Imperative logic is the field of
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
concerned with
imperatives. In contrast to
declaratives, it is not clear whether imperatives denote
propositions
A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...
or more generally what role truth and falsity play in their semantics. Thus, there is almost no consensus on any aspect of imperative logic.
Jørgensen's dilemma
One of a logic's principal concerns is
logical validity. It seems that arguments with imperatives can be valid. Consider:
:P1. Take all the books off the table!
:P2. ''Foundations of Arithmetic'' is on the table.
:C1. Therefore, take ''Foundations of Arithmetic'' off the table!
However, an argument is valid if the conclusion follows from the premises. This means the premises give us reason to believe the conclusion, or, alternatively, the truth of the premises determines truth of the conclusion. Since imperatives are neither true nor false and since they are not proper objects of belief, none of the standard accounts of logical validity apply to arguments containing imperatives.
Here is the dilemma. Either arguments containing imperatives can be valid or not. On the one hand, if such arguments can be valid, we need a new or expanded account of logical validity and the concomitant details. Providing such an account has proved challenging. On the other hand, if such arguments cannot be valid (either because such arguments are all invalid or because validity is not a notion that applies to imperatives), then our logical intuitions regarding the above argument (and others similar to it) are mistaken. Since either answer seems problematic, this has come to be known as Jørgensen's dilemma, named after
Jørgen Jørgensen
Jørgen Jørgensen (name of birth: Jürgensen, and changed to Jorgenson from 1817) (29 March 1780 – 20 January 1841) was a Danes, Danish adventurer during the Age of Revolution. During the action of 2 March 1808, his ship was captured by the ...
(
da).
While this problem was first noted in a footnote by
Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
, it received a more developed formulation by Jørgensen.
Deontic logic takes the approach of adding a modal operator
to an argument with imperatives such that a truth-value can be assigned to the proposition. For example, it may be hard to assign a truth-value to the argument "Take all the books off the table!", but
("Take all the books off the table"), which means "It is obligatory to take all the books off the table", can be assigned a truth-value, because it is in the
indicative mood
A realis mood ( abbreviated ) is a grammatical mood which is used principally to indicate that something is a statement of fact; in other words, to express what the speaker considers to be a known state of affairs, as in declarative sentences. Mo ...
.
Ross's paradox
Alf Ross observed that applying the classical rule of
disjunction introduction
Disjunction introduction or addition (also called or introduction) is a rule of inference of propositional logic and almost every other deduction system. The rule makes it possible to introduce disjunctions to logical proofs. It is the inferen ...
under the scope of an imperative operator leads to unintuitive (or apparently absurd) results. When applied to simple declaratives, the result appears to be valid deduction.
:P1. The room is clean.
:C1. Therefore, the room is clean or grass is green.
However, a similar inference does not seem to be valid for imperatives. Consider:
:P1. Clean your room!
:C1. Therefore, clean your room or burn the house down!
Ross's paradox highlights the challenge faced by anyone who wants to modify or add to the standard account of validity. The challenge is what we mean by a valid imperative inference. For valid declarative inference, the premises give you a reason to believe the conclusion. One might think that for imperative inference, the premises give you a reason to do as the conclusion says. While Ross's paradox seems to suggest otherwise, its severity has been subject of much debate.
The
semantics for deontic logic requires that all obligations in the
domain of discourse
In the formal sciences, the domain of discourse or universe of discourse (borrowing from the mathematical concept of ''universe'') is the set of entities over which certain variables of interest in some formal treatment may range.
It is also ...
be fulfilled in an acceptable possible world. The conclusion "It is obligatory to clean your room or burn the house down" does not falsify the premise "It is obligatory to clean your room." In addition, based on the context, it may also be true that "It is obligatory to not burn the house down", in which case any acceptable possible world must have "Your room is cleaned" and "The house is not burnt down" to be both true.
Some strands of this debate connect it to
Hans Kamp
Johan Anthony Willem "Hans" Kamp (born 5 September 1940) is a Dutch philosopher and Linguistics, linguist, responsible for introducing discourse representation theory (DRT) in 1981.
Biography
Kamp was born in Den Burg. He received a Ph.D. in UC ...
's
paradox of free choice, in which disjunction introduction leads to absurd conclusions when applied under the scope of a possibility modal.
Mixed inferences
The following is an example of a pure imperative inference:
:P1. Do both of the following: wash the dishes and clean your room!
:C1. Therefore, clean your room!
In this case, all the sentences making up the argument are imperatives. Not all imperative inferences are of this kind. Consider again:
:P1. Take all the books off the table!
:P2. ''Foundations of Arithmetic'' is on the table.
:C1. Therefore, take ''Foundations of Arithmetic'' off the table!
Notice that this argument is composed of both imperatives and declaratives and has an imperative conclusion.
Mixed inferences are of special interest to logicians. For instance,
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
held that no imperative conclusion can be validly drawn from a set of premises which does not contain at least one imperative. While
R.M. Hare held that no declarative conclusion can be validly drawn from a set of premises which cannot validly be drawn from the declaratives among them alone.
[Hare, Richard M. (1967). Some Alleged Differences Between Imperatives and Indicatives. '']Mind
The mind is that which thinks, feels, perceives, imagines, remembers, and wills. It covers the totality of mental phenomena, including both conscious processes, through which an individual is aware of external and internal circumstances ...
'', 76, 309-326. There is no consensus among logicians about the truth or falsity of these (or similar) claims and mixed imperative and declarative inference remains vexed.
Applications
Aside from intrinsic interest, imperative logic has other applications. The use of imperatives in moral theory should make imperative inference an important subject for
ethics
Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches inclu ...
and
metaethics
In metaphilosophy and ethics, metaethics is the study of the nature, scope, ground, and meaning of moral judgment, ethical belief, or values. It is one of the three branches of ethics generally studied by philosophers, the others being normativ ...
.
See also
*
Deontic logic
*
Free choice inference
Free choice is a phenomenon in natural language where a linguistic disjunction appears to receive a logical conjunctive interpretation when it interacts with a modal operator. For example, the following English sentences can be interpreted to me ...
*
List of Logical Paradoxes
*
Speech act
In the philosophy of language and linguistics, a speech act is something expressed by an individual that not only presents information but performs an action as well. For example, the phrase "I would like the mashed potatoes; could you please pas ...
s
*
Pragmatics
In linguistics and the philosophy of language, pragmatics is the study of how Context (linguistics), context contributes to meaning. The field of study evaluates how human language is utilized in social interactions, as well as the relationship ...
*
Temporal logic
In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I ...
References
Further reading
*
* Peter B. M. Vranas (2010)
IMPERATIVES, LOGIC OF* Entry for The International Encyclopedia of Ethics
* Covers mostly the approach of
Héctor-Neri Castañeda.
External links
* Mitchell S. Green
Imperative Logic University of Virginia
{{Paradoxes
Deontic logic