11

Further Modal Systems

Modal logic studies arguments whose validity depends on “possible,” “necessary,” and similar notions. The previous chapter presented a basic system that builds on propositional logic. This present chapter considers alternative systems of propositional and quantified modal logic.

11.1 Galactic travel

While logicians usually agree on which arguments are valid, there are more disagreements about modal arguments. Many disputes involve arguments in which one modal operator occurs within the scope of another – like “◇◇A ∴ ◇A” and “☐(A ⊃ ☐B), ◇A ∴ B.” These disputes reflect differences in how to formulate the drop-box rule. So far, we’ve assumed a system called “S5,” which lets us go from any world to any world when we drop a box (§10.2):

Drop box DB

☐A → W ∴ A,

use any world prefix

Here the line with “☐A” and the line with “W ∴ A” can use any world prefixes, the same or different.

This assumes that whatever is necessary in any world is thereby true in all worlds without restriction. A further implication is that whatever is necessary in one world is thereby necessary in all worlds.

Some weaker views reject these ideas. On these views, what is necessary only has to be true in all “suitably related” worlds; so these views restrict the drop-box rule. All the views in question let us go from “☐A” in a world to “A” in the same world. But we can’t always go from “☐A” in one world to “A” in another world; traveling between worlds requires (at least on my way of expressing it) a suitable “travel ticket.”

We get travel tickets when we drop diamonds. Let “W1” and “W2” stand for world prefixes. Suppose we go from “◇A” in world W1 to “A” in new world W2. Then we get a travel ticket from W1 to W2, and we’ll write “W1 ⇒ W2”:

W1 ⇒ W2

We have a ticket to move from world W1 to world W2

0250 Suppose we do a proof with wffs “◇◇A” and “◇B.” We’d get these travel tickets when we drop diamonds (here “#” stands for the actual world):

· 1 ◇◇A

· 2 ◇B

· .……….

· 11 W ∴ ◇A {from 1} # ⇒ W

· 12 WW ∴ A {from 11} W ⇒ WW

· 13 WWW ∴ B {from 2} # ⇒ WWW

Dropping a diamond gives us a travel ticket from the world in the “from” line to the world in the “to” line. So in line 11 we get ticket “# ⇒ W” – because we moved from “◇◇A” in the actual world (“#”) to “◇A” in world W. Tickets are reusable; we can use “W1 ⇒ W2” any number of times.

The rules for using tickets vary. System T lets us use only one ticket at a time, and only in the direction of the arrow; system S4 lets us combine a series of tickets, while system B lets us use them in a backwards direction. Suppose we have “☐A” in world W1 and want to put “A” in world W2:

· System T. We need a ticket from W1 to W2.

· System S4. Like T, but we also can use a series of tickets.

· System B. Like T, but a ticket also works backwards.

Suppose we have three travel tickets:

# ⇒ W

W ⇒ WW

# ⇒ WWW

System T would let us, when we drop boxes, go from # to W, from W to WW, and from # to WWW. The other systems allow these and more. System S4 lets us use a series of tickets in the direction of the arrow; this lets us go from # to WW. System B lets us use single tickets backwards; this lets us go from W to #, from WW to W, and from WWW to #. In contrast, system S5 lets us go from any world to any world; this is equivalent to letting us use any ticket or series of tickets in either direction.

S5 is the most liberal system and accepts the most valid arguments; so S5 is the strongest system. T is the weakest system, allowing the fewest proofs. S4 and B are intermediate, each allowing some proofs that the other doesn’t. The four systems give the same result for most arguments. But some arguments are valid in one system but invalid in another; these arguments use wffs that apply a modal operator to a wff already containing a modal operator.

This argument is valid in S4 or S5 but invalid in T or B: 0251

Line 7 requires that we combine a series of tickets in the direction of the arrow. Tickets “# ⇒ W” and “W ⇒ WW” then let us go from actual world # (line 1) to world WW (line 7). This requires systems S4 or S5.

This next one is valid in B or S5 but invalid in T or S4:

Line 6 requires using ticket “# ⇒ W” backwards, to go from world W (line 5) to the actual world # (line 6). This requires systems B or S5.

This last one is valid in S5 but invalid in T or B or S4:

Line 7 requires combining a series of tickets and using some backwards. Tickets “# ⇒ W” and “# ⇒ WW” then let us go from W (line 5) to WW (line 7). This requires system S5.

S5 is the simplest system in several ways:

· We can formulate S5 more simply. The box-dropping rule doesn’t have to mention travel tickets; we need only say that, if we have “☐A” in any world, then we can put “A” in any world (the same or a different one). 0252

· S5 captures simple intuitions about necessity and possibility: what’s necessary is what’s true in all worlds, what’s possible is what’s true in some worlds, and what’s necessary or possible doesn’t vary between worlds.

