10

Basic Modal Logic

Modal logic studies arguments whose validity depends on “possible,” “necessary,” and similar notions. This chapter covers the basics and the next gets into further modal systems.

10.1 Translations

To help us evaluate arguments, we’ll construct a modal language. This includes propositional logic’s vocabulary, wffs, inference rules, and proofs. It adds symbols for modal operators: “◇” and “☐” (diamond and box):

◇A = It’s possible that A.

A is true in some possible world.

A = It’s true that A.

A is true in the actual world.

☐A = It’s necessary that A.

A is true in all possible worlds.

“Possible” is weaker than “true,” while “necessary” is stronger than “true.” “A is necessary” claims that A has to be true – it couldn’t have been false.

“Possible” here means logically possible (not self-contradictory). “I run a mile in two minutes” may be physically impossible; but it’s logically possible (the idea contains no self-contradiction). Likewise, “necessary” means logically necessary (self-contradictory to deny). “2+2 = 4” and “All bachelors are unmarried” are examples of necessary truths; such truths are based on logic, the meaning of concepts, or necessary connections between properties.

We can rephrase “possible” as true in some possible world – and “necessary” as true in all possible worlds. A possible world is a consistent description of how things might have been or might in fact be. Picture a possible world as a consistent story (or novel). The story is consistent, in that its statements don’t entail self-contradictions; it describes a set of possible situations that are all possible together. The story may or may not be true. The actual world is the story that’s true – the description of how things in fact are.

As before, grammatical formulas are wffs (well-formed formulas). Wffs now are strings we can construct using the propositional rules plus this new rule:

The result of writing “◇” or “☐,” and then a wff, is a wff. 0231

Don’t use parentheses with “◇A” and “☐A”; these forms are incorrect: “◇(A),” “(◇A),” “☐(A),” “(☐A).” Parentheses here would serve no purpose.

We’ll focus now on how to translate English sentences into modal logic. Here are some simpler examples:

· A is possible (consistent, could be true)

· = ◇A

· A is necessary (must be true, has to be true)

· = ☐A

· A is impossible (self-contradictory)

· = ∼◇A = A couldn’t be true

· = ☐∼A = A has to be false

An impossible statement (like “2 ≠ 2”) is one that’s false in every possible world.

These examples are more complicated:

· A is consistent (compatible) with B

· = ◇(A • B)

· It’s possible that A and B are both true

· A entails B

· = ☐(A ⊃ B)

· It’s necessary that if A then B

“Entails” makes a stronger claim than plain “if-then.” Compare these two:

· “There’s rain” entails “There’s precipitation”

· = ☐(R ⊃ P)

· If it’s Saturday, then I don’t teach class

· = (S ⊃ ∼T)

The first is logically necessary; every conceivable situation with rain also has precipitation. The second just happens to be true; we can consistently imagine me teaching on Saturday, even though in fact I never do.

Here are further forms:

· A is inconsistent with B

· = ∼◇(A • B)

· It’s not possible that A and B are both true

· A doesn’t entail B

· = ∼☐(A ⊃ B)

· It’s not necessary that if A then B

· A is a contingent statement¹

· = (◇A • ◇∼A)

· A is possible and not-A is possible

¹ English sometimes uses “possible” to mean what we call “contingent” (true in at least one possible world and false in at least one possible world). In our sense of “possible” (true in at least one possible world), what is necessary is also thereby possible.

· A is a contingent truth

· = (A • ◇∼A)

· A is true but could have been false

Statements are necessary, impossible, or contingent. But truths are only necessary or contingent (since impossible statements are false). 0232

When translating, it’s usually good to mimic the English word order:

· necessary not = ☐∼

· not necessary = ∼☐

· necessary if = ☐(

· if necessary = (☐

Use a separate box or diamond for each “necessary” or “possible.” So “If A is necessary and B is possible, then C is possible” is “((☐A • ◇B) ⊃ ◇C).”

This English form is ambiguous between two meanings:

“If you’re a bachelor, then you must be unmarried.”

Simple necessity

(B ⊃ ☐U)

If you’re a bachelor, then you’re inherently unmarriable (in no possible world would anyone marry you).

If B, then U (by itself) is necessary.

Conditional necessity

☐(B ⊃ U)

It’s necessary that if you’re a bachelor then you’re unmarried.

It’s necessary that if-B-then-U.

Box-inside “(B ⊃ ☐U)” affirms simple necessity: given your bachelorhood, “You’re unmarried” is inherently necessary; this is insulting and presumably false. Box-outside “☐(B ⊃ U)” affirms conditional necessity: what’s necessary isn’t “You’re a bachelor” or “You’re unmarried” by itself, but the connection between the two: it’s necessary that if you’re a bachelor (unmarried man) then you’re unmarried. So our English “If you’re a bachelor, then you must be unmarried” is ambiguous; its wording suggests simple necessity (which denies your freedom to marry) but it’s likely meant as conditional necessity.

