Introduction to Logic: Ch. 7

(Originally written December 11, 2006 in Book 9)

Introduction to Logic
Harry J. Gensler

Chapter 7: Basic Modal Logic

Modal logic studies arguments whose validity depends on 'necessary' and 'possible' notions.

7.1 Translations

1) ◊A = it's possible that A = A is true in some possible world
2) A = It's true that A = A is true in the actual world
3)□A = It's necessary that A = A is true in all possible worlds

"Possible" is weaker then calling something true.

"Necessary" is stronger than calling something true.

Possible means logically possible (not self-contradictory).

Necessary means logically necessary (self-contradictory to deny).

A possible world is a consistent and complete description of how things might have been or might in fact be.

The actual world is the description of how things actually are.

Done' use parentheses with ◊ or □

right: ◊A □A
wrong: ◊(A) □(A) (◊A) (□A)

~◊A = A couldn't be true
□~A = A has to be false

◊(A·B) - it's possible that A and B are true. A is compatible with B.

□(A⊃B) - it's necessary that if A then B. A entails B.

"entails" is a stronger claim then an "if-then"

~◊(A·B) - A is inconsistent with B. It's not possible that A and B are both true.

~□(A⊃B) - A doesn't entail B. It's not necessary that if A then B

(◊A·~◊A) - A is a contingent statement. A is possible and not-A is possible.

(A·◊~A) - A is true but could have been false. A is a contingent truth.

Statements are necessary, impossible or contingent. Truths are only necessary or contingent.

Necessary not = □~
not necessary = ~□
necessary if = □(
if necessary = (□

English sentences can be ambiguous

"If A is true, then its necessary that B" could mean:
1) (A⊃□B) or
2) □(A⊃B)

"If A is true, then it's impossible that B" could mean:
1) (A⊃□~B) or
2) □(A⊃~B)

Inherent necessity: given that the antecedent is true it is necessarily true.

Relative necessity: given that the X is true then the relation between X and Y is true.

Inherent necessity is the "necessity of the consequent". Relative necessity is the "necessity of the consequence"

If A, then B (by itself) is necessary. (A⊃□B)
A entails B. □(A⊃B)
Necessarily, if A then B: □(A⊃B)
It's necessary that if A then B: □(A⊃B)
If A then B is a necessary truth: □(A⊃B)

7.2 Proofs

A world prefix is a string of zero or more instances of "W"

A derived step is now of a line consisting of a world prefix and then "therefore"

An assumption is now a line consisting of a world prefix and 'asm':

Therefore, A (A is true in the actual world)
asm: A (Assume A is true in world W)

WW Therefore, A (A is true in World W)
W asm: A (Assume A is true in world W)

WW therefore, A (A is true in the world WW)

WW asm: A (assume A is ture in world WW)

Reverse squiggle ~□A -> ◊~A
~◊A -> □~A

drop diamond: ◊A -> W therefore, A

drop box - □A -> W therefore, A