· On S5, any string of boxes and diamonds simplifies to its last symbol. So “☐☐” and “◇☐” simplify to “☐,” and “◇◇” and “☐◇” simplify to “◇.”

Which is the best system? This depends on what we take the box and diamond to mean. If we take them to be about the logical necessity and possibility of ideas, then S5 is the best system. If an idea (for example, the claim that 2 = 2) is logically necessary, then it couldn’t have been other than logically necessary. So if A is logically necessary, then it’s logically necessary that A is logically necessary [“(☐A ⊃ ☐☐A)”]. Similarly, if an idea is logically possible, then it’s logically necessary that it’s logically possible [“(◇A ⊃ ☐◇A)”]. Of the four systems, only S5 accepts both formulas. All this presupposes that we use the box to talk about the logical necessity of ideas.

Or we could take the box to be about the logical necessity of sentences. The sentence “2 = 2” just happens to express a necessary truth; it wouldn’t have expressed one if English had used “=” to mean “≠.” So the sentence is necessary, but it’s not necessary that it’s necessary; this makes “(☐A ⊃ ☐☐A)” false. The idea that “2 = 2” now expresses, however, is both necessary and necessarily necessary; a change in our language wouldn’t make this idea false, but it would change how we’d express this idea. So whether S5 is the best system can depend on whether we take the box to be about the necessity of ideas or of sentences.

There are still other ways to take “necessary.” Sometimes calling something “necessary” might mean that it’s “physically necessary,” “proven,” “known,” or “obligatory.” Some logicians like the weak system T because it holds for various senses of “necessary”; such logicians might still use S5 for arguments about the logical necessity of ideas. While I have sympathy with this view, most of the modal arguments I’m interested in are about the logical necessity of ideas. So I use S5 as the standard system of modal logic but feel free to switch to weaker systems for arguments about other kinds of necessity.

Here we’ve considered the four main modal systems. We could invent other systems – for example, ones in which we can combine travel tickets only in groups of three. Logicians develop such systems, not to help us in analyzing real arguments, but rather to explore interesting formal structures.¹

¹ For more on alternative modal systems, consult G. E. Hughes and M. J. Cresswell, A New Introduction to Modal Logic (London: Routledge, 1996).

11.1a Exercise: LogiCola KG

Using system S5, prove each of these arguments to be valid. Also say in which systems the argument is valid: T, B, S4, or S5. 0253

∼☐A

∴ ☐∼☐A

Line 7 combines a series of tickets and uses some backwards. This requires S5.

1. ◇☐A

∴ A

2. ◇A

∴ ◇◇A

3. ◇◇A

∴ ◇A

4. ◇☐A

∴ ☐A

5. (☐A ⊃ ☐B)

∴ ☐(☐A ⊃ ☐B)

6. ☐(A ⊃ B)

∴ ☐(☐A ⊃ ☐B)

7. (◇A ⊃ ☐B)

∴ ☐(A ⊃ ☐B)

8. ☐(A ⊃ ☐B)

∴ (◇A ⊃ ☐B)

9. ◇☐◇A

∴ ◇A

10.◇A

∴ ◇☐◇A

11.☐A

∴ ☐(B ⊃ ☐A)

12.☐◇☐◇A

∴ ☐◇A

13.☐◇A

∴ ☐◇☐◇A

14.☐(A ⊃ ☐B)

◇A

∴ ☐B

15.☐A

∴ ☐☐☐A

11.1b Exercise: LogiCola KG

Fist appraise intuitively. Then translate into logic (using the letters given) and, assuming S5, prove validity. Also say in which systems the argument is valid: T, B, S4, or S5.

1. 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. This argument (which we saw before in §10.3b) is from Charles Hartshorne and St Anselm. Its validity depends on which system of modal logic is correct. Some philosophers defend the argument, often after defending a modal system needed to make it valid. Others argue that the argument is invalid, and so any modal system that would make it valid must be wrong. Still others deny the theological import of the conclusion; they say that a necessary being could be a prime number or the world and needn’t be God.]

2. “There’s a necessary being” isn’t a contingent statement.

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

∴ There’s a necessary being. [Use N. This version of the Anselm–Hartshorne argument is more clearly valid.] 0254

3. Prove that the first premise of argument 1 is logically equivalent to the first premise of argument 2 by showing that each can be deduced from the other. In which systems does this equivalence hold? 3. It’s necessary that if there’s a necessary being then “There’s a necessary being” (by itself) is necessary.