Medievals called the box-inside form “necessity of the consequent” (the second part is necessary) and the box-outside form “necessity of the consequence” (the if-then is necessary). The ambiguity is important; several fallacious philosophical arguments depend on the ambiguity for their plausibility.

It’s not ambiguous if you say that the second part is “by itself” or “intrinsically” necessary or impossible – or if you use “entails” or start with “necessary.” These forms aren’t ambiguous:

· If A, then B (by itself) is necessary = (A ⊃ ☐B)

· If A, then B is intrinsically necessary = (A ⊃ ☐B)

· A entails B = ☐(A ⊃ B)

· Necessarily, if A then B = ☐(A ⊃ B)

· It’s necessary that if A then B = ☐(A ⊃ B)

· “If A then B” is a necessary truth = ☐(A ⊃ B)

The ambiguous forms have if-then with a strong modal term (like “necessary,” 0233 “must,” “impossible,” or “can’t”) in the then-part:¹

¹ There’s an exception to these boxed rules: if the if-part is a claim about necessity or possibility, then just use the box-inside form. So “If A is necessary then B is necessary” is just “(☐A ⊃ ☐B)” – and “If A is possible then B is impossible” is just “(◇A ⊃ ∼◇B).”

“If A is true, then it’s necessary (must be) that B” could mean “(A ⊃ ☐B)” or “☐(A ⊃ B).”

“If A is true, then it’s impossible (couldn’t be) that B” could mean “(A ⊃ ☐∼B)” or “☐(A ⊃ ∼B).”

When you translate an ambiguous English sentence, give both forms. With ambiguous arguments, work out both arguments.

10.1a Exercise: LogiCola J (BM & BT)

Translate these into wffs. Be sure to translate ambiguous forms both ways.

“God exists and evil doesn’t exist” entails “There’s no matter.”

☐((G • ∼E) ⊃ ∼M)

1. It’s necessary that God exists.

2. “There’s a God” is self-contradictory.

3. It’s not necessary that there’s matter.

4. It’s necessary that there’s no matter.

5. “There’s rain” entails “There’s precipitation.”

6. “There’s precipitation” doesn’t entail “There’s rain.”

7. “There’s no precipitation” entails “There’s no rain.”

8. If rain is possible, then precipitation is possible.

9. God exists.

10.If there’s rain, then there must be rain.

11.It’s not possible that there’s evil.

12.It’s possible that there’s no evil.

13.If you get more points than your opponent, then it’s impossible for you to lose.

14.It’s necessary that if you see that B is true then B is true.

15.If B has an all-1 truth table, then B is inherently necessary.

16.Necessarily, if there’s a God then there’s no evil.

17.If there’s a God, then there can’t be evil.

18.If there must be matter, then there’s evil.

19.Necessarily, if there’s a God then “There’s evil” (by itself) is self-contradictory.

20.It’s necessary that it’s heads or tails.

21.Either it’s necessary that it’s heads or it’s necessary that it’s tails.

22.“There’s rain” is a contingent statement.

23.“There’s rain” is a contingent truth.

24.“If there’s rain, then there’s evil” is a necessary truth.

25.If there’s rain, then “There’s evil” (by itself) is logically necessary.

26.If there’s rain, then it’s necessary that there’s evil. 0234

27.It’s necessary that it’s possible that there’s matter.

28.“There’s a God” isn’t a contingent truth.

29.If there’s a God, then it must be that there’s a God.

30.It’s necessary that if there’s a God then “There’s a God” (by itself) is necessary.

10.2 Proofs

For modal proofs, we need world prefixes and modal inference rules.

A world prefix is a string of zero or more instances of “W.” So “ ” (zero instances), “W,” “WW,” and so on are world prefixes; these represent possible worlds, with the blank world prefix (“ ”) representing the actual world. A derived line is now a line consisting of a world prefix and then “∴” and then a wff. And an assumption is now a line consisting of a world prefix and then “asm:” and then a wff. Here are examples of derived lines and assumptions:

· ∴ A (So A is true in the actual world.)

· W ∴ A (So A is true in world W.)

· WW ∴ A (So A is true in world WW.)

· asm: A (Assume A is true in the actual world.)

· W asm: A (Assume A is true in world W.)

· WW asm: A (Assume A is true in world WW.)

Derived lines with W’s are more common.

We can use S- and I-rules and RAA in modal proofs. Unless otherwise specified, we can use an inference rule only within a given world; so if we have “(A ⊃ B)” and “A” in the same world, then we can infer “B” in this same world. RAA needs additional wording (italicized below) for world prefixes:

RAA: Suppose some pair of not-blocked-off lines using the same world prefix has contradictory wffs. Then block off all the lines from the last not-blocked-off assumption on down and infer a line consisting in this assumption’s world prefix followed by “∴” followed by a contradictory of the assumption.

For RAA, lines with contradictory wffs must have the same world prefix. “W ∴ A” and “WW ∴ ∼A” isn’t enough; “A” may be true in one world but false in another. But “WW ∴ A” and “WW ∴ ∼A” is a genuine contradiction. And the derived line must have the same world prefix as the assumption; if “W asm: A” leads to a contradiction in any world, then RAA lets us derive “W ∴ ∼A.”

