=Paper=
{{Paper
|id=Vol-2632/MIREL-19_paper_4
|storemode=property
|title=A Short Note on the Chisholm Paradox
|pdfUrl=https://ceur-ws.org/Vol-2632/MIREL-19_paper_4.pdf
|volume=Vol-2632
|dblpUrl=https://dblp.org/rec/conf/jurix/Governatori19
}}
==A Short Note on the Chisholm Paradox==
https://ceur-ws.org/Vol-2632/MIREL-19_paper_4.pdf
A Short Note on the Chisholm Paradox
Guido Governatori
Data61, CSIRO, Australia
Abstract. We advance an alternative version of the Chisholm Paradox and we
argue that the alternative version (while logically equivalent to the original
version), in its manifestation in the natural language, is not intuitively consistent.
The alternative version of the paradox suggests some requirements for deontic
logics designed for legal reasoning.
Keywords: Chisholm Paradox, Deontic Disjunctive Syllogism
The Chisholm paradox [2] introduced the topic of the so-called contrary-to-duty obli-
gations and the problems related to their formalisation in Deontic Logic. A cornucopia
of research sparkled from the seminal paper and a multitude of logical systems have
been proposed to address the formalisation of CTDs. Chisholm proposed a set of four
statements that seem logically independent from each other when formulated in natural
language but whose formal representation in (Standard) Deontic Logic either leads to
an inconsistency or the statements are no longer logically independent.
The formulation of the paradox reads as follows:
S1 A person ought to help his neighbour.
S2 It ought to be that if the person helps the neighbour he has to tell he is going to
help.
S3 If the person does not help the neighbour then he ought not to tell that he is going
to help.
S4 the person does not help the neighbour.
Literature understood the above set of statements to be intuitively consistent.
There is a widespread agreement in the literature that, form the intuitive point
of view, this set is consistent, and its members are logically independent of
each other; [1, p. 294]
However, when the four statements are encoded in (standard) Deontic Logic by the
following formulae
L1 Oβπππ
L2 O(βπππ β π‘πππ)
Copyright Β©2020 for this paper by its author. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
2 Guido Governatori
L3 Β¬βπππ β OΒ¬π‘πππ
L4 Β¬βπππ
we obtain a contradiction given that from L1 and L2 we derive Oπ‘πππ and from L3 and
L4 we conclude OΒ¬π‘πππ. Moreover, when we replace L2 with βπππ β Oπ‘πππ, then the
alternative version of L2 is a logical consequence of L4, and the four statements are no
longer independent of each other.
Chisholm implicitly implies that the four statements in their natural language man-
ifestation are consistent, though there is no real argument supporting the conclusion
that they must be consistent. Similarly, Γ
qvist [5, pp. 365β6] proposed a set of three
requirements that (r1) the statements are mutually consistent, (r2) none of the state-
ments logically follows from the other three statements and (r3) the obligation of the
contrary-to-duty statement should be obtained. Γ
qvist does not provide a justification
for the first requirement. Similarly, Tomberlin [4, p. 359], who otherwise challenged
the other requirements, accepted, with no discussion, that the four statements are,
plainly, mutually consistent. However, to the best of our knowledge, no real discussion
or empirical evaluation of the claimed consistency has been proposed or carried out.
In this short note, we introduce an alternative formulation of the Chisholm set
that is logically equivalent to the original set when represented in (Standard) Deontic
logic. Nevertheless, there is a fundamental difference: the set of statements is no longer
consistent. Before giving the alternative version and the justification of why that
version is not consistent, we return on the analysis provided by Chisholm. Chisholm [2,
p. 34] suggested that O(π β π) is not adequate for the expression of contrary-to-duty
imperatives, and (ii) advanced the following reading: one should refrain from the joint
action of doing π and not doing π. Hence, Chisholm proposed to read the obligation
as O(Β¬(π β§ Β¬π)). Let us take an extra step, and let us apply the De Morgan Law to
the content of the obligation. This gives us O(Β¬π β¨ π), where the intuitive reading is
that one is obliged to choose between refraining from doing π or doing π. According to
what we have just discussed, the alternative version of the set of statements replaces
the second statement with the following one
S2β it ought to be that the person chooses between not helping the neighbour or to tell
the neighbour that he is going to help.
The statement can be naturally represented by
O(Β¬βπππ β¨ π‘πππ) (L2β)
Clearly, given the logical equivalence of the logical representations, we formally con-
clude that the alternative formal representation is logically inconsistent. What it remains
to do is to investigate whether the alternative version in natural language is intuitively
consistent or not. This leads us to consider the plausibility of what we can call Deontic
Disjunctive Syllogism. The Deontic Disjunctive Syllogism is the inference pattern that
from the obligation of a conjunction derives the obligation of one of the disjuncts when
the other disjunct is forbidden (or when the other disjunct leads to a violation).