“There’s no necessary being” is logically possible.

∴ There’s no necessary being. [Use N. Some object that the first premise of the Anselm–Hartshorne argument just as easily leads to an opposite conclusion.]

4. It’s necessary that 2 + 2 = 4.

It’s possible that no language ever existed.

If all necessary truths hold because of language conventions, then “It’s necessary that 2 + 2 = 4” entails “Some language has sometime existed.”

∴ Not all necessary truths hold because of language conventions. [Use T, L, and N. This attacks the linguistic theory of logical necessity.]

11.2 Quantified translations

We’ll now develop a quantified modal system that combines our quantificational and modal systems. We’ll call this our “naïve” system, since it ignores certain problems; later we’ll add refinements.¹

¹ My understanding of quantified modal logic follows Alvin Plantinga’s The Nature of Necessity (London: Oxford University Press, 1974). For related discussions, see Saul Kripke’s Naming and Necessity (Cambridge, Mass.: Harvard University Press, 1980) and Kenneth Konyndyk’s Introductory Modal Logic (Notre Dame, Ind.: Notre Dame Press, 1986).

Many quantified modal translations are easy. This pair is tricky:

· Everyone could be above average

· = ◇(x)Ax

· It’s possible that everyone is above average

· It’s possible that, for all x, x is above average

· Anyone could be above average

· = (x)◇Ax

· For all x, it’s possible that x is above average

The first is false while the second is true.

Quantified modal logic can express the difference between necessary and contingent properties. Numbers seem to have both kinds of property. The number 8, for example, has the necessary properties of being even and of being one greater than seven; 8 couldn’t have lacked these properties. But 8 also has contingent properties, ones it could have lacked, such as being my favorite number and being less than the number of chapters in this book. We can symbolize “necessary property” and “contingent property” as follows: 0255

· F is a necessary (essential) property of x

· = ☐Fx

· x is necessarily F (x has the necessary property of being F)

· In all possible worlds, x would be F

· F is a contingent (accidental) property of x

· = (Fx • ◇∼Fx)

· x is contingently F (x is F but could have lacked F)

· In the actual world x is F; but in some possible world x isn’t F.

Humans have mostly contingent properties. Socrates had contingent properties, like having a beard and being a philosopher; these are contingent, because he could (without self-contradiction) have been a clean-shaven non-philosopher. But Socrates also had necessary properties, like being self-identical and not being a square circle; every being has these properties of necessity.

Aristotelian essentialism is the controversial view that there are properties that some beings have of necessity but some other beings totally lack. Plantinga, supporting this view, suggests that Socrates had of necessity these properties that some other beings totally lack: not being a prime number, being snub-nosed in W (a specific possible world), being a person (capable of conscious rational activity), and being identical with Socrates. This last property differs from that of being named “Socrates.”

Plantinga explains “necessary property” as follows. Suppose “a” names a being and “F” names a property. Then the entity named by “a” has the property named by “F” necessarily, if and only if the proposition expressed by “a is non-F” is logically impossible. Then to say that Socrates necessarily has the property of not being a prime number is to say that the proposition “Socrates is a prime number” (with the name “Socrates” referring to the person Socrates) is logically impossible. We must use names (like “Socrates”) here and not definite descriptions (like “the entity I’m thinking about”).

We previously discussed the box-inside/box-outside ambiguity. This quantified modal sentence similarly could have either of two meanings:

“All bachelors are necessarily unmarried.”

Simple necessity

(x)(Bx ⊃ ☐Ux)

All bachelors are inherently unmarriable – in no possible world would anyone marry them.

Conditional necessity

☐(x)(Bx ⊃ Ux)

It’s necessarily true that all bachelors are unmarried. (The meaning of “bachelor” makes this true.)

When translating a statement like “All A’s are necessarily B’s,” give both forms. With ambiguous arguments, work out both arguments. As before, fallacies can result from confusing the forms.

Discussions about Aristotelian essentialism frequently involve such modal 0256 ambiguities. This following sentence could have either of two meanings:

“All persons are necessarily persons.”

Simple necessity

(x)(Px ⊃ ☐Px)

Everyone who in fact is a person has the necessary property of being a person.

Conditional necessity

☐(x)(Px ⊃ Px)

It’s necessary that all persons are persons.

The first is controversial and attributes to each person the necessary property of being a person; the medievals called this de re (“of the thing”) necessity. If this first form is true, then you couldn’t have been a non-person – your existing as a non-person is self-contradictory; this excludes the possibility of your being reincarnated as an unconscious doorknob. In contrast, the second form is trivially true and attributes necessity to the proposition (or saying) “All persons are persons”; the medievals called this de dicto (“of the saying”) necessity.

