10. MODAL LOGIC
Outline
James Fieser, UT Martin
updated 12/1/2023
A. MODAL PROPOSITIONS
1. Definition of Modal Logic
Modal logics (broad sense): logic of different aspectsor modesof truth, which builds upon standard systems of propositional and predicate logic.
Modal logic (narrow sense): propositional calculus plus modal operators □p (it is necessary that p) and ◊p (it is possible that p)
2. Types of Modal Logics (in the broad sense)
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)
3. 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 (i.e., impossible for all worlds that share our history up to Lincolns death)
4. Examples of modal propositions
Simple modal propositions (p=Paul is an alien)
◊p (it is possible that Paul is an alien)
□p (it is necessary that Paul is an alien)
Logically equivalent modal propositions (can be substituted for each other)
~◊p same as □~p (it is impossible that Paul is an alien; or, it could not be true that Paul is an alien; or it p must be false that Paul is an alien)
~□p same as ◊~p (it is not necessary that Paul is an alien; or, it is possible that Paul is not an alien)
◊p same as ~□~p (it is possible that Paul is an alien; or it is not necessary that Paul is not an alien
□p same as ~◊~p (it is necessary that Paul is an alien; or it is impossible that Paul is not an alien)
Compound modal propositions
Where
p=Paul is an alien
q=Quincy is an alien hunter
◊p & ◊q (it is possible that Paul is an alien and it is possible that Quincy is an alien hunter)
p v ◊q (Paul is an alien or it is possible that Quincy is an alien hunter)
◊(p → q) (it is possible that if Paul is an alien then Quincy is an alien hunter)
Compound modal propositions of special interest
p & ◊~p (p obtains but it is possible that it might not have; i.e., p is a contingent truth)
e.g., my eyes are blue, but it's possible that they might not have been blue
◊(p & q) (it is possible that p and q are both true; i.e., p and q are compatible)
e.g.,: it is possible both that both Jill has U.S. citizenship and she has British citizenship
~◊(p & q) (it is impossible that P and Q are both true; i.e., P and Q are incompatible)
e.g., it is not possible that both this object is a cube and this object is a sphere
□(p → q) (it is necessary that if p then q; i.e., p necessarily entails q)
e.g., it is necessary that if Joe is a bachelor, then Joe is unmarried
Modal propositions with quantifiers of predicate logic
Ǝx(Fx & Gx) & ◊Cx (there exists some x such that x is F and x is G, and it is possible x is C)
e.g. this frog is green and it is possible that this frog croaks
∀x(Fx → Gx) & ◊Cx (for all x, if x is F then x is G, and it is possible that x is C)
e.g., all frogs are green, and it is possible that they croak
B. POSSIBLE WORLD SEMANTICS
Possible world semantics are rules for determining the truth values of propositions in various worlds. Recall in propositional logic the role that truth tables played in explaining the logical operators of conjunction (&), disjunction (v), negation (~) and conditional (→). The meaning of these logical operators was defined by the specific layout of T and F in their respective truth tables. That is, these truth table assignments constitute the "semantics" (i.e., meaning) of these four logical operators. In modal logic, we have two new logical operators: necessity (□), and possibility (◊). With these new operators, however, truth tables are insufficient for defining their meaning. They are instead defined by a different semantic methodology called "possible word semantics".
1. Four types of accessibility relations between possible worlds
Possible world semantics involves what are called "accessibility relations". As an analogy, think a science fiction of alternate universes. Let's call our actual universe w1 (world 1). Alternate universes would be w2, w3, w4, etc. Suppose that scientists built portals to some of these alternate worlds, such as a portal in w1 that would give us access to w2, where, for example, we could see what was going on there. Consider, next, four possible portal configurations for accessing possible worlds: (a) serial, (b) reflexive, (c) symmetrical, and (d) transitive. These are explained below.
a. Serial relation: every world has access to at least one world
{w1}→ {w2}
e.g., Think about different kinds of permissions we can set on our Facebook accounts for accessing people's photos. If I set up in Facebook only serial access, then, (a) I can access your photos, but (b) I cannot access my own photos, (c) you cannot access mine photos.
b. Reflexive relation: every world can access itself
{w1}
↻
e.g., If I set up in Facebook only a reflexive access, then, (a) I can access my own photos, but (b) I cannot access your or anyone else's photos.