Modal proofs use four new inference rules. The reverse-squiggle rules hold regardless of what pair of contradictory wffs replaces “A” / “∼A”; here “→” means that we can infer whole lines from left to right: 0235

Reverse squiggle RS

∼☐A → ◇∼A

∼◇A → ☐∼A

“Not necessary” entails “possibly false.” And “not possible” entails “necessarily false.” Use these rules only within the same world. Our rules cover reversing squiggles on longer formulas, if the whole formula begins with “∼☐” or “∼◇”:

· ∼◇∼B

· ––––––––

· ∴ ☐∼∼B

· ∼☐(C • ∼D)

· ––––––––––––

· ∴ ◇∼(C • ∼D)

In the first example, we also could conclude “☐B” (dropping “∼∼”). This next example is illegal in our system, since it fits poorly into our proof strategy, even though it’s logically correct:

Don’t do this:

· (P ⊃ ∼☐Q)

· –––––––––

· ∴ (P ⊃ ◇∼Q)

Reverse squiggles whenever you have a wff that begins with “∼” and then a modal operator; this moves an operator to the beginning of the formula, so we can drop it later.

We drop modal operators using the next two rules (which hold regardless of what wff replaces “A”). Here’s the drop-diamond rule:

Drop diamond DD

◇A → W ∴ A,

use a new string of W’s

Here the line with “◇A” can use any world prefix – and the line with “∴ A” must use a new string of one or more W’s (a string not occurring in earlier lines). If “A” is possible, then “A” is true in some possible world; we can give this world a name – but a new name, since “A” needn’t be true in any of the worlds used in the proof so far. We’ll use “W” for the first diamond we drop, “WW” for the second, and so forth. So if we drop two diamonds, then we introduce two new worlds:

· ◇H

· ◇T

· ––––––––

· W ∴ H

· WW ∴ T

Heads is possible, tails is possible; let’s call an imagined world with heads “W,” and one with tails “WW.” It’s OK to use “W” in the first inference, since it occurs in no earlier line. But the second inference must use “WW,” since “W” has now already occurred.

We can drop diamonds from longer formulas, if the diamond begins the wff. So this first inference is fine:

· ◇(A • B)

· ––––––––––

· W ∴ (A • B)

These next two examples are wrong (since the formula doesn’t begin with a diamond – instead, it begins with a left-hand parenthesis):

· (◇A ⊃ B)

· –––––––––––

· W ∴ (A ⊃ B)

· (◇A • ◇B)

· ––––––––––

· W ∴ (A • B)

Drop only initial operators (diamonds or boxes).

Here’s the drop-box rule: 0236

Drop box DB

☐A → W ∴ A,

use any world prefix

The lines with “☐A” and “∴ A” can use any world prefixes, the same or different, including the blank world prefix for the actual world. If “A” is necessary, then “A” is true in all possible worlds, and so we can put “A” in any world. But it’s bad strategy to drop a box into a new world; stay in old worlds. As before, we can drop boxes from longer formulas, as long as the box begins the wff. So this next inference is fine:

· ☐ (A ⊃ B)

· –––––––––––

· W ∴ (A ⊃ B)

These next two example are wrong (since the formula doesn’t begin with a box – instead it begins with a left-hand parenthesis – drop only initial operators):

· (☐A ⊃ B)

· –––––––––––

· W ∴ (A ⊃ B)

· (☐A ⊃ ☐B)

· –––––––––––

· W ∴ (A ⊃ B)

“(☐A ⊃ B)” and “(☐A ⊃ ☐B)” are if-then forms and follow the if-then rules: if we have the first part true, we can get the second true; if we have the second part false, we can get the first false; if we get stuck, we make an assumption.

Here’s a valid modal argument and its proof:

· Necessarily, if there’s rain then there’s precipitation.

· It’s possible that there’s rain.

· ∴ It’s possible that there’s precipitation.

Assume “It’s not possible that there’s precipitation.” Reverse the squiggle to get “It’s necessary that there’s no precipitation” in 4. Drop the diamond in 2, using a new world W, to get “There’s rain” in W in 5. Drop the box in 1 to get “If there’s rain then there’s precipitation” in W in 6. From these two, get “There’s precipitation” in W in 7. Drop a box again in 8 to get contradiction. Our conclusion follows: “It’s possible that there’s precipitation.”

We’ll typically use modal rules in this order: (1) First reverse squiggles; (2) then drop initial diamonds, using a new world each time; (3) lastly, drop each initial box once for each old world. Star when reversing squiggles or dropping a diamond (starred lines have redundant information and can largely can be ignored in deriving further lines):

 * ∼☐A

–––––––––

∴ ◇∼A

 * ◇A

––––––––

W ∴ A

0237 Don’t star when dropping a box; we can never exhaust a “necessary” statement – and we may have to use it again later in the proof.