11.2a Exercise: LogiCola J (QM & QT)

Translate these English sentences into wffs; translate ambiguous forms both ways.

It’s necessary that all mathematicians have the necessary property of being rational.

☐(x)(Mx ⊃ ☐Rx)

Here the first “☐” symbolizes de dicto necessity (“It’s necessary that …”), while the second symbolizes de re necessity (“have the necessary property of being rational”).

1. It’s possible for anyone to be unsurpassed in greatness. [Use Ux.]

2. It’s possible for everyone to be unsurpassed in greatness.

3. John has the necessary property of being unmarried. [Use Ux and j.]

4. All experts are necessarily smart. [Ex, Sx]

5. Being named “Socrates” is a contingent property of Socrates. [Nx, s]

6. It’s necessary that everything is self-identical. [Use “=.”]

7. Every entity has the necessary property of being self-identical.

8. John is necessarily sitting. [Sx, j]

9. Everyone observed to be sitting is necessarily sitting. [Ox, Sx]

10.All numbers have the necessary property of being abstract entities. [Nx, Ax]

11.It’s necessary that all living beings in this room are persons. [Lx, Px]

12.All living beings in this room have the necessary property of being persons.

13.All living beings in this room have the contingent property of being persons.

14.Any contingent claim could be true. [Cx, Tx]

15.“All contingent claims are true” is possible.

16.All mathematicians are necessarily rational. [Mx, Rx]

17.All mathematicians are contingently two-legged. [Mx, Tx]

18.All mathematical statements that are true are necessarily true. [Mx, Tx] 0257

19.It’s possible that God has the necessary property of being unsurpassed in greatness. [Ux, g]

20.Some being has the necessary property of being unsurpassed in greatness. [Ux]

11.3 Quantified proofs

On our initial naïve approach to quantified modal logic (which has defects), we just use the same quantificational and modal inference rules as before. Here’s a quantified modal proof:

· It’s necessary that everything is self-identical.

· ∴ Every entity has the necessary property of being self-identical.

This next modal argument has an ambiguous premise:

· All bachelors are necessarily unmarried.

· You’re a bachelor.

· ∴ “You’re unmarried” is logically necessary.

Premise 1 might assert either simple necessity “(x)(Bx ⊃ ☐Ux)” (“All bachelors are inherently unmarriable”) or conditional necessity ☐(x)(Bx ⊃ Ux)” (“It’s necessary that all bachelors are unmarried”). We’ll work it out both ways:

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

Box-outside version (invalid):

· 1 ☐(x)(Bx ⊃ Ux) Invalid

· 2 Bu

· [ ∴ ☐Uu

· * 3 asm: ∼☐Uu

· 4 ∴ ◇∼Uu {from 3}

· 5 W ∴ ∼Uu {from 4}

· 6 W ∴ (x)(Bx ⊃ Ux) {from 1}

· 7 ∴ (x)(Bx ⊃ Ux) {from 1}

· * 8 W ∴ (Bu ⊃ Uu) {from 6}

· * 9 ∴ (Bu ⊃ Uu) {from 7}

· 10 W ∴ ∼Bu {from 5 and 8}

· 11 ∴ Uu {from 2 and 9}

0258 Both versions are flawed; the first has a false premise while the second is invalid. So another proof that you’re inherently unmarriable fails! Ambiguous modal arguments often have one interpretation with a false premise and another that’s invalid. Such arguments may seem sound until we focus on the ambiguity.

Our refutation has two possible worlds, each with only one entity – you. In the actual world, you’re a bachelor and unmarried; in world W, you’re not a bachelor and not unmarried. In this galaxy, the premises are true (since in both worlds all bachelors are unmarried – and in the actual world you’re a bachelor) but the conclusion is false (since in world W you’re not unmarried).

As with relations, applying our proof strategy mechanically sometimes leads into an endless loop. Here we keep getting new letters and worlds, endlessly:

· It’s possible for anyone to be above average.

· ∴ It’s possible for everyone to be above average.

· 1 (x)◇Ax

· [ ∴ ◇(x)Ax

· * 2 asm: ∼◇(x)Ax

· 3 ∴ ☐∼(x)Ax {from 2}

· * 4 ∴ ◇Aa {from 1} New letter!

· 5 W ∴ Aa {from 4} New world!

· * 6 W ∴ ∼(x)Ax {from 3}

· * 7 W ∴ (∃x)∼Ax {from 6}

· 8 W ∴ ∼Ab {from 7} New letter!

· * 9 ∴ ◇Ab {from 1}

· 10 WW ∴ Ab {from 9} New world!

