# Is there a difference between contradictory and the opposite?

Is there a difference between contradictory and the opposite?

In natural language, we have the idea of opposite such as 'The opposite of good is evil'. In logic, we can represent that symbolically. Let G be good. Using symbols, the negation of good is not good (~G). But contradiction seems to be a type of opposite too.

So, how is 'opposite' defined in logic?

• In non-paraconsistent propositional logic one can only describe negation (contradiction) of a sentential proposition and contradicted propositions cannot coexist conjunctively in this logic. To express coexistent opposites conjunctively such as good and evil you need predicate logic, such as ∃x∃yG(x)∧~G(y)... Dec 18, 2021 at 22:53
• Contradictory is not an opposite. Many people describe contradictions as opposites because most people know what that means. To give the real definition of contradiction would likely confuse people. People use the opposite in slang because it is simpler to Express. Contradiction means both propositions cannot be true AND both proposition cannot be false at the same time. There are other relTions where both propositions cannot be true and are not contradictions. It is important to know the distinctions between propositions. Dec 19, 2021 at 3:27
• We cannot describe opposites in logic alone, it requires more structure. Namely some sort of spectrum along which properties fall, with its extremes being the "opposites". Of course, something cannot have a property along with its opposite, but the same is true for any two properties at distinct points along the spectrum. Contradicting each other is a lot broader than being opposites. Dec 19, 2021 at 7:17
• @Logikal agree, the negation is more appropriate instead of the word contradictory but still the question of the opposite remains.
– Axz
Dec 19, 2021 at 7:43
• @Ddouble knot the expression (∃x)(∃y)G(x) ^~G(y) is to say that there exist an x wich is a G and a y which isn't, x and y are different, i don't see the expression describing the idea of opposite.
– Axz
Dec 19, 2021 at 7:51

Yes. Contradiction just means that not both can be true at the same time. Opposite means the other extreme of the scale. (Of course, if there is no scale and just two possible states, contradiction and opposite coincide.) Also, the notion of contradiction is always applicable whenever there are several mutually exclusive states, whereas the notion of opposite only makes sense when there is also a hierarchical order between these states.

"hot" and "not hot" are contradictory, but "not hot" is not the opposite of "hot"; the opposite would be "cold", and something may be not hot without being cold, for example lukewarm.

"is a cat" and "is a dog" is also contradictory because nothing can be a cat and a dog at the same time, but it doesn't really make sense to say that a dog is the opposite of a cat.

Opposite is not defined in logic though; it's a purely linguistic notion. Contradiction is defined in terms of simple negation and has the same meaning in logic and natural language.

Opposite in logic is generally taken to mean logical negation. A contradiction in logic is generally taken to be two propositions whose truth values are opposites.

Example: P and ~P are opposites, but the set {P, ~P} is a contradiction.

If you're familiar with formal set theory, you know there are different between naive or intuitional definitions and formal definitions. Intuitively, we can talk about 'collections', but in ZFC, 'set' has a rigorous definition. 'Opposite' is a lot like that in logic. It's a matter of fact that 'opposite' doesn't appear in Oxford's dictionary of logic as an entry, although it is used in the dictionary. So, opposite is an intuitional sense. In fact, in mathematical logic, which might be seen as a theory that combines mathematical and logical thinking, opposite has more than one meaning, for instance, when talking about opposites of positive integers (negative integers), opposites of operations (inverses), and of course, opposite meanings (P and ~P). To understand the difference fully, it is helpful to appeal to two "laws" of logic, the principle of bivalence (PB) and the law of noncontradiction (LNC).

PB states:

In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. 1 A logic satisfying this principle is called a two-valued logic3 or bivalent logic.2

LNC states:

In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "p is the case" and "p is not the case" are mutually exclusive. Formally this is expressed as the tautology ¬(p ∧ ¬p). The law is not to be confused with the law of excluded middle which states that at least one, "p is the case" or "p is not the case" holds.

So, PB is saying that for any formula F, it can be assigned one of two truth values, either T or F where T is defined as ~F. LNC says that in a logical system, F can either be T or F but not both. It should be noted that PB implies a logic is binary, that is, has two truth values, and that LNC doesn't tolerate contradictions for proof, unless the contradiction is used to draw a conclusion proof by contradiction is a proof technique.

PB and LNC are relevant to the laws of thought, which is the observation that human discourse seems to obey certain principles be they what they may. Also note, that LNC is not universal to systems of logic. More recent developments in logic, called non-classical logics (See Bumble's answer to In how many and which ways can a logic be non-classical? Are there systems for organizing them? (PhilSE)) might reject a "law" such as LNC.

Yes there is, and the distinction is quite simple: if a and b stand in contradiction, they are said to be contradictories. Contradictories are propositions that can be neither both true nor both false, i.e exactly one of them must be true. However, logical opposites/contraries are propositions which cant be both true, but can be both false. This is the distinction in a logical sense, but when we refer to a broader sense, the distinction becomes that of a universal and a particular. Contradiction would involve any form of negation or mutual exclusion, whilst opposition would basically be that, but also whilst having a "maximum distance" within a given genus. Thus, hotness contradicts coldness (because it is mutually exclusive with it) and it also opposes it (hotness and coldness are the most distant in the genus of, say, temperature), whilst non-coldness contradicts coldness but doesnt necessarily oppose it, such as lukewarmth. Logical sense → contradiction and opposition are distinct and mutually exclusive Broad sense → opposition entails contradiction