Here’s an easy modal proof:

Reverse squiggles to get “◇∼A” in line 3. Drop a diamond to get “W ∴ ∼A” in line 4. Then drop a box to get “W ∴ (A • B)” in line 5.

In this proof, there’s no point to dropping the box into the actual world, to go from “☐(A • B)” in line 1 to “∴ (A • B)” with no initial W’s. Drop a box into the actual world (besides into any W-worlds) only in these two cases:

1. The original premises or conclusion have an unmodalized instance of a letter. (A letter is unmodalized if it doesn’t occur as part a larger wff beginning with “☐” or “◇”; in “(A • ◇A)” only the first “A” is unmodalized.)

2. You’ve done everything else possible (including further assumptions if needed) and still have no other old worlds.

Here are examples:

Case 1: unmodalized letter

Here the original argument has an unmodalized letter. When this happens, drop boxes into the actual world (as in line 3) and also into all W-worlds (if there are any).

Case 2: no other worlds

Here, when you drop the box to get line 3, there are no other old worlds (since you had no diamonds to drop); so use the actual world (with no W’s). Our “standard strategy” here has us drop boxes into the actual world in these two cases, and only these. This always works, but it sometimes gives us lines that we don’t need; we can skip these lines if we see that we don’t need them. 0238

In doing proofs, first assume the conclusion’s opposite; then use modal rules plus S- and I-rules to derive all you can. If you find a contradiction, apply RAA. If you’re stuck and need to break a NOT-BOTH, OR, or IF-THEN, then make another assumption. If you get no contradiction and yet can’t do anything further, then try to refute. Here’s a fuller statement of our strategy’s modal steps:

1. FIRST REVERSE SQUIGGLES: For each unstarred, not-blocked-off line that begins with “∼☐” or “∼◇,” derive a line using the reverse-squiggle rules. Star the original line.

2. THEN DROP DIAMONDS: For each unstarred, not-blocked-off line that begins with a diamond, derive an instance using the next available new world (but don’t drop a diamond if you already have a not-blocked-off-instance in some previous line – so don’t drop “◇A” if you already have “W ∴ A”). Star the original line.

3. LASTLY DROP BOXES: For each not-blocked-off line that begins with a box, derive instances using each old world. Don’t star the original line; you might have to use it again. (Drop boxes into the actual world under the two conditions given on the previous page.)

Drop diamonds before boxes. Introduce a new world each time you drop a diamond, and use the same old worlds when you drop a box. And drop only initial diamonds and boxes.

10.2a Exercise: LogiCola KV

· Prove each of these arguments to be valid (all are valid).

☐(A ⊃ B)

◇∼B

∴ ◇∼A

1. ◇(A • B)

∴ ◇A

2. A

∴ ◇A

3. ∼◇(A • ∼B)

∴ ☐(A ⊃ B)

4. ☐(A ∨ ∼B)

∼☐A

∴ ◇∼B

5. (◇A ∨ ◇B)

∴ ◇(A ∨ B)

6. (A ⊃ ☐B)

◇∼B

∴ ◇∼A

7. ∼◇(A • B)

◇A

∴ ∼☐B 0239

8. ☐A

∴ ◇A

9. ☐A

∼☐B

∴ ∼☐(A ⊃ B)

10.☐(A ⊃ B)

∴ (☐A ⊃ ☐B)

10.2b Exercises: LogiCola KV

First appraise intuitively. Then translate into logic (using the letters given) and prove to be valid (all are valid).

1. “You knowingly testify falsely because of threats to your life” entails “You lie.”

It’s possible that you knowingly testify falsely because of threats to your life but don’t intend to deceive. (Maybe you hope no one will believe you.)

∴ “You lie” is consistent with “You don’t intend to deceive.” [Use T, L, and I; from Tom Carson, who writes on the morality of lying.]

2. Necessarily, if you don’t decide then you decide not to decide.

Necessarily, if you decide not to decide then you decide.

∴ Necessarily, if you don’t decide then you decide. [Use D for “You decide” and N for “You decide not to decide.” This is adapted from Jean-Paul Sartre.]

3. If truth is a correspondence with the mind, then “There are truths” entails “There are minds.”

“There are minds” isn’t logically necessary.

Necessarily, if there are no truths then it is not true that there are no truths.

∴ Truth isn’t a correspondence with the mind. [Use C, T, and M.]

4. There’s a perfect God.

There’s evil in the world.

∴ “There’s a perfect God” is logically compatible with “There’s evil in the world.” [Use G and E. Most who doubt the conclusion would also doubt premise 1.]

5. “There’s a perfect God” is logically compatible with T.

T logically entails “There’s evil in the world.”

∴ “There’s a perfect God” is logically compatible with “There’s evil in the world.” [Use G, T, and E. Here T (for “theodicy”) is a possible explanation of why God permits evil that’s consistent with God’s perfection and entails the existence of evil. T might say: “The world has evil because God, who is perfect, wants us to make significant free choices to struggle to bring a half-completed world toward its fulfillment; moral evil comes from the abuse of human freedom and physical evil from the half-completed state of the world.” This basic argument (but not the specific T) is from Alvin Plantinga.]