· * 11 WW ∴ ∼(x)Ax {from 3}

· 12 WW ∴ (∃x)∼Ax {from 11}

· … and so on endlessly …

Using ingenuity, we can devise a refutation with two entities and two worlds:

Here each person is above average in some world or other – but in no world is every person above average. For now, we’ll assume in our refutations that every world contains the same entities (and at least one such entity).

11.3a Exercise: LogiCola KQ

Say whether valid (and give a proof) or invalid (and give a refutation). 0259

· (x)☐Fx

· ∴ ☐(x)Fx

This is called a “Barcan inference,” after Ruth Barcan Marcus. It’s doubtful that our naïve quantified modal logic gives the right results for arguments like this (see §11.4).

1. (∃x)☐Fx

∴ ☐(∃x)Fx

2. a=b

∴ (☐Fa ⊃ ☐Fb)

3. ∴ ☐(∃x)x=a

4. ∴ (∃x)☐x=a

5. ◇(x)Fx

∴ (x)◇Fx

6. ∴ (x)☐x=x

7. ∴ ☐(x)x=x

8. ☐(x)(Fx ⊃ Gx)

∴ (x)(Fx ⊃ ☐Gx)

9. ◇(∃x)Fx

∴ (∃x)◇Fx

10.(∃x)◇Fx

∴ ◇(∃x)Fx

11.(◇(x)Fx ⊃ (x)◇Fx)

∴ ((∃x)∼Fx ⊃ ☐(∃x)∼Fx)

12.∴ (x)(y)(x=y ⊃ ☐x=y)

13.☐(x)(Fx ⊃ Gx)

☐Fa

∴ ☐Ga

14.∼a=b

∴ ☐∼a=b

11.3b Exercise: LogiCola KQ

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. I have a beard.

∴ “Whoever doesn’t have a beard isn’t me” is a necessary truth. [Use Bx and i. G. E. Moore criticized such reasoning, which he saw as essential to idealistic metaphysics and its claim that every property of a thing is necessary. The conclusion entails that “I have a beard” is logically necessary. Moore would see “Whoever doesn’t have a beard isn’t me” as only a contingent truth.]

2. “Whoever doesn’t have a beard isn’t me” is a necessary truth.

∴ “I have a beard” is logically necessary. [Use Bx and i.]

3. Aristotle isn’t identical to Plato.

If some being has the property of being necessarily identical to Plato but not all beings have the property of being necessarily identical to Plato, then some beings have necessary properties that other beings lack.

∴ Some beings have necessary properties that other beings lack. [Use a, p, and S (for “Some beings have necessary properties that other beings lack”). This defense of Aristotelian essentialism is essentially from Alvin Plantinga.] 0260

4. All mathematicians are necessarily rational.

Paul is a mathematician.

∴ Paul is necessarily rational. [Mx, Rx, p]

5. Necessarily there exists something unsurpassed in greatness.

∴ There exists something that necessarily is unsurpassed in greatness. [Ux]

6. The number that I’m thinking of isn’t necessarily even.

8 = the number that I’m thinking of.

∴ 8 isn’t necessarily even. [Use n, E, and e. Does our naïve quantified modal logic correctly decide whether this argument is valid?]

7. “I’m a thinking being, and there are no material objects” is logically possible.

Every material object has the necessary property of being a material object.

∴ I’m not a material object. [Use Tx, Mx, and i; from Alvin Plantinga.]

8. All humans are necessarily rational.

All living beings in this room are human.

∴ All living beings in this room are necessarily rational. [Use Hx, Rx, and Lx; from Aristotle, who was the first logician and the first to combine quantification with modality.]

9. It’s not necessary that all cyclists are rational.

Paul is a cyclist.

Paul is rational.

∴ Paul is contingently rational. [Cx, Rx, p]

10.“Socrates has a pain in his toe but doesn’t show pain behavior” is consistent.

It’s necessary that everyone who has a pain in his toe is in pain.

∴ “All who are in pain show pain behavior” isn’t a necessary truth. [Use s, Tx for “x has a pain in his toe,” Bx for “x shows pain behavior,” and Px for “x is in pain.” This attacks a behaviorist analysis of the concept of “pain.”]

11.If Q (the question “Why is there something and not nothing?”) is a meaningful question, then it’s possible that there’s an answer to Q.

Necessarily, every answer to Q refers to an existent that explains the existence of other things.

Necessarily, nothing that refers to an existent that explains the existence of other things is an answer to Q.

∴ Q isn’t a meaningful question. [M, Ax, Rx]

12.The number of apostles is 12.

12 is necessarily greater than 8.

