due to premise 1) has an external cause. All of physical reality is contingent, and since the physical universe is a subset of the contingent universe, the external cause of the contingent universe would have to be nonphysical. We can then employ the same sort of reasoning in LCA 1 to conclude that the nonphysical external cause of the contingent universe is a transcendent personal cause. Since we’re looking for an external cause of the contingent universe (i.e. the totality of contingent things), that which is the external cause of the contingent universe cannot itself be contingent but must be metaphysically necessary. So the transcendent personal entity that is the external cause of the contingent universe is metaphysically necessary. What is metaphysically necessary is also eternal, since at no times and in no circumstances can metaphysically necessary entities fail to exist. So the transcendent, metaphysically necessary, personal entity that is the external cause of the contingent universe is also eternal. We thus end up with an eternal, transcendent, metaphysically necessary, personal entity that is the external cause of the contingent universe if the contingent universe has an explanation of its existence. Consequently, “If God does not exist, then the contingent universe has no explanation of its existence” is very likely true, and since “If God does not exist, then the contingent universe has no explanation of its existence” is logically equivalent to “If the contingent universe has an explanation of its existence, then God exists” we have strong reason to believe that premise 3 is true.

As before, if the atheist for whatever reason insists that God can’t be used as an explanation, then we can construct a tone downed version of the argument.
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The contingent universe exists.
  3. If the contingent universe has an explanation for its existence, that explanation is is an eternal, transcendent, metaphysically necessary, personal entity.
  4. Therefore, the contingent universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation of the contingent universe is an eternal, transcendent, metaphysically necessary, personal entity (from 3 and 4).
Arguably, any atheism that accepts the existence of an eternal, transcendent, metaphysically necessary, personal entity that is the cause of the universe would not be worthy of the name “atheism.”

The question “Why is there something rather than nothing?” is one of the most fundamental questions in philosophy. The atheist can say, “Platonism is true and abstract objects exist necessarily, and if abstract objects exist something exists.” But if the atheist is not a Platonist, it seems that atheism cannot offer a viable explanation for why there is something rather than nothing whereas theism does. Moreover, even if we were to grant Platonism, there is a “Why does the contingent universe exist?” question that Platonism cannot answer (since abstract objects cannot cause anything) and we end up with an eternal, transcendent, metaphysically necessary, personal entity causing the universe. This is enough to make atheism less plausible.

While one could believe that the transcendent personal cause for the physical universe is different from the transcendent personal cause of the contingent universe, Ockham’s razor suggests we not multiply explanatory entities unnecessarily and it is simpler to posit the same transcendent personal cause for both the physical and contingent universe (I suspect there is extremely large overlap between the physical universe and the contingent universe anyway). Even apart from that, if we justifiably believe that there is a transcendent personal cause of the physical universe and a transcendent personal cause of the contingent universe, this is enough to be intellectually dangerous to atheism.

In any case, God (or at least an eternal, transcendent, metaphysically necessary, personal entity as the external cause of the physical/contingent universe) explains why there is something rather than nothing, why the physical universe exists, and why the contingent universe exists. If nothing else, theism has a certain explanatory scope for things that the atheist has no satisfactory explanation for (assuming the atheist is unwilling to concede e.g. a transcendent personal cause of the physical universe).

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Tuesday, May 15, 2012

The Leibnizian Cosmological Argument for God (Page 4)

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The Leibnizian Cosmological Argument for God
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From Before

Some forms the LCA can take:
  1. The argument that asks, “Why does the physical universe exist?” and offers theism as an explanation.
  2. The argument that asks “Why is there something rather than nothing?” and offers theism as an explanation. One weakness: Platonists believe that abstract objects (like numbers) exist independently of the mind and have necessary existence, and so Platonists won’t see the need for God as an explanation because Platonic abstract objects exist by the necessity of their own nature. If you’re not a Platonist though, this is potentially a good argument because “Why is there something rather than nothing?” is an excellent question and it seems to be the sort of thing that we should look for an explanation for.
  3. The argument that asks, “Why do contingent things exist?” and offers theism as an explanation. This bypasses the problem of Platonism because abstract objects (like the number six) can’t cause anything, including contingent objects. This type of argument is also called “the argument from contingency.”
Let’s label the above three LCA arguments LCA 1, LCA 2, and LCA 3, respectively. In part 1 of the series I argued for a few versions of LCA 1, and in so doing I also argued for a form of the principle of sufficient reason (viz. “Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause”), an important and controversial part of the LCA. In this blog entry I’ll deal with the other two LCA varieties.

