3.3 Mere propositions
We have seen that the propositions-as-types logic has both good and bad properties. Both have a common cause: when types are viewed as propositions, they can contain more information than mere truth or falsity, and all “logical” constructions on them must respect this additional information. This suggests that we could obtain a more conventional logic by restricting attention to types that do not contain any more information than a truth value, and only regarding these as logical propositions.
Such a type will be “true” if it is inhabited, and “false” if its inhabitation yields a contradiction (i.e. if is inhabited). What we want to avoid, in order to obtain a more traditional sort of logic, is treating as logical propositions those types for which giving an element of them gives more information than simply knowing that the type is inhabited. For instance, if we are given an element of , then we receive more information than the mere fact that contains some element. Indeed, we receive exactly one bit more information: we know which element of we were given. By contrast, if we are given an element of , then we receive no more information than the mere fact that contains an element, since any two elements of are equal to each other. This suggests the following definition.
A type is a mere proposition if for all we have .
Note that since we are still doing mathematics in type theory, this is a definition in type theory, which means it is a type — or, rather, a type family. Specifically, for any , the type is defined to be
Thus, to assert that “ is a mere proposition” means to exhibit an inhabitant of , which is a dependent function connecting any two elements of by a path. The continuity/naturality of this function implies that not only are any two elements of equal, but contains no higher homotopy either.
If is a mere proposition and , then .
Define by , and by . The claim follows from the next lemma, and the observation that is a mere proposition by Theorem 2.8.1 (http://planetmath.org/28theunittype#S8.Thmthm1). ∎
If and are mere propositions such that and , then .
Suppose given and . Then for any , we have since is a mere proposition. Similarly, for any we have since is a mere proposition; thus and are quasi-inverses. ∎
In homotopy theory, a space that is homotopy equivalent to is said to be contractible. Thus, any mere proposition which is inhabited is contractible (see also §3.11 (http://planetmath.org/311contractibility)). On the other hand, the uninhabited type is also (vacuously) a mere proposition. In classical mathematics, at least, these are the only two possibilities.
Mere propositions are also called subterminal objects (if thinking categorically), subsingletons (if thinking set-theoretically), or h-propositions. The discussion in §3.1 (http://planetmath.org/31setsandntypes) suggests we should also call them -types; we will return to this in http://planetmath.org/node/87580Chapter 7. The adjective “mere” emphasizes that although any type may be regarded as a proposition (which we prove by giving an inhabitant of it), a type that is a mere proposition cannot usefully be regarded as any more than a proposition: there is no additional information contained in a witness of its truth.
Note that a type is a set if and only if for all , the identity type is a mere proposition. On the other hand, by copying and simplifying the proof of Lemma 3.1.8 (http://planetmath.org/31setsandntypes#Thmprelem1), we have:
Every mere proposition is a set.
Suppose ; thus for all we have . Fix and define . Then for any and we have . Hence by Lemma 1 (http://planetmath.org/211identitytype#Thmlem1), we have , which is to say that . Thus, for any , we have . ∎
In particular, this implies:
For any type , the types and are mere propositions.
Suppose . By function extensionality, to show it suffices to show for any . But and are both paths in , and hence are equal because, by either or , we have that is a mere proposition, and hence by Lemma 3.3.4 (http://planetmath.org/33merepropositions#Thmprelem3) is a set. Similarly, suppose , which is to say that for all and , we have and . But by then since is a set (by either or ), and hence a 1-type, it follows that ; hence by function extensionality. ∎
We have seen one other example so far: condition (3.) (http://planetmath.org/24homotopiesandequivalences#I1.i3) in §2.4 (http://planetmath.org/24homotopiesandequivalences) asserts that for any function , the type should be a mere proposition.
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