∴ The number of apostles is necessarily greater than 8. [Use n, t, e, and Gxy. Does our naïve system correctly decide whether this argument is valid?]

13.All (well-formed) cyclists are necessarily two-legged.

Paul is a (well-formed) cyclist.

∴ Paul is necessarily two-legged. [Cx, Tx, p] 0261

14.Something exists in the understanding than which nothing could be greater. (In other words, there’s some x such that x exists in the understanding and it’s not possible that there be something greater that x.)

Anything that exists in reality is greater than anything that doesn’t exist in reality. Socrates exists in reality.

∴ Something exists in reality than which nothing could be greater. (In other words, there’s some x such that x exists in reality and it’s not possible that there be something greater than x.) [Use Ux for “x exists in the understanding,” Rx for “x exists in reality,” Gxy for “x is greater than y,” and s for “Socrates.” Use a universe of discourse of possible beings – including fictional beings like Santa Claus in addition to actual beings. (Is this legitimate?) This is a form of St Anselm’s first ontological argument for the existence of God.]

15.“Someone is unsurpassably great” is logically possible.

“Everyone who is unsurpassably great is, in every possible world, omnipotent, omniscient, and morally perfect” is necessarily true.

∴ Someone is omnipotent, omniscient, and morally perfect. [Use Ux and Ox. This is a simplified form of Alvin Plantinga’s ontological argument for the existence of God. Plantinga regards the second premise as true by definition; he sees the first premise as controversial but reasonable.]

16.Anything could cease to exist.

∴ Everything could cease to exist. [Use Cx for “x ceases to exist.” Some see Aquinas’s third argument for the existence of God as requiring this inference.]

11.4 A sophisticated system

Our naïve quantified modal logic has defects. Dealing with these will push us to question established logical and metaphysical ideas.

First, our system mishandles definite descriptions (terms of the form “the so and so”). We’ve been translating definite descriptions using small letters, as in the following example:

The number I’m thinking of is necessarily odd = ☐On

But this English sentence is ambiguous; it could mean either of two things (where “Tx” means “I’m thinking of number x”):

· (∃x)((Tx • ∼(∃y)(∼x=y • Ty)) • ☐Ox)

· I’m thinking of just one number, and it has the necessary property of being odd.

· ☐(∃x)((Tx • ∼(∃y)(∼x=y • Ty)) • Ox)

· This is necessary: “I’m thinking of just one number and it’s odd.”

The first form (box inside) might be true – if, for example, the number 7 has the necessary property of being odd and I’m thinking of just the number 7. The 0262 second form (box outside) is definitely false, since it’s possible that I’m thinking of no number, or more than one number, or an even number.

So our naïve way to translate “the so and so” is ambiguous. To fix this problem, our sophisticated system will require that we symbolize “the so and so” using Russell’s “there is just one …” analysis (§9.6) – as in the above boxes. This analysis also blocks the proof of invalid arguments like this one:

· 8 is the number I’m thinking of.

· It’s necessary that 8 is 8.

· ∴ It’s necessary that 8 is the number I’m thinking of.

· e=n

· ☐e=e

· ∴ ☐e=n

This is invalid – since it may be only contingently true that 8 is the number I’m thinking of. The argument is provable in naïve quantified modal logic, since the conclusion follows from the premises by the substitute-equals rule (§9.2). Our sophisticated system avoids this by requiring the longer analysis of “the number I’m thinking of.” So “8 is the number I’m thinking of” gets changed into “I’m thinking of just one number and it is 8” – and the above argument becomes this:

· I’m thinking of just one number and it is 8.

· It’s necessary that 8 is 8.

· ∴ This is necessary: “I’m thinking of just one number and it is 8.”

· (∃x)((Tx • ∼(∃y)(∼x=y • Ty)) • x=e) Invalid ☐e=e

· ∴ ☐(∃x)((Tx • ∼(∃y)(∼x=y • Ty)) • x=e)

So translated, the argument becomes invalid and not provable.

The second problem is that our naïve system assumes that the same entities exist in all possible worlds. This leads to implausible results; for example, it makes Gensler (and everyone else) into a logically necessary being:

· ∴ In every possible world, there exists a being who is Gensler.

But Gensler isn’t a logically necessary being; there are impoverished possible worlds without me. So something is wrong here.

There are two ways out of the problem. One way changes how we take “(∃x).” The provable “☐(∃x)x=g” is false if we take “(∃x)” to mean “for some existing being x.” But we might take “(∃x)” to mean “for some possible being x”; then “☐(∃x)x=g” would mean the more plausible: “In every possible world, there’s a possible being who is Gensler.” Perhaps there’s a possible being Gensler in every 0263 world; in some of these worlds Gensler exists, and in others he doesn’t. This view would need an existence predicate “Ex” to distinguish between possible beings that exist and those that don’t; we could then use “(∃x)∼Ex” to say that there are possible beings that don’t exist. This view is paradoxical, since it posits non-existent beings.