6. “There’s a perfect God and there’s evil in the world and God has some reason for permitting the evil” is logically consistent.

∴ “There’s a perfect God and there’s evil in the world” is logically consistent. [Use G, E, and R. This is Ravi Zacharias’s version of Plantinga’s argument.] 0240

7. God is omnipotent.

“You freely always do the right thing” is logically possible.

If “You freely always do the right thing” is logically possible and God is omnipotent, then it’s possible for God to bring it about that you freely always do the right thing.

∴ It’s possible for God to bring it about that you freely always do the right thing. [Use O, F, and B; from J. L. Mackie. He thought God had a third option besides making robots who always act rightly and free beings who sometimes act wrongly: he could make free beings who always act rightly.]

8. “God brings it about that you do A” is inconsistent with “You freely do A.”

“God brings it about that you freely do A” entails “God brings it about that you do A.”

“God brings it about that you freely do A” entails “You freely do A.”

∴ It’s impossible for God to bring it about that you freely do A. [Use B, F, and G. This attacks the conclusion of the previous argument.]

9. “This is a square” entails “This is composed of straight lines.”

“This is a circle” entails “This isn’t composed of straight lines.”

∴ “This is a square and also a circle” is self-contradictory. [S, L, C]

10.“This is red and there’s a blue light that makes red things look violet to normal observers” entails “Normal observers won’t sense redness.”

“This is red and there’s a blue light that makes red things look violet to normal observers” is logically consistent.

∴ “This is red” doesn’t entail “Normal observers will sense redness.” [Use R, B, and N; from Roderick Chisholm.]

11.“All brown dogs are brown” is a necessary truth.

“Some dog is brown” isn’t a necessary truth.

“Some brown dog is brown” entails “Some dog is brown.”

∴ “All brown dogs are brown” doesn’t entail “Some brown dog is brown.” [Use A for “All brown dogs are brown,” X for “Some dog is brown,” and S for “Some brown dog is brown.” This attacks a doctrine of traditional logic (§2.8), that “all A is B” entails “some A is B.”]

12.It’s necessary that, if God exists as a possibility but does not exist in reality, then there could be a being greater than God (namely, a similar being that also exists in reality).

“There could be a being greater than God” is self-contradictory (since “God” is defined as “a being than which no greater could be”).

It’s necessary that God exists as a possibility.

∴ It’s necessary that God exists in reality. [Use P for “God exists as a possibility,” R for “God exists in reality,” and G for “There’s a being greater than God.” This is a modal version of St Anselm’s ontological argument.] 0241

13.If “X is good” and “I like X” are interchangeable, then “I like hurting people” logically entails “Hurting people is good.”

“I like hurting people but hurting people isn’t good” is consistent.

∴ “X is good” and “I like X” aren’t interchangeable. [Use I, L, and G. This argument attacks subjectivism.]

14.“You sin” entails “You know what you ought to do and you’re able to do it and you don’t do it.”

It’s necessary that if you know what you ought to do then you want to do it.

It’s necessary that if you want to do it and you’re able to do it then you do it.

∴ It’s impossible for you to sin. [S, K, A, D, W]

15.Necessarily, if it’s true that there are no truths then there are truths.

∴ It’s necessary that there are truths. [Use T for “There are truths.”]

10.3 Refutations

Applying our proof strategy to an invalid argument leads to a refutation:

· It’s possible that it’s heads.

· It’s possible that it’s tails.

· ∴ It’s possible that it’s both heads and tails.

· * 1 ◇H Invalid

· * 2 ◇T

· [ ∴ ◇(H • T)

· * 3 asm: ∼◇(H • T)

· 4 ∴ ☐∼(H • T) {from 3}

· 5 W ∴ H {from 1}

· 6 WW ∴ T {from 2}

· * 7 W ∴ ∼(H • T) {from 4}

· * 8 WW ∴ ∼(H • T) {from 4}

· 9 W ∴ ∼T {from 5 and 7}

· 10 WW ∴ ∼H {from 6 and 8}

Reverse a squiggle (line 4). Drop two diamonds, using a new world each time (lines 5 and 6). Drop the box twice, using W and WW (lines 7 and 8). Getting no contradiction, we gather simple wffs for a refutation. We get a little galaxy of two possible worlds: one with heads-and-not-tails and another with tails-and-not-heads. The argument is invalid, since this galaxy makes the premises both true (since it’s heads in one possible world and tails in another) but the conclusion false (since no possible world has both heads and tails).

If we try to prove an invalid argument, we’ll instead be led to a refutation – a galaxy of possible worlds that make the premises all true and conclusion false. In evaluating premises and conclusion, use these rules to evaluate each formula or subformula that starts with a modal operator: 0242

“◇A” is true if and only if at least one world has “A” true.

“☐A” is true if and only if all worlds have “A” true.

