MODAL LOGIC

 

Definitions

Modal logics (broad sense): logic of different aspects—or “modes”—of truth, which builds upon standard systems of propositional and predicate logic.

Modality         Operator          Expression

Alethic            □                      □p (it is necessary that p)

                        ◊                      ◊p (it is possible that p)

Deontic           O                     Op (it is obligatory that p)

                        P                      Pp (it is permitted that p)

                        F                      Fp (it is forbidden that p)

Temporal         G                     Gp (it will always be the case that p)

                        F                      Fp (it will be the case that p)

                        H                     Hp (it has always been the case that p)

                        P                      Pp (it was the case that p)

Epistemic        K                     Kp (it is known that p)

Doxastic          B                     Bp (it is believed that p)

Counterfactual □→                 p □→ q (if p had obtained then q would have obtained)

Modal logic (narrow sense): propositional calculus plus modal operators □p (it is necessary that p) and ◊p (it is possible that p)

Types of possibility (modal logic can use any of these)

Logical possibility: it is logically possible for a colored object to not be red, but impossible for a red object to not be colored

Metaphysical possibility: it is metaphysically possible for a substance to be in the form of a sphere, but impossible for a substance to have no form whatsoever

Physical possibility: it is physically possible to change water into hydrogen and oxygen, but impossible to change water into gold

Technological possibility: it is technologically possible to travel to Mars in one year, but impossible in one day

Temporal (historical) possibility: it is temporally possible for war to break out tomorrow, but impossible for Abraham Lincoln to become president tomorrow (impossible for all worlds that share our history up to Lincoln’s death)

Examples of modal propositions

Simple modal propositions (p=Bob is an alien)

◊p (it is possible that Bob is an alien)

□p (it is necessary that Bob is an alien)

Logically equivalent modal propositions (can be substituted for each other)

~◊p same as □~p (it is impossible that Bob is an alien; or, p could not be true; or p must be false)

~□p same as ◊~p (it is not necessary that Bob is an alien; or, it is possible that Bob is not an alien)

◊p same as ~□~p (it is possible that Bob is an alien; or it is not necessary that Bob is not an alien

□p same as ~◊~p (it is necessary that Bob is an alien; or it is impossible that Bob is not an alien)

Compound modal propositions

p & ◊~p (p obtains but it is possible that it might not have; i.e., p is a contingent truth)

◊(p & q) (it is possible that p and q are both true; i.e., p and q are compatible)

~◊(p & q) (it is impossible that P and Q are both true; i.e., P and Q are incompatible)

□(p → q) (it is necessary that if p then q; i.e., p necessarily entails q)

 

Possible World Semantics (rules for determining the truth values of propositions in various worlds)

Necessity: □p is true in world w1 if and only if p is true in every world accessible to w1

{w1 □p}          ———→ {w2 p=true}

———→ {w3 p=true}

Possibility: ◊p is true in world w1 if and only if p is true in some world accessible to w1

{w1 ◊p}          ———→ {w2 p=false}

———→ {w3 p=true}

Four types of accessibility relations between possible worlds

Serial relation: every world has access to at least one world

{w1}———→ {w2}

e.g., spying: I can spy on you, but that doesn’t mean that (a) I can spy on myself, or (b) I can spy on your friends, or (c) you all can spy on me

Reflexive relation: every world can access itself

{w1}

  ↻

e.g., mind reading: I can read my thoughts, but that doesn’t mean (a) I can read your thoughts, or (b) your friend’s thoughts, or (c) you all can read my thoughts

Symmetric relation: for all worlds, w1, w2, if w1 has access to w2, then w2 has access to w1

{w1} ←———→ {w2}

e.g., visual range: if I am in visual range of you, then are in visual range of me, but that doesn’t mean that people in your visual range are in my visual range

Transitive relation: For all worlds, w1, w2, w3, if w1 has access to w2, and w2 has access to w3, then w1 has access to w3

{w1}   ———→ {w2} ———→ {w3}

    ————————————

e.g., exclusive party: if I can invite you to my party and you can invite a friend, then I can invite that friend; but that doesn’t mean that you or your friend can invite me to the party

