Jerzy Kalinowski

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Jerzy Kalinowskis NORMATIVE LOGIC (Systems K1
and K2)
Robert Trypuz trypuz_at_loa-cnr.it
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PART I General intuitions and Language of
Normative Logic
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The subject of normative logic
AGENT
ACTION
normative relation
REALITY (NORMATIVE SITUATION)
NORMATIVE SENTENCE
NATURAL LANGUAGE
NORMATIVE LOGIC
LOGIC FORMULA
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The main normative relations
Obligation Permission Prohibition Indifference
AGENT
ACTION
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Example
TARIQ
PRAY
obligation
REALITY (NORMATIVE SITUATION)
Tariq ought to pray.
NATURAL LANGUAGE
NORMATIVE LOGIC
O(Tariq, pray) O(x,p)
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Atomic formulas of the System K1
Obligation (O) Permission (P) Prohibition
(F) Indifference (I)
AGENT (x)
ACTION (p)
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The truth-value of norms and logical inference

• According to definition of logical inference, the
  parts of its may be only sentences (in indicative
  mood) which are true or false.
• It is clear that someone who wants to use norms
  in such type of inferences has two solutions
• she accepts that norms may be true (or false)
• she doesnt accept that norms may be true (or
  false) and then she has to make some modification
  of norms. In order to use the logical inference,
  changes each of norms N into normative
  statements N exists or N exists for the sake
  of set of norms. The normative statements are
  sentences in the indicative mood and thereby are
  true or false.

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Example
The normative sentence (norm) Tariq ought to
pray is true
If and only if
PRAY
TARIQ
obligation
If and only if
PRAY is essentially good
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PART II System K1
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The truth table of the normative logic
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Example
P(x,Np) ? P(x,p) is a thesis of Normative Logic,
because For p1 P(x,N1) ? P(x,1)
P(x,0) ? P(x,1) 0 ? 1 1 ? 1 1 For p
1/2 P(x,N1/2) ? P(x,1/2) P(x,1/2) ?
P(x,1/2) 1 ? 1 1 Dla p0 P(x,N0) ?
P(x,0) P(x,1) ? P(x,0) 1 ? 0 0 ? 0 1
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System K1

• Axioms
• Axioms of Classical Calculus
• A. P(x,Np) ? P(x,p)
• 2. Definitions
• D1. O(x,p) P(x,Np)
• D2. F(x,p) P(x,p)
• D3. I(x,p) P(x,p) ? P(x,Np)
• 3. Rules
• Classical Calculus Rules
• For p can be substituted only Np

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Thesis of The System K1
T1 O(x,p) ? P(x,p) T2 P(x,p) ? P(x,Np) T3 O(x,p)
? F(x,p) T4 I(x,p) ? P(x,p) T5 I(x,p) ?
P(x,Np) T6 F(x,p) ? Ip T7 O(x,p) ? I(x,p) T8
O(x,p) ? F(x,p) ? I(x,p)
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PART III System K2 (Normative Syllogistic)
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System K2 (Normative syllogistic)
To the language of the system K1 are added 1.
Two formulas Xx x is X (or x belongs to X) Ap
p is A (or p belongs to A) 2. Two quantifiers
? (for all), ? (there exist) bind the variables
p and x ?p, ?x, ?x, ?p System K2 1. System
K1 2. Substitutions of thesis of first-order
predicate calculus
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Example
Thesis of K2 Tg1 ?p Ap ? ?x(Xx ? O(x,p) ??p
(Bp ? Ap) ? ?p Bp ? ?x(Xx ? O(x,p)
Deduction scheme bases on thesis Tg1 ?p Ap ?
?x(Xx ? O(x,p) ?p (Bp ? Ap) ?p Bp ? ?x(Xx ?
O(x,p)
Example of inference bases on the scheme ?p p
is Salat Ul Khamsa ? ?x (x is Muslim ? O(x,p) ?p
(p is Isha ? p is Salad Ul Khamsa) ?p p is Isha
? ?x (x is Muslim ? O(x,p)
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Thesis of The System K2
Tg1 ?p Ap ? ?x(Xx ? O(x,p) ??p (Bp ? Ap) ? ?p
Bp ? ?x(Xx ? O(x,p) Tg2 ?x Xx ? ?p(Ap ?
P(x,p) ? ?x (Yx?Xx) ? ?x Yx ? ?p(Ap ?
R(x,p) Tg3 ?x Xx ? ?p(Ap ? P(x,Np) ? ?xXx ?
?p(Ap ? P(x,p)
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CONCLUSIONS

• External and Internal operators in deontic logics
• (We dont need internal operators!)
• The meaning of permission

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References
Kalinowski, Jerzy (Georges), Teoria zdan
normatywnych. In Studia logica, 1 (1953), pp.
113-146. Traduzione francese Théorie des
propositions normatives. In Studia logica, 1
(1953), pp. 147-182. Kalinowski, Jerzy
(Georges), Norms and Logic, in The American
Journal of Jurisprudence, 18(1973), pp. 59-75.
Aristotle, Nicomachean Ethics Aristotle,
Metaphysic