Alvin Plantinga defends an opposing view, which he calls “actualism.” Actualism holds that to be a being and to exist is the same thing; there neither are nor could have been non-existent beings. Of course there could have been beings other than those that now exist. But this doesn’t mean that there now are beings that don’t exist. Actualism denies the latter claim.

Since I favor actualism, I’ll avoid non-existent beings and continue to take “(∃x)” to mean “for some existing being.” On this reading, “☐(∃x)x=g” means “It’s necessary that there’s an existing being who is Gensler.” This is false, since I might not have existed. So we must reject some line of the above proof.

The faulty line seems to be 5 (and its derivation from 4):

· In W, every existing being is distinct from Gensler.

· ∴ In W, Gensler is distinct from Gensler.

· 4 W ∴ (x)∼x=g

· 5 W ∴ ∼g=g {from 4}

This inference shouldn’t be valid – unless we presuppose the additional premise “W ∴ (∃x)x=g” – that Gensler is an existing being in world W.

Rejecting line 5 requires moving to a free logic – one free of the assumption that individual constants like “g” always refer to existing beings. Recall our drop-universal rule DU of §8.2:

Drop universal DU

(x)Fx → Fa,

use any constant

Every existing being is F.

∴ a is F.

Suppose that every existing being is F; “a” might not denote an existing being, and so “a is F” might not be true. So we need to modify the rule to require the premise that “a” denotes an existing being:

Drop universal DU*

(x)Fx, (∃x)x=a → Fa,

use any constant

Every existing being is F.

a is an existing being.

∴ a is F.

Here we symbolize “a is an existing being” by “(∃x)x=a” (“For some existing being x, x is identical to a”). With this change, “☐(∃x)x=g” (“Gensler is a necessary being”) is no longer provable.

If we weaken DU, we need to strengthen our drop-existential rule DE:

Drop existential DE*

(∃x)Fx → Fa, (∃x)x=a,

use a new constant

Some existing being is F.

∴ a is F.

∴ a is an existing being.

0264 When we drop an existential using DE*, we get an existence claim (like “(∃x)x=a”) that we can use in dropping universals with DU*. The resulting system can prove almost everything we could prove before – except that proofs are now longer. The main effect is to block a few proofs; we can no longer prove that Gensler exists in all possible worlds.

Our free-logic system also blocks the proof of this Barcan inference:

· Every existing being has the necessary property of being F.

· ∴ In every possible world, every existing being is F.

· 1 (x)☐Fx Invalid

· [ ∴ ☐(x)Fx

· * 2 asm: ∼☐(x)Fx

· * 3 ∴ ◇∼(x)Fx {from 2}

· * 4 W ∴ ∼(x)Fx {from 3}

· * 5 W ∴ (∃x)∼Fx {from 4}

· 6 W ∴ ∼Fa {from 5}

· 7 W ∴ (∃x)x=a {from 5}

Our new rule for dropping “(∃x)” tells us that “a” denotes an existing being in world W (line 7). But we don’t know if “a” denotes an existing being in the actual world; so we can’t conclude “☐Fa” from “(x)☐Fx” in line 1. With our naïve system, we could conclude “☐Fa” – and then put “Fa” in world W to contradict line 6; but now the line is blocked, and the proof fails.

While we don’t automatically get a refutation, we can invent one on our own. Our refutation lists which entities exist in which worlds; it uses “a exists” for “(∃x)x=a.” Here “Every existing being has the necessary property of being F” is true – since entity b is the only existing being and in every world it is F. But “In every possible world, every existing being is F” is false – since in world W there is an existing being, a, that isn’t F.

Here’s another objection to the argument. Suppose only abstract objects (numbers, sets, etc.) existed and all these had the necessary property of being abstract. Then “Every existing being has the necessary property of being abstract” would be true. But “In every possible world, every existing being is abstract” could still be false – if other possible worlds had concrete entities.¹

¹ Or suppose God created nothing and all uncreated beings had the necessary property of being uncreated. Then “Every existing being has the necessary property of being uncreated” would be true. But “In every possible world, every existing being is uncreated” could still be false – since there could have been possible worlds with created beings.