Consider the rules of sudoku (9 Γ 9) that state that
1. for every cell, for every row, column or block the cell must contain one digit from
1 to 9; and
A Short Note on the Chisholm Paradox 3
2. for every row, column or block, if a cell contains a digit, no other cell in the same
row, column or block can contain that digit.
9 5 8 7 6 4 2 3
3 8 7 5
1 7 3 5 8 4
6 7 9 4 2 8 5 1 3
8 5 7 3 6 2 9
5 2 3 6 4 8 7
9 2 8 5 3
5 3 1 2
3 2 7 9 5
Fig. 1. A Partially Solved Sudoku Puzzle
Consider now a situation where two cells in a column have not been filled (see the
4th column in the sudoku diagram in Figure 1, where the missing digits are 1 and 9).
This means that only the remaining two digits can occur in these cells. Accordingly,
we can say, that it the cell in the 3rd row, and 4th column must contain either the digit
1 or the digit 9. The same applies to the cell in column 4 and row 2. However, in one of
the rows for the empty cells (namely the cells in the third row in the diagram), one cell
contains one of the two digits (i.e., 1). Hence, to obtain a legal solution, the other digit
must be in the cell. Alternatively, we can say that, to get a valid solution, it is forbidden
to put the digit 1 in the fourth cells in the third row. Thus, the digit that must occur in
that cell is 9.
Based on the discussion we had so far, the Deontic Disjunctive Syllogism appears to
be a reasonable, sound and intuitive inference pattern for reasoning with (disjunctive)
obligations. This view is also shared by Horty [3, p. 430β431] who proposed the example
of two norms βfight in the army or perform alternative serviceβ and βdonβt fight in the
armyβ (possibly from two difference sources), where he claims that the obligation to
perform the alternative service follows, from an intuitive standpoint, from the two (par-
tially) conflicting norms. Before going back to the alternative version of the Chisholm
paradox, we quickly investigate the plausibility of disjunctive obligations from a formal
point of view. To this end, we study how to model the sudoku rules in (Standard)
Deontic Logic, using disjunctive obligations as the main means for the formalisation.
First of all we assume the following set of atomic propositions {πππ |π, π, π β {1, . . . 9}}
where the meaning of πππ is that digit π appears in the cell whose coordinates are π (for
the column) and π (for the row); in addition, we partition the 81 coordinates ππ into
nine blocks (each as a set of 9 coordinates forming a 3 Γ 3 square), and we use ππ β π΅
to indicate that the cell with coordinate ππ is in block π΅. Then, the two sudoku rules
4 Guido Governatori
are encoded as follows (where π, π and π range from 1 to 9):
Γ
O πππ (1)
1β€π β€9
Γ Γ Γ
πππ β OΒ¬πππ 0 β§ OΒ¬ππ 0π β§ OΒ¬ππ 0π 0 (2)
π β π 0 πβ π 0 ππ β π 0π 0
ππ,π 0π 0 βπ΅
Γ Γ Γ
Oπππ β OΒ¬πππ 0 β§ OΒ¬ππ 0π β§ OΒ¬ππ 0π 0 (3)
π β π 0 πβ π 0 ππ β π 0π 0
ππ,π 0π 0 βπ΅
Γ
0 0
πππ β Β¬πππ β§ OΒ¬πππ (4)
πβ π 0
A valid sudoku is a set of initial clues (providing the placement of some digits in the
grid) that has a unique solution that satisfies the two rules above. The clues are a set
of propositions of the form πππ that generates a set of 81 obligations of the from Oπππ ,
derived from the initial placement and the formulas above. A valid (or legal) solution is
a set of 81 (atomic) propositions, such that all the 81 obligations have been fulfilled.
Alternatively, we can define a legal/valid solution as a set of 81 (atomic) propositions
containing the initial clues and satisfying the two rules/formulas above.