Premise “◇H” is true because world W has “H” true, and premise “◇T” is true because world WW has “T” true.¹ But conclusion “◇(H • T)” is false because no world has “(H • T)” true:

¹ POSSIBLE is like OR: something holds in this world OR that world OR that world … – so a single true case makes a POSSIBLE true. NECESSARY is like AND: something holds in this world AND that world AND that world … – so a single false case makes a NECESSARY false.

In W: (H • T) = (1 • 0) = 0

In WW: (H • T) = (0 • 1) = 0

Always check that your refutation works. If you don’t get premises all 1 and conclusion 0, then you did something wrong; look at what you did with the wff that came out wrong (a premise that’s 0 or ?, or a conclusion that’s 1 or ?).

These two rules are crucial for working out proofs and refutations:

· For each initial diamond, introduce a new world.

· For each initial box, derive an instance for each old world.

If you have two diamonds, don’t drop both using the same world – and don’t drop just one diamond. And if you have two worlds, then drop any box using both worlds; if in our example we dropped the box in “☐∼(H • T)” using “W” but not “WW,” then our attempted refutation would fail:

Since “H” is unknown in WW, our conclusion “◇(H • T)” would also be unknown (because the second case with “WW” is unknown):

In W: (H • T) = (1 • 0) = 0

In WW: (H • T) = (? • 1) = ?

The “It’s possible that it’s both heads and tails” conclusion is unknown, since our world WW doesn’t exclude it being heads (besides being tails). We avoid such problems if we drop each initial box using each old world; here we’d go from “☐∼(H • T)” to “WW ∴ ∼(H • T),” which would lead to “WW ∴ ∼H.”

As we refute arguments, we’ll often have to evaluate premises or conclusions that don’t start with boxes or diamonds, such as these wffs: 0243

Identify any subformulas that start with a boxes or diamonds (as highlighted here). Evaluate each subformula to be 1 or 0, and then apply “∼” to reverse the result. On our heads-tails refutation, “☐H” = 0, and so “∼☐H” = 1. Likewise, “☐(H ∨ T)” = 1, and so “∼☐(H ∨ T)” = 0; and “◇(H • T)” = 0, and so “∼◇(H • T)” = 1. In evaluating a wff that starts with a squiggle and then a box-or-diamond, evaluate the wff without the squiggle and then give the original wff the opposite value. Divide and conquer!

Here’s another invalid argument:

· 1 (☐A ⊃ ☐B) Invalid

· [∴ (A ⊃ B)

· * 2 asm: ∼(A ⊃ B)

· 3 ∴ A {from 2}

· 4 ∴ ∼B {from 2}

· ** 5 asm: ∼☐A {break 1}

· ** 6 ∴ ◇∼A {from 5}

· 7 W ∴ ∼A {from 6}

Our refutation has an actual world and a possible world W. To evaluate the premise, first identity and evaluate subformulas that start with a box or diamond (these are highlighted here), and then plug in 1 or 0 for these:

The conclusion is “(A ⊃ B),” which uses unmodalized letters; these should be evaluated in the actual world. So conclusion (A ⊃ B) = (1 ⊃ 0) = 0. Since we have true premises and a false conclusion, our argument is invalid.

As we refute invalid arguments, we’ll often have complex premises or conclusions to evaluate, such as these wffs:

As above, first identity subformulas that start with boxes or diamonds (as highlighted). Evaluate each such subformula to be 1 or 0, replace it with 1 or 0, and figure out whether the whole formula is 1 or 0. Divide and conquer! 0244

This English argument has an ambiguous first premise, which could have two different meanings:

· If you’re a bachelor, then you must be unmarried.

· You’re a bachelor.

· ∴ It’s logically necessary that you’re unmarried.

· (B ⊃ ☐U) If you’re a bachelor, then you’re inherently unmarriable.

· ☐(B ⊃ U) It’s necessary that if you’re a bachelor then you’re unmarried.

Work out both versions:

Box-inside version (valid but premise 1 is false):

· * 1 (B ⊃ ☐U) Valid

· 2 B

· [ ∴ ☐U

· 3 ⌈ asm: ∼☐U

· 4 ⌊ ∴ ☐U {from 1 and 2}

· 5 ∴ ☐U {from 3; 3 contradicts 4}

Box-outside version (invalid):

· 1 ☐(B ⊃ U) Invalid

· 2 B

· [∴ ☐U

· * 3 asm: ∼☐U

· * 4 ∴ ◇∼U {from 3}

· 5 W ∴ ∼U {from 4}

· * 6 W ∴ (B ⊃ U) {from 1}

· * 7 ∴ (B ⊃ U) {from 1}

· 8 W ∴ ∼B {from 5 and 6}

· 9 ∴ U {from 2 and 7}

Both versions are flawed: the first has a false premise, while the second is invalid. So the proof that you’re inherently unmarriable fails. Arguments with a modal ambiguity often have one interpretation with a false premise and another that’s invalid; such arguments often seem sound until we focus on the ambiguity.

