In
philosophical logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophic ...
, the concept of an impossible world (sometimes called a non-normal world)
is used to model certain
phenomena that cannot be adequately handled using ordinary
possible world
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their met ...
s. An
impossible world,
, is the same sort of thing as a possible world
(whatever that may be),
except that it is in some sense "impossible." Depending on the context, this may mean that some
contradiction
In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
s, statements of the form
are true at
, or that the normal laws 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 ...
,
metaphysics
Metaphysics is the branch of philosophy that examines the basic structure of reality. It is traditionally seen as the study of mind-independent features of the world, but some theorists view it as an inquiry into the conceptual framework of ...
, and
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, fail to
hold at
, or both. Impossible worlds are controversial objects in
philosophy
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
,
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 ...
, and
semantics
Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
. They have been around since the advent of possible world semantics for
modal logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields
it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
, as well as world based semantics for non-classical logics, but have yet to find the ubiquitous acceptance, that their possible counterparts have found in all walks of philosophy.
Argument from ways
Possible worlds
Possible worlds are often regarded with suspicion, which is why their proponents have struggled to find arguments in their favor. An often-cited argument is called the argument from ways. It defines possible worlds as "ways how things could have been" and relies for its premises and inferences on assumptions from
natural language
A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and change. It can take different forms, typically either a spoken language or a sign language. Natural languages ...
,
for example:
:(1)
Hillary Clinton
Hillary Diane Rodham Clinton ( Rodham; born October 26, 1947) is an American politician, lawyer and diplomat. She was the 67th United States secretary of state in the administration of Barack Obama from 2009 to 2013, a U.S. senator represent ...
could have won the
2016 US election.
:(2) So there are other ways how things could have been.
:(3) Possible worlds are ways how things could have been.
:(4) So there are other possible worlds.
The central step of this argument happens at ''(2)'' where the plausible ''(1)'' is interpreted in a way that involves
quantification over "ways". Many philosophers, following
Willard Van Orman Quine
Willard Van Orman Quine ( ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century" ...
, hold that quantification entails
ontological commitment
Ontology is the philosophical study of being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of reality and every ...
s, in this case, a commitment to the existence of possible worlds. Quine himself restricted his method to scientific theories, but others have applied it also to natural language, for example,
Amie L. Thomasson in her paper entitled ''Ontology Made Easy''.
The strength of the ''argument from ways'' depends on these assumptions and may be challenged by casting doubt on the quantifier-method of ontology or on the reliability of natural language as a guide to ontology.
Impossible worlds
A similar argument can be used to justify the thesis that there are impossible worlds,
for example:
:(a) Hillary Clinton couldn't have both won and lost the 2016 US election.
:(b) So there are ways how things couldn't have been.
:(c) Impossible worlds are ways how things couldn't have been.
:(d) So there are impossible worlds.
The problem for the defender of possible worlds is that language is ambiguous concerning the meaning of ''(a)'': does it mean that this is a way how things couldn't be or that this is not a way how things could be.
It is open to critics of impossible worlds to assert the latter option, which would invalidate the argument.
Applications
Non-normal modal logics
Non-normal worlds were introduced by
Saul Kripke
Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American analytic philosophy, analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emer ...
in 1965 as a purely technical device to
provide semantics for
modal logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields
it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
s weaker than the system K — in particular, modal logics that reject
the rule of necessitation:
:
.
Such logics are typically referred to as "non-normal." Under the standard interpretation of modal vocabulary in
Kripke semantics
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André ...
, we have
if and only if in each model,
holds in all worlds. To construct a model in which
holds in all worlds but
does not, we need either to interpret
in a non-standard manner (that is, we do not just consider the truth of
in every accessible world), or we reinterpret the condition for being ''valid''. This latter choice is what Kripke does. We single out a class of worlds as ''normal'', and we take ''validity'' to be truth in every normal world in a model. in this way we may construct a model in which
is true in every normal world, but in which
is not. We need only ensure that this world (at which
fails) have an accessible world which is not ''normal.'' Here,
can fail, and hence, at our original world,
fails to be necessary, despite being a truth of the logic.
These non-normal worlds are ''impossible'' in the sense that they are not constrained by what is true according to the logic. From the fact that
, it does not follow that
holds in a non-normal world.
For more discussion of the interpretation of the language of modal logic in models with worlds, see the entries on
modal logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields
it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
and on
Kripke semantics
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André ...
.
Avoiding Curry's paradox
Curry's paradox
Curry's paradox is a paradox in which an arbitrary claim ''F'' is proved from the mere existence of a sentence ''C'' that says of itself "If ''C'', then ''F''". The paradox requires only a few apparently-innocuous logical deduction rules. Since '' ...
is a serious problem for
logicians who are interested in developing formal languages that are "semantically closed" (i.e. that can express their own semantics). The paradox relies on the seemingly obvious principle of
contraction:
:
.
There are ways of using non-normal worlds in a semantical system that invalidate contraction. Moreover, these methods can be given a reasonable philosophical justification by construing non-normal worlds as worlds at which "the laws of logic fail."
Counternecessary statements
A counternecessary statement is a
counterfactual conditional
Counterfactual conditionals (also ''contrafactual'', ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be h ...
whose antecedent is not merely false, but ''necessarily'' so (or whose consequent is necessarily true).
For the sake of argument, assume that either (or both) of the following are the case:
:1.
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...
is false.
:2. The
law of excluded middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and t ...
is true.
Presumably each of these statements is such that if it is true (false), then it is
necessarily true (false).
Thus one (or both) of the following is being assumed:
:1′. Intuitionism is false at every possible world.
:2′. The law of excluded middle is true at every possible world.
Now consider the following:
:3. If intuitionism is true, then the law of excluded middle holds.
This is intuitively false, as one of the fundamental tenets of intuitionism is precisely that the LEM ''does not'' hold. Suppose this statement is cashed out as:
:3′. Every possible world at which intuitionism is true is a possible world at which the law of excluded middle holds true.
This holds vacuously, given either (1′) or (2′).
Now suppose impossible worlds are considered in addition to possible ones. It is
compatible with (1′) that there are ''impossible'' worlds at which intuitionism is true, and with (2′) that there are ''impossible'' worlds at which the LEM is false. This yields the interpretation:
: 3*. Every (possible or impossible) world at which intuitionism is true is a (possible or impossible) world at which the law of excluded middle holds.
This does not seem to be the case, for intuitively there are ''impossible'' worlds at which intuitionism is true and the law of excluded middle does not hold.
See also
*
Possible world
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their met ...
*
Modal realism
*
Extended modal realism
References
Bibliography
* Kripke, Saul. 1965. Semantical analysis of modal logic, II: non-normal modal propositional calculi. In J.W. Addison, L. Henkin, and A. Tarski, eds., ''The Theory of Models''. Amsterdam: North Holland.
*
Priest, Graham (ed.). 1997. ''Notre Dame Journal of Formal Logic'' 38, no. 4. (Special issue on impossible worlds.
Table of contents* Priest, Graham. 2001. ''
An Introduction to Non-Classical Logic''. Cambridge: Cambridge University Press.
External links
*
*
*
Edward N. ZaltaA classically-based theory of impossible worlds(PDF)
{{Logic
Possible world
Concepts in logic