Let us examine the diagram in Figure 1: in the fourth column the digits 1 and 9
are missing; hence either 1 or 9 should be in the cell with coordinate 42 or 43. This
means, that with repeated use of (2) and the Deontic Disjunctive Syllogism we can
derive (i) O(142 β¨ 143 ) and (ii) O(942 β¨ 943 ). Moreover, the digit 1 appears in the cell with
coordinate 23, 123 and from (2) we conclude O¬143 , which, together with (i) implies
O142
Harmed with the discussion we had about the reasonableness of the Deontic Dis-
junctive Syllogism, we can address the issue if the alternative version of the Chisholm
Paradox we have proposed is consistent or not. Clearly not helping the neighbour is
the opposite of helping, thus we can use the Deontic Disjunctive Syllogism on 1. and 2β.
to conclude that the person ought to tell the neighbour that he is going to help. On
the other hand, from 3. and 4., by Modus Ponens, we conclude that the person ought
to refrain from telling that he is going to help. Consequently, in a situation where
the person does not comply with the obligation to help, the person ought to tell and
ought not to tell, thus no matter what the person does, the person cannot comply with
the requirements about informing the neighbour about his intention (or lack of it) of
helping him. Finally, when we assume, as it is typically the case in Deontic Logic, that
a set of norms is consistent (encoded by the axiom Oπ β Β¬OΒ¬π, we conclude that the
statements in the alternative version of the paradox are not mutually consistent.
What are the consequences of the alternative version?
1. We have argued that the set of statements in the alternative version of the Chisholm
paradox appears to be, from an intuitive analysis, not consistent. Accordingly, we
can argue that the (original) Chisholm set is not necessarily consistent, or at least
its consistency requires the use of mechanisms to handle conflicting obligations,
in particular under the reading proposed by Chisholm for O(π β π).
A Short Note on the Chisholm Paradox 5
2. S2β and S3 require different representations, and then, since L2β and L2 are logically
equivalent S2 and S3 require different representations as well.
3. L2β cannot be used to justify the switch to a deontic conditional or a dyadic
obligation to maintain that the four statements are logically independent. To
achieve the same result one can replace S4, the non-compliant behaviour, with the
behaviour complying with S1: βthe person helps the neighbourβ, formalised as
L4β βπππ
which then makes L3 derivable from it using material implication.
4. A deontic logic should accept both factual detachment for a normative conditional
(or a dyadic obligation operator), to be able to derive Oπ from π and π βO π
(or, in case of a dyadic obligation O(π/π), and deontic detachment for material
implication (or deontic disjunctive syllogism) to handle disjunctive obligations.
5. In addition to the previous remark, Independently from whether one admits the
deontic disjunctive syllogism, the set of OΒ¬π, OΒ¬π and O(π β¨ π) cannot be consis-
tently complied with; and it will require a mechanism to solve the conflict among
the three obligations (e.g., by considering the three obligations as prima facie
obligations)
Acknowledgements
I thank Silvano Colombo Tosatto, Nick van Beest and Antonino Rotolo for the nice
discussions we had on this topic. I am also grateful to Leon van der Torre and Xavier
Parent for their comments on an earlier draft of the paper. As it is customary, the author
is obliged to thank people who help them with the paper. Similarly, the author has
the choice of not thanking people who helped them or to provide a disclosure that
the errors are entirely the responsibility of the author. I fully comply with these two
customs; hence I hereby acknowledge that any error, omission and misunderstanding
is only mine.
References
1. JosΓ© Carmo and Andrew J. I. Jones. Deontic Logic and Contrary-to-Duties, pages 265β343.
Springer Netherlands, Dordrecht, 2002.
2. Rodrick M. Chisholm. Contrary-to-Duty Imperatives and Deontic Logic. Analysis, 24(2):33β36,
1963.
3. John Horty. Deontic modals: Why abbandon the classical semantics. Pacific Philosophical
Quarterly, 95:424β460, 2014.
4. James E. Tomberlin. Contrary-to-duty imperatives and conditional obligation. NoΓ»s, 15(3):357β
375, 1981.
5. Lennart Γ
qvist. Good Samaritan, contrary-to-duty imperatives, and epistemic obligations.
NoΓ»s, 1(4):361β379, 1967.
6 Guido Governatori
A The Initial Configuration of the Sudoku Puzzle
For the readers who enjoy solving a sudoku puzzle, here is the initial configuration of
the puzzle discussed in the previous pages.
5 8 7 4
3
7 3 5 4
6 8 5 1
8 7 2
2 3 6 7
9 8 5 3
2
2 7 9 5
The solution is on the next page.
A Short Note on the Chisholm Paradox 7
B The Solution of the Sudoku Puzzle
9 5 8 7 6 4 2 3 1
3 6 4 1 8 2 7 9 5
2 1 7 9 3 5 8 4 6
6 7 9 4 2 8 5 1 3
4 8 1 5 7 3 6 2 9
5 2 3 6 1 9 4 8 7
7 9 2 8 5 1 3 6 4
8 4 5 3 9 6 1 7 2
1 3 6 2 4 7 9 5 8