Metaphysics Study Guide: Modality (Possibility,
Necessity, & Possible Worlds)
Based on Alyssa Ney, Metaphysics: An Introduction (Chapter 7)
1. Possibility and Necessity: Modes of Truth
In metaphysics, modality refers to the mode or way in which a statement or proposition is true
or false. Instead of merely asking what is true (actuality), modality asks what could be true
(possibility) and what must be true (necessity).
Core Modal Categories & Operators
● Contingent Truth: A statement that happens to be true in our actual world, but could
have been false (e.g., "Mount Everest is the tallest mountain on Earth").
● Possible Statement (◇P): A statement that is true, or could have been true. (Note:
Every true statement is automatically possible).
● Necessary Statement (□P): A statement that is true and could not possibly have been
false (e.g., "2 + 2 = 4" or "All bachelors are unmarried").
● Impossible Statement (¬◇P): A statement that is false and could not possibly be true.
Modal Equivalence Rules (Dualities)
Just as universal (∀) and existential (∃) quantifiers interdefine each other in predicate logic,
the box (□) and diamond (◇) operators are interdefinable:
1. □P ≡ ¬◇¬P (To say P is necessary is equivalent to saying it is not possible that
not-P).
2. ◇P ≡ ¬□¬P (To say P is possible is equivalent to saying it is not necessary that
not-P).
, 2. Species of Possibility and Necessity
Not all "musts" and "cans" mean the same thing. Philosophers distinguish between restricted
and unrestricted varieties of modality:
Species of Definition & Scope Example Statement
Modality
Epistemic Relative to what an agent or "It is possible that it is
Modality community knows at a given time raining outside right
("For all I know..."). now."
Deontic Modality Relative to moral rules, laws, or "It is necessary that you
institutional obligations. stop at a red light."
Nomological Consistent with the laws of nature "It is impossible for a
(Physical) of our actual universe (W_actual). massive object to
Modality accelerate past the
speed of light."
Logical Modality Consistent with the laws of logic "It is logically possible
alone (contains no formal that gravity repels
self-contradiction, ¬(P ∧ ¬P)). rather than attracts."
Metaphysical Possibility tout court "It is metaphysically
Modality (absolute/unrestricted possibility). impossible for water to
Consistent with the fundamental be anything other than
nature and essences of things. H₂O."
High-Yield Exam Distinction (Nomological vs. Metaphysical):
Every nomologically possible event is metaphysically possible, because our laws of nature are
themselves metaphysically possible. However, metaphysical possibility is broader than
nomological possibility: a universe with slightly weaker gravity violates our physical laws
(nomologically impossible), yet it is still metaphysically possible. Conversely, nomological
necessity is weaker than metaphysical necessity.
Necessity, & Possible Worlds)
Based on Alyssa Ney, Metaphysics: An Introduction (Chapter 7)
1. Possibility and Necessity: Modes of Truth
In metaphysics, modality refers to the mode or way in which a statement or proposition is true
or false. Instead of merely asking what is true (actuality), modality asks what could be true
(possibility) and what must be true (necessity).
Core Modal Categories & Operators
● Contingent Truth: A statement that happens to be true in our actual world, but could
have been false (e.g., "Mount Everest is the tallest mountain on Earth").
● Possible Statement (◇P): A statement that is true, or could have been true. (Note:
Every true statement is automatically possible).
● Necessary Statement (□P): A statement that is true and could not possibly have been
false (e.g., "2 + 2 = 4" or "All bachelors are unmarried").
● Impossible Statement (¬◇P): A statement that is false and could not possibly be true.
Modal Equivalence Rules (Dualities)
Just as universal (∀) and existential (∃) quantifiers interdefine each other in predicate logic,
the box (□) and diamond (◇) operators are interdefinable:
1. □P ≡ ¬◇¬P (To say P is necessary is equivalent to saying it is not possible that
not-P).
2. ◇P ≡ ¬□¬P (To say P is possible is equivalent to saying it is not necessary that
not-P).
, 2. Species of Possibility and Necessity
Not all "musts" and "cans" mean the same thing. Philosophers distinguish between restricted
and unrestricted varieties of modality:
Species of Definition & Scope Example Statement
Modality
Epistemic Relative to what an agent or "It is possible that it is
Modality community knows at a given time raining outside right
("For all I know..."). now."
Deontic Modality Relative to moral rules, laws, or "It is necessary that you
institutional obligations. stop at a red light."
Nomological Consistent with the laws of nature "It is impossible for a
(Physical) of our actual universe (W_actual). massive object to
Modality accelerate past the
speed of light."
Logical Modality Consistent with the laws of logic "It is logically possible
alone (contains no formal that gravity repels
self-contradiction, ¬(P ∧ ¬P)). rather than attracts."
Metaphysical Possibility tout court "It is metaphysically
Modality (absolute/unrestricted possibility). impossible for water to
Consistent with the fundamental be anything other than
nature and essences of things. H₂O."
High-Yield Exam Distinction (Nomological vs. Metaphysical):
Every nomologically possible event is metaphysically possible, because our laws of nature are
themselves metaphysically possible. However, metaphysical possibility is broader than
nomological possibility: a universe with slightly weaker gravity violates our physical laws
(nomologically impossible), yet it is still metaphysically possible. Conversely, nomological
necessity is weaker than metaphysical necessity.
