Opaque context

An intensional or (referential) opaque context (Latin: opacus "shadowed, opaque") is used in the philosophy of language , logic and semantics to describe a linguistic context in which the truth value of the expressed statement may change by replacing partial expressions with the same scope of meaning . In normal, non-opaque contexts, the replacement of expressions with the same extension is always truth-preserving, i.e. salva veritate is possible.

The representation of opaque contexts is one of the essential challenges for any model of formal semantics , since it shows that neither the ordinary objects in the world nor private ideas can directly be the meaning of linguistic expressions. Gottlob Frege identified, among other things, the reproduction of other people's utterances in indirect speech as an opaque context (referred to by him as "odd speech"). The study of opaque contexts plays a role in the philosophical theory of proper names , the theory of labels, as well as modal logic and epistemic logic .

Explanations

Extension and Intension

Since the logic of Port-Royal (1662), it has been customary to use linguistic expressions to differentiate between their subject-matter (reference or extension) and their content (meaning or intention). In modern semantics, which is based on Gottlob Frege's essay On Meaning and Meaning (1892) and essentially goes back to Alfred Tarski and Rudolf Carnap , the following classification has established itself as the standard:

Expression type	Extension	Intension
Proper names	Bearer of the name	Individual term
single-digit predicates	Sets of individuals	Terms
multi-digit predicates	Sets of n- tuples	Relations
sentences	Truth values	Propositions

Coextensionality

Coextensionality is a semantic property of expressions: proper names or labels are coextensional if they denote the same thing. Accordingly, the expressions “the highest mountain on earth” and “ Mount Everest ”, for example, are coextensional , ie a label and a proper name that both designate the same object.

A classic and now historically outdated example comes from Willard Van Orman Quine . The expressions “the number 9” and “the number of planets (within our solar system)” could be considered coextensional in Quine's time, since the current list of planets at that time still included Pluto (see article Planet ).

One speaks of coextensional names for concepts or predicates when everything to which one concept is assigned also includes the other concept and vice versa. The concepts then have the same scope, that is, the sets of things that satisfy the predicates are identical. According to an example by Carnap, coextension terms would be “() is a living being with a heart” and “() is a living being with kidneys”, since Carnap assumes that all living beings that have a heart also have kidneys and vice versa.

Substitutability salva veritate

Under normal circumstances, the truth or falsity of a statement does not change if one replaces an expression in it with a co-extensional one, which is why one speaks of substitutability salva veritate . The meaning of complex linguistic expressions is in this case directly dependent on the meaning of the simple linguistic expressions occurring in them. For example, if we replace the expression "Mount Everest" with "the highest mountain on earth" in " Sir Edmund Hillary climbed Mount Everest", we get "Sir Edmund Hillary climbed the highest mountain on earth".

These two sentences now have the same truth value , since the meaning should not have changed. I.e. if the first sentence is true, the second sentence must also be true and vice versa. It is analogous with: "All living beings with a heart are mammals" and "All living beings with kidneys are mammals", these sentences are either both true or both false.

Opaque contexts

Opaque contexts are special linguistic constructions in which the substitution principle, i.e. H. the commonly applicable substitutability of co-extensional expressions salva veritate is suspended. With "Peter believes that Sir Edmund Hillary climbed Mount Everest" and "Peter believes that Sir Edmund Hillary climbed the highest mountain on earth" it can actually be that the first sentence is true and the second false; for example when Peter doesn't know that Mount Everest is the highest mountain on earth. It is very similar with the sentences: "Peter believes that all animals with a heart are mammals." And "Peter believes that all animals with kidneys are mammals."

If a sentence is part of a more complex sentence in which an attitude or intention of a person is reported on that sentence, then, just like indirect speech, it is an opaque context. These cases of opaque contexts are indicated by expressions such as “believes that”, “is pleased that”, “reports that” etc. In special cases such as “mistakenly assumes that” there is a superposition of opaque and non-opaque context, since with a sentence of the form “It is wrong that p ” the truth value already depends on that of the statement p , for a sentence of the form “Assumes that p ” however does not.

Another class of expressions that open up opaque contexts are modal expressions such as “necessary” and “possible”. So of the two sentences “The number eight is necessarily even.” And “The number of planets in our solar system is necessarily even.” The first sentence is true because it expresses a mathematical truth (no circumstances are conceivable under which the Number eight is not even); however, the second sentence is wrong, the number of planets is even, but this is by no means necessarily the case, since it is a simple empirical fact. It would be possible to imagine without contradiction that there could be one planet more or less. This shows that even in modal contexts, the substitutability of co-extensional expressions is not generally guaranteed, that is, opaque contexts are involved.

Applications

The theory of opaque contexts is relevant to epistemic logic , that is, the sub-discipline of logic that deals with the analysis of expressions such as “believes that” or “knows that”. When formulating the laws that apply here, it must be taken into account that the conclusion of a statement such as “Peter believes that Sir Edmund Hillary climbed Mount Everest” to “Peter believes that Sir Edmund Hillary climbed the highest mountain on earth” is not generally valid is, but provided that Peter believes Mount Everest is the highest mountain on earth. The same applies to modal logic , the sub-discipline of logic that deals with the expressions “necessary” and “possible”.

The following conclusion is logically valid because the premises and the conclusion are extensional:

1. Sir Edmund Hillary climbed Mount Everest.

2. Mount Everest is the highest mountain on earth.

3. So: Sir Edmund Hillary has climbed the highest mountain on earth.

The following conclusion, however, is an example of an intensional fallacy :

1. Peter believes that Sir Edmund Hillary climbed Mount Everest.

2. Mount Everest is the highest mountain on earth.

3. Peter believes that Sir Edmund Hillary climbed the highest mountain on earth.

This is a fallacy, because if Peter knows 1. and 2. is true, he therefore does not have to know 3. yet. However, the conclusion can be corrected by using other premises about Peter's belief system.