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Course 14 · Lesson 7.1

Possible worlds and modal logic in plain language

9 minute read

Lessons in this course

Course 14 outline

Module 1: Arguments, Existence and Necessity

  1. What natural theology attempts and what it does not
  2. Necessary and contingent existence
  3. How to read and test an argument for God
  4. Philosophy of religion today: a map of the field

Module 2: The Ontological Argument

  1. Anselm and the argument of the Proslogion
  2. Gaunilo and the perfect island objection
  3. Kant: existence is not a real predicate
  4. What the ontological argument would show if it worked

Module 3: Cosmological Arguments

  1. Aquinas: the first three ways
  2. Infinite regress: what is the problem?
  3. Leibniz, the principle of sufficient reason and the argument from contingency
  4. Objections: the brute fact reply and the fallacy of composition

Module 4: The Kalam Cosmological Argument

  1. The kalam tradition: Philoponus, al-Ghazali and Bonaventure
  2. The modern argument: whatever begins to exist has a cause
  3. Did the universe begin? Philosophy and cosmology
  4. Objections and replies, and what follows about the cause

Module 5: Teleological and Fine-Tuning Arguments

  1. From Aquinas’s fifth way to Paley’s watch
  2. Hume and the Dialogues Concerning Natural Religion
  3. Darwin and the change in the design argument
  4. The fine-tuning argument: what is claimed and what is debated
  5. The multiverse and other replies

Module 6: The Moral Argument and Divine Hiddenness

  1. The moral argument: Kant, Lewis and contemporary forms
  2. The Euthyphro question and moral realism without God
  3. Evolution and moral knowledge
  4. The argument from divine hiddenness and Christian replies

Module 7: Contemporary Modal Arguments and Evaluation

  1. Possible worlds and modal logic in plain language
  2. Plantinga’s modal ontological argument
  3. Assessing the modal argument: what it needs and what it shows
  4. Weighing the arguments together, and what none of them prove

All courses

Program / Course 14: Arguments for the Existence of God / Lesson 7.1

What you will learn

  • Explain what a possible world is as philosophers use the term.
  • Define necessary, possible and impossible in terms of possible worlds.
  • Read the two modal symbols and the key principle S5.
  • Recognize why possible worlds are a tool and not a claim that other worlds exist.

The big idea

Modal logic studies reasoning about what must be, what might be and what cannot be. Philosophers find it useful to speak of possible worlds: complete ways things could have been. A statement is possible if it is true in at least one possible world, and necessary if it is true in all of them. This language, developed in the twentieth century by Saul Kripke and others, allows precise statements of old ideas about necessary being. It is a tool for clarity, and most philosophers do not take it to claim that other worlds literally exist.

Key terms

Modality
The way a statement is true or false: as necessary, possible or impossible, as opposed to merely true or false.
Possible world
A complete and consistent way the universe could have been, used to explain possibility and necessity.
Actual world
The possible world that is the way things really are.
Necessarily true
True in every possible world.
Possibly true
True in at least one possible world.
S5
A system of modal logic in which whatever is possibly necessary is necessary. Plantinga’s argument uses it.
Maximal greatness
In Plantinga’s argument, having maximal excellence (knowledge, power and goodness) in every possible world.

Background

Modal ideas are old. Aristotle discussed possibility and necessity, and the medieval philosophers developed them in detail. Leibniz spoke of many possible worlds among which God chose the best. In the twentieth century, logicians such as C. I. Lewis built formal systems for reasoning about necessity and possibility. In the 1950s and 1960s, Saul Kripke and others gave these systems an interpretation in terms of possible worlds, which made them much easier to understand and use.

The language of worlds is a tool. Some philosophers, like David Lewis, held that other worlds are as real as the actual world. Most, including Plantinga and Kripke, treat worlds as abstract ways things might be, like stories or descriptions that are complete. For the purposes of this course, you do not need to decide this question.

Step-by-step explanation

Step 1: Three kinds of statement

Some statements are true but could have been false, such as the statement that it rained in Tulsa on a given day. Some are false but could have been true, such as the statement that you were born a year earlier. Some are true and could not have been false, such as that two plus two is four. Some are false and could not have been true, such as that there is a square circle. Modal logic gives a way to talk about these differences.

Step 2: Worlds as ways things could be

Imagine a vast collection of complete descriptions of the universe. Each one is a possible world. In some, you have a different job. In others, no humans exist. In one, the description is true of how things actually are. That is the actual world. To say something is possible is to say it is true in some world. To say it is necessary is to say it is true in all. To say it is impossible is to say it is true in none.

Step 3: The symbols

Logicians write a diamond for possibly and a box for necessarily. So a diamond with P means P is possibly true, and a box with P means P is necessarily true. The two are linked: P is necessary exactly when it is not possible that P is false. This is the same as saying that something is true in all worlds exactly when there is no world where it is false.