c. Symmetric relation: for all worlds, w1, w2, if w1 has access to w2, then w2 has access to w1
{w1} ←→ {w2}
e.g., If I set up in Facebook only a symmetrical access, then, (a) you and I can access each other's photos, but (a) I cannot access my own photos.
d. 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., If I set up in Facebook only a transitive access, then, (a) I can access your photos, and (b) I can access your friends' photos, but (c) none of you can access mine, and (d) I cannot access my own.
e. Serial, reflexive, symmetrical, and transitive relations combined: every world accesses every other world
e.g., If I set up in Facebook only an access that exhibits all four of these, then, (a) I can access your photos, (b) I can access my own photos, (c) I can access your friends' photos, (d) you can access my photos, and (e) your friends can access my photos. In essence, everyone has access to everyone else's photos.
2. Determining the truth value of modal propositions like ◊p using possible world semantics
Informally, based on possible world semantics, ◊p is true on our world (w1) if and only if it is true in at least one world to which w1 has access (e.g., to w2, w3, w4, and maybe even w1 itself). By contrast, □p is true in our world (w1) if and only if it is true in every world to which w1 has access (e.g., to w2, w3, w4, and maybe even w1 itself).
a. Example: consider the following serial relationship between three (and only three) possible worlds:
{w1} → {w2 ~p}
→ {w3 p}
In this, ◊p is true only in w1, □p is not true in any. For, w1 has access to both w2 and w3, but p obtains in only w3 and not in w2.
b. Formal definitions of necessity and possibility using possible world semantics
1. Necessity: □p is true in world w1 if and only if p is true in every world accessible to w1
{w1 □p} iff → {w2 p=true}
→ {w3 p=true}
2. Possibility: ◊p is true in world w1 if and only if p is true in some world accessible to w1 (i.e., at least one must be true, others can be false but don't need to be)
{w1 ◊p} iff → {w2 p=false}
→ {w3 p=true}
C. NORMAL MODAL LOGIC SYSTEMS (K or stronger)
Axioms in modal logic attempt to express common sense truths about possibility and necessity, such as □p→p (whatever is necessary is the case). Six such axioms are discussed below. Systems of modal logic have been developed that incorporate different combinations of axioms, six of which are presented here. Philosophers commonly use systems S4 or S5 to model real life applications and arguments involving possibility and necessity, where S5 goes further in its assumptions (or is "stronger") than S4. The first few systems (K, D, T, and B), are so minimal that they do not reflect what we mean by possibility and necessity in real life, and are of interest only abstractly to logicians. Further, to create system like S4 and S5 which do reflect our real life views, we must build them incrementally upon the simpler systems of K, D, T and B. We begin with K, which is the simplest, and progress to more complex ones.
1. System K
Special Features
Axiom K: □(p → q) → (□p → □q) (distribution: the operator outside the parentheses can be distributed to both the antecedent and consequent inside the parentheses)
Relation: none
Total features
All Axioms: K
All relations: none
2. System D
Special Features
Axiom D: □p→◊p (whatever is necessary is possible, e.g., if it is necessary that 2+2=4, then it is possible that 2+2=4)
Relation: serial
Total features
All Axioms: K + D
All relations: serial
3. System T (or M)
Special Features
Axiom T: □p→p (whatever is necessary is the case; e.g., if it is necessary that 2+2=4, then 2+2=4)
Relation: reflexive
Total features
All Axioms: K + T
All relations: reflexive
Important corollary: p → ◊p (whatever is the case is possible; e.g., if 2+2=4, then it is possible that 2+2=4)
4. System B
Special Features
Axiom B: p→□◊p (whatever is the case, is necessarily possible; e.g., if 2+2=4, then, of necessity, it is possible that 2+2=4)
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; e.g., if it is possible that of necessary 2+2=4, then 2+2=4)
5. System S4
Special Features
Axiom 4: □p→□□p (whatever is necessary is necessarily necessary; e.g., if it is necessary that 2+2=4, then, of necessity, it is necessity that 2+2=4)
Relation: transitive
Total features
All Axioms: K + T + 4
All relations: reflexive and transitive
Important corollary: ◊◊p→◊p (reduces strings of similar operators; e.g., if it is possible that it is possible that 2+2=4, then it is possible that 2+2=4)
6. System S5
Special Features
Axiom 5: ◊p→□◊p (whatever is possible is necessarily possible; e.g., if it is possible that 2+2=4, then it is necessary that it is possible that 2+2=4)
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; e.g., if it is possible that it is necessary that 2+2=4, then it is necessary that 2+2=4)
D. MODAL ONTOLOGICAL ARGUMENT FOR GOD
The ontological argument for God's existence is a famous proof devised by medieval philosopher Anselm which shows that the attribute of existence is built into the definition of God as "the greatest possible being" (or, in Anselm's words, God is "that than which nothing greater can be conceived). Logicians have attempted to represent Anslem's argument in different ways. We will look at two here: a version based solely on the logical operators in propositional logic, and one based on the S5 system of modal logic.