10.3a Exercise: LogiCola KI

Prove each of these arguments to be invalid (all are invalid).

· ☐ (A ⊃ B)

· ◇ A

· ∴ ☐ B

· 1 ☐(A ⊃ B) Invalid

· * 2 ◇A

· [∴ ☐B

· * 3 asm: ∼☐B

· * 4 ∴ ◇∼B {from 3}

· 5 W ∴ ∼B {from 4}

· 6 WW ∴ A {from 2}

· * 7 W ∴ (A ⊃ B) {from 1}

· * 8 WW ∴ (A ⊃ B) {from 1}

· 9 W ∴ ∼A {from 5 and 7}

· 10 WW ∴ B {from 6 and 8}

0245

1. ◇A

∴ ☐A

2. A

∴ ☐A

3. ◇A

◇B

∴ ◇(A • B)

4. ☐(A ⊃ ∼B)

B

∴ ☐∼A

5. (☐A ⊃ ☐B)

∴ ☐(A ⊃ B)

6. ◇A

∼☐B

∴ ∼☐(A ⊃ B)

7. ☐(C ⊃ (A ∨ B))

(∼A • ◇∼B)

∴ ◇∼C

8. ☐(A ∨ ∼B)

∴ (∼◇B ∨ ☐A)

9. ☐((A • B) ⊃ C)

◇A

◇B

∴ ◇C

10.∼☐A

☐(B ≡ A)

∴ ∼◇B

10.3b Exercise: LogiCola KC

First appraise intuitively. Then translate into logic (using the letters given) and say whether valid (and give a proof) or invalid (and give a refutation). Translate ambiguous English arguments both ways; prove or disprove each symbolization separately.

1. If the pragmatist view of truth is right, then “A is true” entails “A is useful to believe.”

“A is true but not useful to believe” is consistent.

∴ The pragmatist view of truth isn’t right. [Use P, T, and B.]

2. You know.

“You’re mistaken” is logically possible.

∴ “You know and are mistaken” is logically possible. [Use K and M.]

3. Necessarily, if this will be then this will be.

∴ If this will be, then it’s necessary (in itself) that this will be. [Use B. This illustrates two senses of “Que será será” – “Whatever will be will be.” The first sense is a truth of logic while the second is a form of fatalism.]

4. I’m still.

If I’m still, then it’s necessary that I’m not moving.

If it’s necessary that I’m not moving, then whether I move is not a matter of my free choice.

∴ Whether I move is not a matter of my free choice. [Use S, M, and F. This is adapted from the medieval thinker Boethius, who used a similar example to explain the box-inside/box-outside distinction.]

5. It’s necessarily true that if you’re morally responsible for your actions then you’re free.

It’s necessarily true that if your actions are uncaused then you aren’t morally responsible for your actions.

∴ “You’re free” doesn’t entail “Your actions are uncaused.” [Use R, F, and U; from A. J. Ayer.] 0246

6. If “One’s conscious life won’t continue forever” entails “Life is meaningless,” then a finite span of life is meaningless.

If a finite span of life is meaningless, then an infinite span of life is meaningless.

If an infinite span of life is meaningless, then “One’s conscious life will continue forever” entails “Life is meaningless.”

∴ If it’s possible that life is not meaningless, then “One’s conscious life won’t continue forever” doesn’t entail “Life is meaningless.” [C, L, F, I]

7. If you have money, then you couldn’t be broke.

You could be broke.

∴ You don’t have money. [Use M and B. Is this argument just a valid instance of modus tollens: “(P ⊃ Q), ∼Q ∴ ∼P”?]

8. If you know, then you couldn’t be mistaken.

You could be mistaken.

∴ You don’t know. [Use K and M. Since we could repeat this reasoning for any alleged item of knowledge, the argument seems to show that genuine knowledge is impossible.]

9. It’s necessary that if there’s a necessary being then “There’s a necessary being” (by itself) is necessary.

“There’s a necessary being” is logically possible.

∴ “There’s a necessary being” is logically necessary. [Use N for “There’s a necessary being” or “There’s a being that exists of logical necessity”; this being is often identified with God; from Charles Hartshorne and St Anselm; it’s sometimes called “Anselm’s second ontological argument.” The proof raises logical issues that we’ll deal with in the next chapter.]

10.It’s necessary that either I’ll do it or I won’t do it.

If it’s necessary that I’ll do it, then I’m not free.

If it’s necessary that I won’t do it, then I’m not free.

∴ I’m not free. [Use D for “I’ll do it.” Aristotle and the Stoic Chrysippus discussed this argument. This argument’s flaw relates to a point made by Chrysippus, that “☐(D ∨ ∼D) ∴ (☐D ∨ ☐∼D)” is invalid and is like arguing “Everything is either A or non-A; therefore either everything is A or everything is non-A.”]

11.“This agent’s actions were all determined” is consistent with “I describe this agent’s character in an approving way.”

“I describe this agent’s character in an approving way” is consistent with “I praise this agent.”

