Logical possibility
Encyclopedia
A logically possible proposition
is one that can be asserted without implying a logical contradiction
. This is to say that a proposition is logically possible if there is some coherent way for the world to be, under which the proposition would be true. Thus, "the sky
is blue
" (and all other actually true propositions) is logically possible: there exists some logically coherent way for the world to be such that it is true, viz., the way that the world actually is. But this "way for the world to be" need not be the way the world actually is; it need only be logically coherent. So, for example, the false proposition the sky is green
is also logically possible, so long as we are able (as we indeed seem to be) to conceive of some logically coherent world in which the sky is green. Philosophers generally consider logical possibility to be the broadest sort of subjunctive possibility
in modal logic
.
Logical possibility should be distinguished from other sorts of subjunctive possibilities. For example, it may be logically possible for the universe
's physical laws to be different from what they actually are. If it is, then many things that we would normally consider to be demonstrably impossible can be logically possible: for example, that travel might be possible at speeds faster-than-light
or that escape from black holes is not impossible. Many philosophers, then, have held that these scenarios are logically possible but nomologically impossible (impossible under the actual laws of nature).
These propositions are also to be contrasted with logically impossible propositions, i.e., propositions which could not possibly be true under any circumstances in any universe because they are formal contradictions. While it is logically possible for the sky to be green, it is not logically possible for a square to be circular in shape. Some combinations of physical laws are also known to result in contradictions. For instance, if a given universe's physical laws are invariant through time
, then the law of conservation of energy
holds in that universe. This is a consequence of Noether's theorem
, which can be proven mathematically
. Thus, a universe whose physical laws do not vary with time and which does not exhibit conservation of energy is not logically possible.
With this understanding of logical possibility in mind, other logical modalities may be defined in terms of it: a proposition is logically necessary if it is not logically possible for it to be false, logically impossible if it is not logically possible for it to be true, and logically contingent if it is logically possible for it to be true and also logically possible for it to be false.
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
is one that can be asserted without implying a logical contradiction
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
. This is to say that a proposition is logically possible if there is some coherent way for the world to be, under which the proposition would be true. Thus, "the sky
Sky
The sky is the part of the atmosphere or outer space visible from the surface of any astronomical object. It is difficult to define precisely for several reasons. During daylight, the sky of Earth has the appearance of a pale blue surface because the air scatters the sunlight. The sky is sometimes...
is blue
Blue
Blue is a colour, the perception of which is evoked by light having a spectrum dominated by energy with a wavelength of roughly 440–490 nm. It is considered one of the additive primary colours. On the HSV Colour Wheel, the complement of blue is yellow; that is, a colour corresponding to an equal...
" (and all other actually true propositions) is logically possible: there exists some logically coherent way for the world to be such that it is true, viz., the way that the world actually is. But this "way for the world to be" need not be the way the world actually is; it need only be logically coherent. So, for example, the false proposition the sky is green
Green
Green is a color, the perception of which is evoked by light having a spectrum dominated by energy with a wavelength of roughly 520–570 nanometres. In the subtractive color system, it is not a primary color, but is created out of a mixture of yellow and blue, or yellow and cyan; it is considered...
is also logically possible, so long as we are able (as we indeed seem to be) to conceive of some logically coherent world in which the sky is green. Philosophers generally consider logical possibility to be the broadest sort of subjunctive possibility
Subjunctive possibility
Subjunctive possibility is the form of modality most frequently studied in modal logic...
in modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
.
Logical possibility should be distinguished from other sorts of subjunctive possibilities. For example, it may be logically possible for the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
's physical laws to be different from what they actually are. If it is, then many things that we would normally consider to be demonstrably impossible can be logically possible: for example, that travel might be possible at speeds faster-than-light
Faster-than-light
Faster-than-light communications and travel refer to the propagation of information or matter faster than the speed of light....
or that escape from black holes is not impossible. Many philosophers, then, have held that these scenarios are logically possible but nomologically impossible (impossible under the actual laws of nature).
These propositions are also to be contrasted with logically impossible propositions, i.e., propositions which could not possibly be true under any circumstances in any universe because they are formal contradictions. While it is logically possible for the sky to be green, it is not logically possible for a square to be circular in shape. Some combinations of physical laws are also known to result in contradictions. For instance, if a given universe's physical laws are invariant through time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, then the law of conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
holds in that universe. This is a consequence of Noether's theorem
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
, which can be proven mathematically
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
. Thus, a universe whose physical laws do not vary with time and which does not exhibit conservation of energy is not logically possible.
With this understanding of logical possibility in mind, other logical modalities may be defined in terms of it: a proposition is logically necessary if it is not logically possible for it to be false, logically impossible if it is not logically possible for it to be true, and logically contingent if it is logically possible for it to be true and also logically possible for it to be false.
See also
- Modal logicModal logicModal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
- ParadoxParadoxSimilar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
- Subjunctive possibilitySubjunctive possibilitySubjunctive possibility is the form of modality most frequently studied in modal logic...
- Possibility theoryPossibility theoryPossibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. D. Dubois and H. Prade further...
- Possible worldPossible worldIn philosophy and logic, the concept of a possible world is used to express modal claims. The concept of possible worlds is common in contemporary philosophical discourse and has also been disputed.- Possibility, necessity, and contingency :...
External links
- Do Modal Claims Imply the Existence of Possible Worlds?, paper criticizing David Lewis' theory of possibility.