Logical Analysis of an Italian Verdict
What is illogical is also unjust: this principle struggles to assert itself in the world of law and particularly in the world of Justice. We have witnessed innumerable judgements issued by judges of various types that to define contradictory, illogical, unreasonable does not give a good idea. The author has been studying for years how logic, the science of reasoning, can help to have a ” right justice “, because it conforms to logic.
Do you not believe that an illogical verdict is unjust and therefore to be discarded ? Here below the formal logic “rejects” the “motivation” of that verdict issued by an Italian Court of Appeal where three judges (all women ! In Italy there isn’t the Jury in criminal trial) ) have absolved the defendants of rape writing (in the reasons of the verdict) that the victim, a woman, was too ugly and masculine to be raped, to be desired sexually and that therefore it was impossible for someone to rape her. What follows, while in the scientific rigor, will be exposed by the author in a slightly divulgative way in order to allow to all the interested ones an approach to such topics that I will be more deeply developing in a work of mine of next publication.
We will use the so called semantic tableau; a Beth tree (or semantic tableau) is a binary rooted tree in which each node is labeled with one or more formulas: they are
tree graphs consisting of a flat arrangement of nodes each of which containing one or more statements; the first node contains the statement that must be refuted.
Nodes can be added to a branch according to rules that can be traced to following:
Rule α Rule β
X∧Y X∨Y
↓ ↓
⏐ ⏐ ⏐
X X Y
Let’s analyze these rules.
Given a set Γ of formulas, the procedure for building a tableau for Γ is as follows :
(0) Place Γ at the root of the tree.
(n + 1) For each branch R of the tree and for each node n of R.
– if n contains α, extend R into one branch R ∪ {α1, α2}
– if n contains β, extend R into two branches R ∪ {β1} and R ∪ {β2}
If Γ is finite, the procedure always terminates.
A branch is said to be terminated if it can no longer be extended. A tree, therefore, is said terminated if all its possibly open branches are terminated. A tableau is terminated if it is no longer possible to apply one of the rules of the proof system to any of the formulas that label its nodes. A closed branch is such as it contains a contradiction.
Thus , the method of semantic tableaux ends after a finite number of steps:
if the tableau is complete for the statement P, this statement will not be satisfactory if and only if the Tableax is “closed”.
When is a Tableax “closed” ? When there will be in the same node both a statement and its negation, that is contradictory. Below is an example:
the statement “ A∧ (¬A) ” is obviously false, is the negation (absurd) of the Law of Non-Contradiction [the LNC formally this is expressed as the tautology ¬ (A ∧ ¬ A) ] and of the Law of excluded middle (TND acronyme from Latin for tertium non datur – no third choice is given, and it is a tautology too).
Well, by applying the above rules (a and b) in a tableau we will place its component statements (A and negation of A, i.e. ¬A) in the same node (as indicated by rule α).
The presence in the same node of an statement and its negation is therefore connected to an inevitable situation of falsity and absurdity.
Therefore, the simultaneous presence of A and ¬A in the same node makes it useless to proceed with the analysis and the branch to which the node belongs is said to be closed.
Now all that remains is to translate into formal logic the motivation of that verdict (that to call illogical is an understatement); here is its translation:
￢ [(A→B)∧ (B→C)]→(A→C) (1)
it reads, with reference to that “world” of “judgers”:
It is not true that If A then B and if B then C therefore if A then C
I translate it into words comprehensible even by a three-year-old (as the lawyer would say in the movie Philadelphia):
It is not true ( ¬ ) that If a woman is ugly (A) Then (→ ) she can be equally sexually desirable (B) AND (∧ ) If she is not sexually desirable (B) then there can be no sexual rape ( C ) ; therefore (→ ) the ugly woman was not been raped by anyone as she cannot be raped by anyone as she is undesirable.
We must essentially refute, if we can, such a statement (1) and we will do this in 5 steps using the semantic tableaus and their rules above:
the opposite of (1) is naturally and logically the following statement
{ [(A→B)∧(B→C)]→(A→C) } (2)
meaning “an ugly woman can be sexually desirable and be raped”, the opposite of the statement (1).
Let us try to refute the negation of (2) i.e. (1), the statement ” that we do not like”:
1)
¬ {[(A→B)∧(B→C)]→(A→C)}
⏐
(A→B)∧(B→C)
¬ (A→C)
2)
¬ {[(A→B)∧(B→C)]→(A→C)}
⏐
(A→B)∧ (B→C)
¬ (A→C)
⏐
A→ B
B→ C
¬ (A→C)
3)
¬ {[(A→B)∧(B→C)]→(A→C)}
⏐
(A→B)∧(B→C)
¬(A→C)
⏐
A→B
B→C
¬ (A→C)
⏐
A→B
B→C
A
¬ C
4)
¬ [(A→B)∧(B→C)]→(A→C)}
⏐
(A→B)∧(B→C)
¬ (A→C)
⏐
A→B
B→C
¬ (A→C)
⏐
A→B
B→C
A
¬ C
⏐ ⏐
B→C B→C
A A
¬ C ¬ C
¬ A B
♦
The branch that ends with “ ¬ A ” is closed (and we indicate it with the rhomboidal symbol ♦ not to be confused with the symbol of modal logic “diamond”), because in the last node we find both “A” and “¬ A” (the negation of “A” ). Let’s continue:
5)
¬ {[(A→B)∧(B→C)]→(A→C)}
⏐
(A→B)∧(B→ C)
¬ (A→C)
⏐
A→B
B→C
¬ (A→C)
⏐
A→B
B→C
A
¬ C
⏐ ⏐
B→C B→C
A A
¬ C ¬ C
¬ A B
♦ | |
A A
¬ C ¬ C
B B
¬ B C
♦ ♦
As you can see the branches created are both closed: the one on the left due to the simultaneous presence of B and ¬B in the last node; the one on the right due to the presence of C and ¬C in the last node.
The initial statement (1) is therefore disproved and this means that its negation (2), [(A→B)∧(B→C)]→(A→C), is a tautology, that is a right and correct reasonment and a logical law.
It represents a law and a logical structure, a well-formed formula or a series of well-formed formulas and therefore is right, while its negation (1) is wrong because it is illogical.
What is illogical is wrong and injust: Q.E.D. ( for the uninitiated it stands for “quod erat demonstrandum” Latin for “whic was to be demonstrated”).
It will still take a long time to subject Justice to logical-formal controls. Perhaps with the effort of all the experts in the field, tomorrow we will reach a verification through special complex computer algorithms of the right meaning and of judicial reasoning of the Courts of Justice. The effort is not only scientific and logical, but it also requires the modification of the legal systems, both substantive and procedural law. No one lives in the “best of all possible worlds”: improving the world of Justice is the first step to take in order to approach a world that can access all other possible worlds, according to the “dialectic” of modal logic of Leibnizian teaching. In other works of mine I am developing this new different from other seemingly similar studies research between logic and law.
The world is full of unjust verdicts: logic and computer science in a joint effort can help the human race to progress.
—o—
References
Irving H. Anellis in From semantic tableaux to Smullyan trees: a history of the development of the falsifiability tree method (1990)
W. Beth in Semantic entailment and formal derivability (1955)
E.Hughes, M.J.Cresswell in A New Introduction to Modal Logic (1996)
Smullyan in First-order logic (1968)
Soeteman in Logic in Law: Remarks on Logic and Rationality in Normative Reasoning. Especially in Law (1989)
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