∴ “This agent’s actions were all determined” is consistent with “I praise this agent.” [D, A, P]

12.If thinking is just a chemical brain process, then “I think” entails “There’s a chemical process in my brain.”

“There’s a chemical process in my brain” entails “I have a body.”

“I think but I don’t have a body” is logically consistent.

∴ Thinking isn’t just a chemical brain process. [Use J, T, C, and B. This argument attacks a form of materialism.] 0247

13.If “I did that on purpose” entails “I made a prior purposeful decision to do that,” then there’s an infinite chain of previous decisions to decide.

It’s impossible for there to be an infinite chain of previous decisions to decide.

∴ “I did that on purpose” is consistent with “I didn’t make a prior purposeful decision to do that.” [Use D, P, and I; from Gilbert Ryle.]

14.God knew that you’d do it.

If God knew that you’d do it, then it was necessary that you’d do it.

If it was necessary that you’d do it, then you weren’t free.

∴ You weren’t free. [Use K, D, and F. This argument is the focus of an ancient controversy. Would divine foreknowledge preclude human freedom? If it would, then should we reject human freedom (as did Luther) or divine foreknowledge (as did Charles Hartshorne)? Or perhaps (as the medieval thinkers Boethius, Aquinas, and Ockham claimed) is there a flaw in the argument that divine foreknowledge precludes human freedom?]

15.If “good” means “socially approved,” then “Racism is socially approved” logically entails “Racism is good.”

“Racism is socially approved but not good” is consistent.

∴ “Good” doesn’t mean “socially approved.” [Use M, S, and G. This argument attacks cultural relativism.]

16.Necessarily, if God brings it about that A is true, then A is true.

A is a self-contradiction.

∴ It’s impossible for God to bring it about that A is true. [Use B and A, where B is for “God brings it about that A is true.”]

17.If this is experienced, then this must be thought about.

“This is thought about” entails “This is put into the categories of judgments.”

∴ If it’s possible for this to be experienced, then it’s possible for this to be put into the categories of judgments. [Use E, T, and C; from Immanuel Kant, who argued that our mental categories apply, not necessarily to everything that exists, but rather to everything that we could experience.]

18.Necessarily, if formula B has an all-1 truth table then B is true.

∴ If formula B has an all-1 truth table, then B (taken by itself) is necessary. [Use A and B. This illustrates the box-outside versus box-inside distinction.]

19.Necessarily, if you mistakenly think that you exist then you don’t exist.

Necessarily, if you mistakenly think that you exist then you exist.

∴ “You mistakenly think that you exist” is impossible. [Use M and E. This relates to Descartes’s “I think, therefore I am” (“Cogito ergo sum”).]

20.If “good” means “desired by God,” then “This is good” entails “There’s a God.”

“There’s no God, but this is good” is consistent.

∴ “Good” doesn’t mean “desired by God.” [Use M, A, and B. This attacks one form of the divine command theory of ethics. Some (see 9 and 26 of this section and 12 of §10.2b) say, against premise 2, that “There’s no God” is logically impossible.] 0248

21.If Plato is right, then it’s necessary that ideas are superior to material things.

It’s possible that ideas aren’t superior to material things.

∴ Plato isn’t right. [P, S]

22.“I seem to see a chair” doesn’t entail “There’s an actual chair that I seem to see.”

If we directly perceive material objects, then “I seem to see a chair and there’s an actual chair that I seem to see” is consistent.

∴ We don’t directly perceive material objects. [S, A, D]

23.“There’s a God” is logically incompatible with “There’s evil in the world.”

There’s evil in the world.

∴ “There’s a God” is self-contradictory. [G, E]

24.If you do all your homework right, then it’s impossible that you get this problem wrong.

It’s possible that you get this problem wrong.

∴ You don’t do all your homework right. [R, W]

25.“You do what you want” is compatible with “Your act is determined.”

“You do what you want” entails “Your act is free.”

∴ “Your act is free” is compatible with “Your act is determined.” [W, D, F]

26.It’s necessarily true that if God doesn’t exist in reality then there’s a being greater than God (since then any existing being would be greater than God).

It’s not possible that there’s a being greater than God (since “God” is defined as “a being than which no being could be greater”).

∴ It’s necessary that God exists in reality. [Use R and B. This is a simplified modal form of St Anselm’s ontological argument.]

27.It was always true that you’d do it.

If it was always true that you’d do it, then it was necessary that you’d do it.

If it was necessary that you’d do it, then you weren’t free.

∴ You weren’t free. [Use A (for “It was always true that you’d do it” – don’t use a box here), D, and F. This argument is much like problem 14. Are statements about future contingencies (for example, “I’ll brush my teeth tomorrow”) true or false before they happen? Should we do truth tables for such statements in the normal way, assigning them “1” or “0”? Does this preclude human freedom? If so, should we then reject human freedom? Or should we adopt a many-valued logic that says that statements about future contingencies aren’t “1” or “0” but must instead have some third truth value (maybe “½”)? Or is the argument fallacious?]