Our new approach lets different worlds have different existing entities. Gensler might exist in one world but not another. We shouldn’t picture existing in different worlds as spooky; it’s just a way of talking about different possibilities. I might not have existed. We can tell consistent stories where my parents didn’t meet and where I never came into existence. If the stories had been true, then I wouldn’t have existed. So I don’t exist in these stories (although I might exist in other stories). Existing in a possible world is much like existing in a story; a “possible world” is a technical analogue of a “consistent story.” “I exist in world W” just means “If world W had been actual, then I would have existed.” 0265

We also could allow possible worlds with no entities. In such worlds, all wffs starting with existential quantifiers are false and all those starting with universal quantifiers are true.

Should we allow this as a possible world when we do our refutations?

It seems incoherent to claim that “a has property F” is true while a doesn’t exist. It seems that only existing beings have positive properties; in a consistent story where Gensler doesn’t exist, Gensler couldn’t be a logician or a backpacker. So if “a exists” isn’t true in a possible world, then “a has property F” isn’t true in that world either. We can put this idea into an inference rule PE*:

Property existence PE*

Fa → (∃x)x=a

a has property F.

∴ a is an existing being.

Rule PE* holds regardless of what capital letter replaces “F,” what constant replaces “a,” and what variable replaces “x.” By PE*, “Descartes thinks” entails “Descartes exists.” Conversely, the falsity of “Descartes exists” entails the falsity of “Descartes thinks.” Rule PE* expresses that it’s a necessary truth that only existing objects have properties. Plantinga calls this view “serious actualism”; actualists who reject PE* are deemed frivolous.

The first example below isn’t a correct instance of PE* (since the wff substituted for “Fa” in PE* can’t contain “∼”), but the second is:

This one is wrong:

· ∼Fa

· –––––––––

· ∴ (∃x)x=a

· a isn’t F

· –––––––––

· ∴ a exists

This one is right:

· Fa

· –––––––––

· ∴ (∃x)x=a

· a is F

· ––––––––

· ∴ a exists

This point is confusing because “a isn’t F” in English can have two different senses. “Descartes doesn’t think” could mean either of these:

· Descartes is an existing being who doesn’t think

· = (∃x)(x=d • ∼Td)

· It’s false that Descartes is an existing being who thinks

· = ∼(∃x)(x=d • Td)

The first form is de re (about the thing); it affirms the property of being a non-thinker of the entity Descartes. Taken this first way, “Descartes doesn’t think” entails “Descartes exists.” The second form is de dicto (about the saying); it denies the statement “Descartes thinks” (which may be false either because Descartes is a non-thinking entity or because Descartes doesn’t exist). Taken this second way, “Descartes doesn’t think” doesn’t entail “Descartes exists.”

One might object to PE* on the grounds that Santa Claus has properties (such as being fat) but doesn’t exist. But various stories predicate conflicting properties 0266 to Santa; they differ, for example, on which day he delivers presents. Does Santa have contradictory properties? Or is one Santa story uniquely “true”? What would that mean? When we say “Santa is fat,” we mean that in such and such a story (or possible world) there’s a being called Santa who is fat. We shouldn’t think of Santa as a non-existing being in our actual world who has properties such as being fat. Rather, what exists in our actual world is stories about there being someone with certain properties – and children who may believe these stories. So Santa needn’t make us give up PE*.

We need to modify our current definition of “necessary property”:

· F is a necessary property of a

· = ☐Fa

· In all possible worlds, a is F

Let’s grant that Socrates has properties only in worlds where he exists – and that there are worlds where he doesn’t exist. Then there are worlds where Socrates has no properties – and so there aren’t any properties that Socrates has in all worlds. By our definition, Socrates would have no necessary properties.

Socrates still might have some necessary combinations of properties. Perhaps it’s true in all worlds that if Socrates exists then Socrates is a person. This suggests a more refined definition of “necessary property”:

· F is a necessary property of a

· = ☐((∃x)x=a ⊃ Fa)

· It’s necessary that if a exists then a is F

· In all possible worlds where a exists, a is F

This reflects better what philosophers mean when they speak of necessary properties. It also lets us claim that Socrates has the necessary property of being a person. This would mean that Socrates is a person in every possible world where he exists; equivalently, in no possible world does Socrates exist as anything other than a person. Here’s an analogous definition of “contingent property”:

· F is a contingent property of a

· = (Fa • ◇((∃x)x=a • ∼Fa))

· a is F; but in some possible world where a exists, a isn’t F

This section sketched a sophisticated quantified modal logic. Its refinements overcome some problems but also make the system harder to use. We seldom need the refinements. So we’ll keep the naïve system of earlier sections as our “official system” and build on it in later chapters. But we’ll be aware that this system is oversimplified in some ways. If our naïve system gives questionable results, we can appeal to the sophisticated system to clear things up.