Reflexive, symmetrical, and transitive relations combined: every world accesses every other world

e.g., viral pandemic: I can infect myself (reflexive); if I can infect you then you can infect me (symmetrical); if I can infect you, and you can infect your friends, then I can infect your friends (transitive)

 

Normal modal logic systems (K or stronger)

System K

Special Features

Axiom K: □(p → q) → (□p → □q) (distribution)

Relation: none

Total features

All Axioms: K

All relations: none

System D

Special Features

Axiom D: □p→◊p (whatever is necessary is possible)

Relation: serial

Total features

All Axioms: K + D

All relations: serial

System T (or M)

Special Features

Axiom T: □p→p (whatever is necessary is the case)

Relation: reflexive

Total features

All Axioms: K + T

All relations: reflexive

Important corollary: p → ◊p (whatever is the case is possible)

System B

Special Features

Axiom B: p→□◊p (whatever is the case, is necessarily possible)

Relation: symmetric

Total features

All Axioms: K + T + B

All relations: reflexive and symmetric

Important Corollary: ◊□p→p (whatever is possibly necessary is the case)

System S4

Special Features

Axiom: □p→□□p (whatever is necessary is necessarily necessary)

Relation: transitive

Total features

All Axioms: K + T + 4

All relations: reflexive and transitive

Important corollary: ◊◊p→◊p (reduces strings of similar operators)

System S5

Special Features

Axiom: ◊p→□◊p (whatever is possible is necessarily possible)

Relation: Euclidian

Total features

All Axioms: K + T + 5 (or K + B + 4)

All relations: reflexive, symmetric, and transitive, i.e., every world accesses every other world

Important Corollary: ◊□p→□p (whatever is possibly necessary is necessary; reduces strings of operators to the last one)

 

Modal Ontological Argument for God

Simple version in propositional logic

Intuition: the idea of God is that of a necessary being (the highest level of existence)

p=it is possible that a necessary being exists,

q=a necessary being exists

1. ~p v q

[either it is impossible that a necessary being exists, or a necessary being exists]

2. p

[it is possible that a necessary being exists]

3. ˫ q

[a necessary being exists (1, 2 DS)]

Simple version in modal logic

Where p=God exists

1. ◊□p→□p

[if it is possible that God necessarily exists, then God necessarily exists (from corollary axiom in modal system s5)]

2. ◊□p

[it is possible that God necessarily exists (from intuition – it is possible)]

3. □p

[God necessarily exists (1, 2 MP)]

4. □p → p

[if God necessarily exists then God exists (from system T axiom)]

5. ˫ p

[God exists (4, 3 MP)]

Controversy between whether S4 or S5

Should modal logic system S4 or S5 be the default system of modal logic that represents our normal intuitions about logical possibility

S5 has an axiom with establishes the symmetry relation, but S4 does not have that axiom and symmetry

The ontological argument works in S5 because of the symmetry axiom, but the argument does not work in S4 which lacks the symmetry axiom.

It comes down to whether the symmetry axiom should be part of our default modal logic system

 

Modal Proofs

Rules and Axioms

Necessitation Rule (NEC): if wff A is a proved theorem (e.g., truth table tautology such as “p v ~p”), then we may infer □A

Change Modal Operator Rule (CMO)

◊p ↔ ~□~p

□p ↔ ~◊~p

~□p ↔ ◊~p

□~p ↔ ~◊p

Major Axioms

A1: ◊P ↔ ~□~P

A2: □(P→Q) → (□P → □Q)

A3: □P→ P

A4: ◊P → □◊P

Example of Proof

 (1) ˫ P → ◊P

1. ˫ P → ◊P

2. □~P→ ~P [A3]

3. ~~P → ~□~P [2 TRANS]

4. P → ~□~P [3 DN]

5. ◊P ↔ ~□~P [A1]

6. ~□~P → ◊P [5 ↔E]

7. ˫ P → ◊P [4, 6 HS]