LCA 3

Yeah, I’m going kind of out of order, but that’s what you get when you read an article written by a maverick Christian (sometimes). For LCA 3 we can use the same sort of reasoning as LCA 1 except we replace the “physical universe” with the “totality of contingent things,” where we can call the “totality of contingent things” the “contingent universe” (if you’re wondering what the difference is between the contingent universe and the physical universe, the contingent universe would include contingent nonphysical entities like angels and souls if they exist). LCA 3 suggests that it’s possible for the contingent universe to not exist, i.e. that there is a possible worlds where no contingent thing exists. Thus, the contingent universe does not exist by the necessity of its own nature. We could justify that assertion with an argument from subtraction; upon reflection it seems there is a possible world where only a thousand contingent things exist, and it seems there is a possible world where only fifty contingent things exist etc. all the way down to zero contingent things existing. If nothing else, we can point out that it’s possible for the contingent universe to have had a different collection of contingent things it now has, similar to how our own physical universe could have had a different collection of fundamental physical units.

Beyond the argument from subtraction, there is one other thing to consider. It seems clear that there is no physical thing that exists necessarily, but one could propose that while there is no creative force that exists necessarily (like God), in all possible worlds there is a contingent thing. That is, it is necessary that some contingent thing or other exists, but there is no physical (or contingent) thing in particular that exists necessarily. One problem with this view: why is it that a contingent thing exists in every possible world? After all, on this view there’s no necessarily existing creative force to bring them about in every possible world. The proponent of this view would have to say that it’s inexplicable; it just happens to be the case that in every possible world there is a contingent thing. But the number of possible worlds is quite literally infinite; the odds that by chance there just happens to be a contingent thing in every possible world is infinitesimally small. That there is a possible world in which there is no contingent thing seems much more plausible than its denial.

Bringing out the whole argument more explicitly:
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The contingent universe exists.
  3. If the contingent universe has an explanation for its existence, that explanation is God.
  4. Therefore, the contingent universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation of the contingent universe is God (from 3 and 4).
The physical universe (all of physical reality) is contingent, and since the physical universe is a subset of the contingent universe, the external cause of the contingent universe would have to be nonphysical. We can then employ the same sort of reasoning in LCA 1 to conclude that the nonphysical external cause of the contingent universe is a transcendent personal cause.

Further reasoning shows that the transcendent personal cause of the contingent universe is also metaphysically necessary (it exists in all possible worlds and cannot fail to exist) and eternal. If it’s possible that no contingent thing exist, you can’t sensibly appeal to a contingent thing to explain the contingent universe, because any contingent thing is part of the contingent universe, so it wouldn’t be an explanation for why the contingent universe exists as opposed to no contingent things existing. Similarly, one can’t say that the physical universe causing itself (as if by time travel) gives us an adequate explanation, because the entire causal-loop universe would (as a contingent thing) require explanation for its existence. But if no contingent thing can explain the existence of the contingent universe, the only hope left is to appeal to a necessarily existing entity. But then the explanation for the universe’s existence is also eternal, since at no time and in no circumstances can necessarily existing entities fail to exist. So we have an eternal, transcendent, metaphysically necessary, personal entity that is the external cause of the contingent universe. This sounds even more like God. Similar to the case of LCA 1, if the atheist insists that we can’t use God as any type of explanation (for whatever reason) or does not like God being a part of the third premise etc., we can tone down the argument a bit as follows:
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The contingent universe exists.
  3. If the contingent universe has an explanation for its existence, that explanation is is an eternal, transcendent, metaphysically necessary, personal entity.
  4. Therefore, the contingent universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation of the contingent universe is an eternal, transcendent, metaphysically necessary, personal entity (from 3 and 4).
Once one grants the existence of an eternal, transcendent, metaphysically necessary personal entity as the cause of the universe, atheism becomes much less plausible to say the least. Arguably, any atheism that accepts the existence of an eternal, transcendent, metaphysically necessary personal entity that is the cause of the universe would not be worthy of the name “atheism.”

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Monday, May 14, 2012

The Leibnizian Cosmological Argument for God (Page 3)

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The Leibnizian Cosmological Argument for God
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The Argument (LCA 1A)
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The universe exists.
  3. If the universe does have an explanation for its existence, that explanation is God.
  4. Therefore, the universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation for the existence of the universe is God (from 3 and 4).
Justifying Premise 1: the PSR

This brings us to the premise that’s the most controversial: premise 1. Why think that “Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause”? Here I’ll borrow a bit from philosopher Richard Taylor’s illustration of finding a translucent ball in the woods. “How did it get there?” you ask. I reply, “There is no explanation for it being in the woods; the ball just exists inexplicably.” My response seems less plausible than the idea that there is some explanation for the ball’s existence. What if we enlarged the ball to the size of a car? Same problem: some explanation seems to be needed. How about a city? Same problem. A planet? Same problem. A galaxy? Same problem; increasing the size does nothing to remove the need for an explanation. How about if the ball were as big as the universe? Same problem. All things considered, it seems intuitively plausible that if a contingent thing exists, there is some reason why it exists, since it could have failed to exist.

