Truth Tables and Oppositional Solids

Frédéric Sart

Corte — June 18, 2010

2nd World Congress on the Square of Opposition

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Truth table method

• The truth table method is used to calculate the values of a truth function

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Truth function

true/false

true/false

true/false

true/false

true/false

Logical configurations

Truth function

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Logical configurations (1)

• Classical propositional logic

Function which maps n elementary propositions to true/false

Logical configuration

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TLP 4.31

Logical configurations (2)

• Deontic propositional logic

Logical configuration

? ? ?

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Back to basics (1)

• Two primitives – The alethic values “true” and “false”

• Proposition – What is either true or false

• Alethic configuration over a finite set N of propositions – Function which maps the elements of N to true/false

• Alethic space E0(N) – The totality of alethic configurations over the set N

• Truth function on the alethic space E0(N) – Function which maps the elements of E0(N) to true/false

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Back to basics (2)

• Two more primitives – The deontic values “permitted” and “forbidden”

• Action – What is either permitted or forbidden

• Postulate – Any action is an alethic configuration over a finite set of propositions

• Deontic configuration over the set N – Function which maps the elements of E0(N) to permitted/forbidden

• Alethico-deontic configuration over the set N – Ordered pair (alethic configuration over N, deontic configuration over N)

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Back to basics (3)

• Alethico-deontic space E1(N) – The totality of alethico-deontic configurations over the set N

• Truth function on the alethico-deontic space E1(N) – Function which maps the elements of E1(N) to true/false

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Illustration (1)

• Let us consider the following single proposition

p I close the window

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Illustration (2)

• Two alethic configurations can be constructed

a1 It is the case that p

a2 It is not the case that p

• Four deontic configurations can be constructed

D1 Both a1 and a2 are permitted

D2 a1 is permitted and a2 is forbidden

D3 a1 is forbidden and a2 is permitted

D4 Both a1 and a2 are forbidden

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Illustration (3)

• Eight alethico-deontic configurations emerge

∙ The first one is

(a1, D1) Both a1 and a2 are permitted and I choose a1

∙ The last two are

(a1, D4) Both a1 and a2 are forbidden and I choose a1

(a2, D4) Both a1 and a2 are forbidden and I choose a2

Evil configurations

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Number of logical configurations

Number of elem. prop.

Classical logic

Deontic logic (evil conf. incl.)

Deontic logic (evil conf. excl.)

1 2 8 6

n 2n 2n x 22n 2n x (22n

- 1)

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Number of truth functions (1)

Number of elem. prop.

Classical logic

Deontic logic (evil conf. incl.)

Deontic logic (evil conf. excl.)

1 4 256 64

n 22n 22n×22n

22n×(22n-1)

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Number of truth functions (2) T T T T T T

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f64

F F F F F F

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f1

62 non-trivial truth functions

F T T T T T

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f63

F F F F F T

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f7

F F F T T T

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f42

F F T T T T

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f57

F F F F T T

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f22

T T T T T F

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f58

T F F F F F

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f2

T T T T F F

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f43

T T F F F F

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f8

T T T F F F

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) f23

. . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . .

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Link with Moretti’s deontic solid

• Claim

There is a natural way to couple the 62 formulas decorating Moretti’s deontic solid with the above specified 62 truth functions

• How

Via truth tables

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Truth tables in deontic logic (1)

• Truth conditions

∙ For propositional letters and classical operators • Defined as usual

∙ For deontic operators

• (a, D) ⊨ Oj iff for all b that D maps to permitted, (b, D) ⊨ j

• (a, D) ⊨ Pj iff for some b that D maps to permitted, (b, D) ⊨ j

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• Example

Truth tables in deontic logic (2)

(a1, D1) (a2, D1) (a1, D2) (a2, D2) (a1, D3) (a2, D3)

(a, D) Pp ∧ (Op ∨ ¬p)

T F T F T F

T F T F T F

T F T F T F

T T T T F F

F F T T F F

F T F T F T

F T T T F T

F T T T F F

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Back to Moretti’s deontic solid

• Theorem

The truth tables of the 62 formulas decorating Moretti’s deontic solid coincide with the above specified 62 truth functions

• Proof

By constructing the 62 truth tables

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Conclusion

• The truth table method can be extended to modal logic

• It provides a systematic way to generate what Moretti calls “oppositional solids” (classical or modal)

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References

• MORETTI, A. (2009) “The Geometry of Standard Deontic Logic”. Logica Universalis 3, 19-57.

• SART, F. (2009) “A Purely Combinatorial Approach to Deontic Logic”. Logique et Analyse 206, 131-138.

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