1. Simple version in propositional logic
Intuition: the idea of God is that of a necessary being (the highest level of existence).
Abbreviations:
p=it is possible that a necessary being exists,
q=a necessary being exists
The argument
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; alternately ~~p it is not the case that it is impossible that a necessary being exists]
3. ˫ q
[a necessary being exists (1, 2 DS)]
2. 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 about possibility)]
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)]
3. Possible world diagram of modal ontological argument, using S5 which has the three relations of being reflexive, symmetrical and transitive (i.e., what happens in one world happens in all worlds)
Where w1 is the actual world, assume that ◊□p (i.e., it is possible that God necessarily exists, premise 2 above); thus, □p obtains in some world w2):
↷ ↷ ↷ ↷
{w1}←→ {w2 □p} ←→ {w3}←→ {w4}
↖↗
↖↗
↖↗
Now, if □p obtains in w2, this means that p must also obtain in every world accessible to w2, including w1 and w3 (through symmetry), and w4 (through transitivity). Since w1 has access to every world that w2 has access to (and p obtains in all of these worlds) then □p also obtains in w1. Thus, if □p obtains in w1, then p also obtains in w1.
4. Controversy between whether S4 or S5 in the ontological argument
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.
E. MODAL PROOFS
The following are some modal logic proofs that are superimposed on propositional calculus (but not predicate calculus). It consists of (1) the necessitation rule (NEC), which allows adding the necessity box "□" to any tautology in propositional logic, (2) the change modal operator rule (CMO) which allows that allow replacing ◊ with □, and vice versa, and (3) the axioms of the S5 modal system (i.e., K, T and 5). In these proofs, no premises are given, only the conclusion is stipulated. The proof consists of creating premises through the axioms, and using the rules of propositional calculus to draw inferences until the conclusion is reached.
1. 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
Explanation: recall the rule "theorem introduction" (TI) in propositional calculus, which allows introducing into a proof any tautology, such as "p v ~p, and "~(p & ~p)". We may do that here too, but with the added feature of adding to these the necessity operator, such as "□(p v ~p), and "□[~(p & ~p)]". That is, if a proposition A is a tautology (i.e., it is true in every truth table scenario), then that proposition A is necessary (i.e., □A). In short, the necessitation rule allows creating out of thin air a modal proposition that may be inserted into a modal proof as needed.
Change Modal Operator Rule (CMO)
◊p ↔ ~□~p
□p ↔ ~◊~p
~□p ↔ ◊~p
□~p ↔ ~◊p
Major Axioms (for the S5 modal system)
A1: ◊p ↔ ~□~p [CMO]
A2: □(p →q) → (□p → □q) [Axiom K, distribution]
A3: □p → p [Axiom T]
A4: ◊p → □◊p [Axiom 5]
2. Example Proofs
(1) Prove the following proposition using modal rules and axioms: ˫ ~◊~B→ B
Answer (modal rules/axioms highlighted):
1. ˫ ~◊~B→ B
2. □B → B [A3]
3. ˫ ~◊~B→ B [2 CMO]
(2) Prove the following proposition using modal rules and axioms: ˫ ~□◊D → □~D
Answer:
1. ˫ ~□◊D → □~D
2. ◊D → □◊D [A4]
3. ~□◊D → ~◊D [2 TRANS]
4. ˫~□◊D → □~D [3 CMO]
(3) Prove the following proposition using modal rules and axioms: ˫ F → ◊F
Answer:
1. ˫ F → ◊F
2. □~F→ ~F [A3]
3. ~~F → ~□~F [2 TRANS]
4. F → ~□~F [3 DN]
5. ◊F ↔ ~□~F [A1]
6. ~□~F → ◊F [5 ↔E]
7. ˫ F → ◊F [4, 6 HS]