One way a thing can have an explanation for its existence is to exist by the necessity of its own nature, but that isn’t an option for contingent things. The other way to explain why something exists is by reference to an external cause. But as far as the PSR is concerned, a thing that has an external cause as its explanation need not have begun to exist. For example, one idea is that the universe’s existence is being eternally sustained by God, similar to an eternal flutist eternally sustaining a note of music; God is the ground of all being and if God were to cease existing so would the universe. So the PSR does not require that contingent things begin to exist. That said, with the finite age of the known physical universe (among other considerations) it seems more likely to me that God created the universe as opposed to (merely) sustaining it in being.

Why do we need an external cause? Why not have an explanation from within, especially if the universe is infinitely old? While I think the universe is most likely of a finite age, not everybody agrees, especially atheists. With an infinitely old universe, we could conceive of each part of an infinitely long causal chain having an explanation for its existence, and one idea is that if we explain each individual element in the physical universe, then we explain the whole.

But this thinking commits the fallacy of composition, unjustifiably asserting something is true for the whole because it is true for the parts, e.g. because each part of an elephant is light in weight, therefore the whole elephant is light in weight. The fallacy in the case of explaining the universe reveals itself by considering the following scenario, borrowing largely from Leibniz’s illustration of copied geometry books. Suppose there have always been geometry books and that each geometry book has been copied from a previous book. So throughout the infinite past, there have been an infinite series of geometry books, each one copied from its predecessor. Although this is an explanation for each geometry book, it does not explain why we have an infinite series of geometry books rather than e.g. an infinite series of books on propositional logic, since that series of books could have been copied from eternity past as well, if geometry books could have. Nor does it explain why there is an infinite series of geometry books rather than no infinite series of books at all. Consequently, it does not explain the existence of the infinite series as a whole, even though it explains the existence of each individual book. That sort of thinking behind the infinite series of geometry books also applies to a physical universe with an infinitely long causal chain. Even though we have an explanation for each link in the chain, we can still ask why this infinite series exists rather than some other infinite series, and we can still ask why there exists an infinite series rather than no series at all. Explaining each part does not entail explaining the whole, and to think otherwise commits the fallacy of composition.

What about a causal-loop universe? One could argue that the universe has an internal cause of its existence in the sense of the universe causing itself, but that idea is metaphysically incoherent; for the universe to cause itself it would have to already exist. I know, time travel is a time-honored tradition of science fiction, but in the real world an agent going back in time to affect its own existence (say, a robot named Bob existing because it goes back in time to create itself) faces severe if not fatal problems (e.g. if such a thing were possible, it seems the aforementioned Bob the robot could also have gone back in time to destroy its younger self, thereby entailing that it doesn’t go back in time to destroy its younger self, thereby producing a self-contradictory state of affairs).

Still, suppose we believe that a causal-loop universe is metaphysically coherent. The causal-loop universe would itself be contingent and could have failed to exist, and the sort of problems affecting the infinitely causal chain afflict the causal-loop universe, in part because a causal-loop universe just is a specialized instance of an infinite past (albeit one that loops forever rather than a more linear infinite past). We can still ask why this particular circular causal chain exists rather than some other circular chain, and we can still ask why a circular causal chain exists rather than there being no such chain. Pointing to any component of the causal-loop universe (say, a time-travelling cause) won’t work to explain the universe’s existence because any such component is part of the very thing to be explained. It would be like trying to explain the existence of a circularly moving river (if such a thing were possible) by pointing to one half of the river in which water flows from that half to the other half of the river; this would do nothing to explain why the river as a whole exists rather than not (e.g. it wouldn’t explain why the river as a whole has water rather than there being no water to circulate). Similarly, pointing to any causal-flowing segment of a circularly-causal universe wouldn’t explain why the causal-loop universe as a whole exists rather than not.

Another way to look at premise 1 is that it’s the nature of rational inquiry to look for explanations for why things exist. We seek explanations for the existence of humans, of planets, of stars, and of galaxies. Avoiding all that and saying, “It all just exists inexplicably” would cripple science.

Challenging the PSR

Even if it is the nature of rational inquiry to look for explanations, a critic could say that this methodological aim to look for explanations doesn’t entail that explanations will always be found. A gold prospector might have the attitude of “Always look for gold” but that method doesn’t entail she will always find gold. So maybe not everything that exists has an explanation for its existence.