Step 4: The principle S5

A key principle, in a system called S5, says that if something is possibly necessary, then it is necessary. In words: if there is some world where P is necessarily true, then P is true in all worlds. The reason is that in S5 what is necessary does not vary from world to world. S5 is widely used but not accepted by everyone. Some philosophers doubt it for certain kinds of possibility. Plantinga’s argument depends on it, so the point is important in Lesson 7.2.

Step 5: The cautions

Talk of worlds can mislead. It does not show that anyone knows what is possible. To say that a thing exists in some world needs a prior reason to think the thing is possible. Conceivability is a rough guide to possibility, but history shows many cases where people thought something was possible and it was not. Philosophers argue about how modal truths at all.

  1. Can you give one example of each of the three kinds of statement?
  2. Is it possible that you were never born? What kind of possibility is that?
  3. Does talk of other worlds seem helpful, or confusing?
  4. Why would it matter whether necessary truths are the same in every world?
  5. How do you decide whether something is really possible?
  6. Is conceivable the same as possible?

Examples

Socrates and the other worlds

It is true that Socrates was a philosopher. There are possible worlds where he was a stonemason, as his father was. In the language of worlds, this means that his being a philosopher is true in some worlds and false in others, so it is contingent.

A mathematical truth

Philosophers typically say that 7 is a prime number is true in every possible world. There is no way things could have been in which 7 was not prime. This is how modal logic expresses what it means for a truth to be necessary.

The impossible

A married bachelor, or a triangle with four sides, is true in no possible world. Such descriptions are impossible because they contradict themselves.

Different perspectives

Catholic

Catholic philosophers use modal logic freely, and the Thomist tradition has its own account of possibility in terms of God’s power and ideas. Aquinas held that God can do whatever is not contradictory. The language of worlds fits within this account, if understood as ways God could have made things.

Orthodox

Orthodox theology has not developed this language, and some Orthodox writers are suspicious of applying abstract logic to God. They would place little weight on a proof that rests on it, preferring the experience of God in prayer and worship.

Protestant

Plantinga, a Reformed Protestant, and many analytic philosophers use possible worlds as a standard tool. Other Protestant philosophers, such as William Hasker, apply it to questions of free will and providence.

Philosophical views

Philosophers divide on the nature of worlds. David Lewis held they are concrete and as real as the actual world, a view few share. Plantinga and Kripke take them as abstract. Others reject the language and prefer other ways of talking about possibility.

Skeptical views

Skeptics such as Quine doubted that modal talk was clear, and some think there is no reliable way to know what is possible beyond logic. They are therefore wary of arguments that depend on claims about all possible worlds.

Why it matters

Plantinga’s argument and many contemporary discussions use this language. A student who understands it will find the next lessons much clearer, and will be able to spot where an argument slides from the logical to the metaphysical meaning of possible.

Common misunderstandings

Misunderstanding: Possible worlds are other universes that exist somewhere.

Correction: For most philosophers, worlds are abstract ways things could have been. Only David Lewis held that they are as real as the actual world.

Misunderstanding: If it can be imagined, it is possible.

Correction: Imagination is an imperfect guide. People once imagined a perfect squaring of the circle, which is proved impossible.

Misunderstanding: Modal logic is a form of mysticism.

Correction: It is a formal system studied in mathematics and philosophy, with exact rules.

Misunderstanding: Possible means likely.

Correction: In this context possible means not contradictory or not ruled out. Many possible things are very unlikely.

Knowledge check

What is a possible world?

It is a complete and consistent way the universe could have been. The actual world is the one that is how things really are. Philosophers use the idea to explain possibility and necessity: something is possible if true in at least one world, necessary if true in all, and impossible if true in none. Most treat worlds as abstract, not as other places.

How are necessary and possible related?

A statement is necessary exactly when it is not possible that it is false. Equivalently, it is true in every world exactly when there is no world where it is false. The two ideas are linked, so a system of modal logic can use one to define the other.

What does S5 say, in plain words?

It says that whatever is possibly necessary is necessary. If there is some world in which a statement is necessarily true, then it is true in all worlds. This means that what is necessary does not differ from world to world. Plantinga’s argument uses it, and some philosophers question it.

Why is it a mistake to think imagining something proves it is possible?

Because people have imagined things that turn out to be impossible, and the mind can fail to notice contradictions that are hidden. Conceivability gives some evidence of possibility, but not proof. This matters for arguments that move from something being conceivable to it being possible, as some arguments for God do.

Think it through

  1. Is it helpful to think of a necessary truth as true in all worlds?
  2. Which of yhuman beliefs about what is possible could be wrong?
  3. What would it mean for God to exist in all worlds?
  4. Does the language of worlds make arguments clearer, or hide problems?

Independent assignment

Sorting statements by modality

In your own notebook, write 300 to 450 words. List ten statements, such as that you are reading this, that 5 is greater than 3, that dogs are mammals, and that there are unicorns. For each, say whether it is necessary, contingent or impossible, and give a one-sentence reason. Note the two you are least sure about. Do not send this to anyone.

Further study