Maybe that’s true. In fact, maybe some things we think have explanations don’t have explanations at all. It’s logically possible (in the sense of not being self-contradictory) that the big bang theory is false and that the things that the big bang theory explains, like the cosmic microwave background radiation, simply exist with no explanation for their existence.

Still, that doesn’t seem to be the most rational way to go. The same holds true for a giant translucent ball (no matter how large it is). We should not exempt the universe from requiring an explanation if we have no good reason to do so, since it seems more rational to accept that there are explanations for the existence of things if we don’t have good reason to believe otherwise, especially if we have an explanation readily available and no evidence for the explanation being false. In regards to finding an explanation for the universe’s existence, if the only reason one rejects the proposed explanation in favor of “there is no explanation” is that one just doesn’t like the proposed explanation very much, it doesn’t seem like such a person has a good enough reason (imagine if someone rejected the big bang theory for that reason!). One could argue that, all things considered (the nature of rational inquiry etc.) we should accept PSR unless we have good reason to believe that it is false, including when it comes to the existence of the physical universe.

Another consideration is to imagine the shoe being on the other foot; if the best known explanation for the existence of the physical universe were devastating to theism instead of devastating to atheism, I suspect atheists would criticize a theist’s “the universe has no explanation” response, and they would be right to do so. The rational thing to do is to accept that there is an explanation for the existence of the universe if we don’t have good reason to think otherwise.

Do we have such a reason? Here’s the best atheist position I can think of. While contingent things that begin to exist require an explanation of their existence via an external cause, contingent things that exist eternally don’t necessarily have external causes and thus don’t necessarily require an explanation of their existence. Such contingent things exist eternally and inexplicably. The atheist can argue that the physical universe fits this category: it is contingent but eternal and doesn’t have an explanation of its existence.

The first thing to say is that it isn’t necessarily the case that if a contingent thing exists eternally that no explanation for its existence is needed. For example, suppose we humans learned of an eternally existing monument at the center of the universe that says, “I, the Lord thy God, am the sustainer of the universe and have sustained it throughout all eternity” (if questions of different languages bother you, imagine further that it displays this message through a kind of mechanical telepathy such that anybody who looks at the monument sees the message in her own language). A response like, “Well, the monument existed eternally, so no explanation is needed for why this monument exists” doesn’t seem convincing. Or to use a more decidedly nontheistic example, imagine a three-dimensional hologram projection that exists eternally and is also eternally sustained by a hologram projector. Even though the hologram exists eternally, we have an explanation for the hologram’s eternal existence. The hologram is contingent and it could have failed to exist, and there is an explanation readily available for why it exists rather than it not existing. With that in mind, suppose some contingent thing X meets the following conditions:
  1. X is eternal but contingent (it could have failed to exist).
  2. There is an explanation for why X exists .
  3. It is the only explanation of X’s existence that is a live option.
  4. There is no reason to believe that this explanation for X’s existence is false.
I think that if we know that all four conditions are met for thing X, then we should accept that explanation for X if we have no good reason not to. In considering these conditions it might also help to envisage the shoe being on the other foot. Suppose we had an explanation for the physical universe’s existence that was devastating to theism and the explanation met conditions 1 through 4 (it explains why the universe exists, there is no other viable explanation etc.). I have a hard time believing that atheists wouldn’t use this devastating-to-theism explanation as evidence against theism. Moreover, it seems they would be right to do so if their explanation for why the universe exists rather than not is the only viable explanation, there is no evidence against the explanation etc. But then rationality dictates we be consistent and recognize that a transcendent personal cause meeting these conditions poses a serious intellectual problem for atheism.

It’s possible that the atheist thinks there is a good reason to believe the transcendent personal cause explanation is false and thus that condition 4 is not met, though I have yet to see such a reason. In any case, it seems clear that for the atheist to rationally reject the explanation, some good reason will need to be given, i.e. the ball is in the atheist’s court. Such atheists don’t have to swing at the ball, but if they don’t, they’ll lose the game.

At the end of the day, the theist can grant for sake of argument that maybe some eternal contingent things don’t require explanations for their existence and still argue that all things considered (e.g. the four conditions mentioned earlier regarding eternal contingent things, and what would be rational to do if the shoe were on the other foot) we are rational to accept that the universe has an explanation of its existence, giving us this version of the LCA (call it LCA 1C):
  1. If the universe exists it has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The universe exists.
  3. If the universe does have an explanation for its existence, that explanation is a transcendent personal cause.
  4. Therefore, the universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation for the existence of the universe is a transcendent personal cause (from 3 and 4).
If all three premises are rational and justified, we still have a conclusion that makes atheism considerably less plausible even with the weakened version of premise 1.

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