In the context of modal logic and specifically in epistemic modal logic, two key modalities that one wants to consider are those of necessity and possibility.
The idea is to consider for any proposition $p$
a proposition labeled $\Box p$ expressing the idea that “$p$ is necessarily true”;
a proposition labeled $\lozenge p$ expressing the idea that “$p$ is possibly true”.
For every comonadic modality $\Box$, one typically demands that it preserves implication, in the sense that
This preservation of implication is called the K-axiom. In traditional non-categorical logic this is often all that is considered, because in traditional classical modal logic, the operators are considered dual, i.e. $\Box A= \neg (\lozenge (\neg A))$. In an intuitionistic modal logic, one would also demand that $\lozenge$ preserves implication in a suitable sense, such as
One may also ask for a further compatibility such as $(\lozenge p \to \Box q) \to \Box(p\to q)$, as Simpson does.
Similarly, the axiom $\Box(p \to q) \to (\Box p \to \Box q)$ implies that $\Box$ preserves conjunctions, i.e. $\Box(p\wedge q) \leftrightarrow (\Box p \wedge \Box q)$. The nullary version $\Box(\top)\leftrightarrow \top$ follows from the “necessitation rule” which says that if $p$ is provable in the empty context, then so is $\Box p$. (This is an additional rule assumed in modal logics.) From duality, $\Box A= \neg (\lozenge (\neg A))$ it follows that $\lozenge$ preserves disjunctions (both binary and nullary), while intuitionistically one may want to ask for this separately (Simpson does, but Biermann and de Paiva don’t).
A minimum further requirement on a formalization of $\Box$ and $\lozenge$ to have the interpretation of “necessity” and “possibility” is arguably that there are implications
$\Box p \rightarrow p$;
$p \rightarrow \lozenge p$,
expressing that if something is necessarily true, then that should mean that it is true in all instances, and that if something is true in one instance, then it is evidently possible for it to be true. With these additions to the above K modal logic, one speaks of T modal logic.
By similar plausibility arguments one often demands that
$\Box p \to \Box \Box p$
$\lozenge \lozenge p \to \lozenge p$
which may be read as expressing that iterating the previous reasoning does not yield any new insight. This additional enhancement to T modal logic yields S4 modal logic.
In terms of categorical logic interpreting propositional logic into a Heyting algebra, the S4 axioms just say (Bierman & de Paiva 92) that
$\Box$ is a (necessarily idempotent) comonad which is monoidal, hence product-preserving; and
$\lozenge$ is a (necessarily idempotent) monad which is a “$\Box$-strong functor”, and (perhaps) preserves coproducts.
As usual, we can also replace the Heyting algebra by a cartesian closed category, thereby obtaining a “proof-relevant” system; in this case monoidality of $\Box$ doesn’t imply directly that it preserves products, although one might reasonably ask that it does. See monad in computer science.
The above reasoning makes plausible that any operators expressing “necessity” and “possibility” should at least satisfy these (co)monad axioms. However, not every (co)monad is sensibly interpreted this way.
For example, there is
the (idempotent) comonad $\emptyset$ which sends every proposition to false (the non-being-modality);
the (idempotent) monad $\ast$ which sends every proposition to true (the being-modality).
These $\emptyset \dashv \ast$ satisfy all of the above axioms (as well as more axioms that are being considered, such as those called S5 modal logic) but they are not a very good interpretation of the informal concepts of “possibility” and “necessity”. Under this interpretation nothing is necessarily true, and everything is possibly true. While this is not nonsensical, it is not very interesting and doesn’t correspond well to our intuitive meanings of “necessary” and “possible”.
This issue becomes more pronounced as one generalizes from the small realm of propositional logic to include both or either of:
types more general than propositions (in modal type theory);
first-order logic with quantifiers, hence hyperdoctrines/dependent types.
There are many modal operators in such contexts which are all modeled by (idempotent) (co)monads but which do not have the interpretation of expressing the modes of “necessity” or “possibility”. See at modal operator for some examples.
Therefore it makes sense to ask which additional axioms on a modal operator make it an accurate formalization of the informal concepts of necessity and possibility. The answer to this may depend on context and intention (after all one is trying to find a precise formulation of an a priori informal idea).
One common philosophical interpretation of “necessarily” and “possibly” is in terms of a collection of “possible worlds” that are similar to the “real world”, but not the same. Under this interpretation, a proposition is necessarily true if it is true in all possible worlds, and possibly true if it is true in some possible world.
Now the idea of a proposition being true “necessarily in all possible cases” or “possibly at least in one case” is formally very well established in predicate logic: it is just the interpretation of the universal quantifier “for all” $\forall$ and of the existential quantifier “there exists” $\exists$. In categorical logic, these quantifiers (see there for details) are part of an adjoint triple whose middle piece is context extension, and as such they naturally induce a comonad and a monad. Thus, if we interpret “necessarily” and “possibly” in terms of possible worlds, we can model them by this base change adjoint triple.
In more detail, let $W$ be the context type of variables/terms on which the propositions depend, i.e. the collection “of all possible worlds”. Any specific choice of $W$ may be taken as specifying what is to be understood as a “possible world”.
Writing $\mathbf{H}_{\ast}$ for the category of all context-free types under consideration and writing $\mathbf{H}_{/W}$ for the category of types in context “$W$”, then in categorical logic (for instance $\mathbf{H}_{/(-)}$ might be a hyperdoctrine over a category of contexts containing objects $W$ and $\ast$) the quantifiers $\forall_{x\colon X}$ and $\exists_{x\colon X}$ participate in a base change adjoint triple
In a context of pure logic this would be called
existential quantifier$\dashv$ context extension $\dashv$ universal quantifier
whereas in a context of dependent type theory this would be called
dependent sum$\dashv$ context extension $\dashv$ dependent product.
In either case, under the suitable version of propositions as types (and using bracket types etc. if desired), the operations $\forall$ and $\exists$ have the usual interpretation of “for all” and “there exists”.
Note, however, that these operations change the context from $W$ to $\ast$. In other words, if a proposition $P$ depends on $W$, so that it may be true in some worlds and false in others, then $\exists_W P$ and $\forall_W P$ no longer depend on $W$. The idea of a necessity and a possibility modality is to send a proposition in some context to a proposition in the same context, so that we can for instance say that $\Box P \to P$ and so on. Thus we need to make $\exists_W P$ and $\forall_W P$ into propositions that again depend on $W$ — even if they now depend trivially on $W$, being extended back from the absolute context $\ast$ to $W$.
This is just what is accomplished by passing from the above adjoint triple to the induced adjoint pair on $\mathbf{H}_{/W}$ by forming composites with the context extension operation $W^\ast$
With this, if $p\in \mathbf{H}_{/W}$ is a proposition about terms $w$ of $W$ (a $W$-dependent type) then
$\underset{W}{\lozenge}(p)$ is true/inhabited precisely if $\underset{w \colon W}{\exists} p(w)$ is true/inhabited, hence (that is the standard interpretation of the quantifier) if it is possible for $p(w)$ to be true for some $w$;
$\underset{W}{\Box}(p)$ is true/inhabited precisely if $\underset{w \colon W}{\forall} p(w)$ is true/inhabited, hence (that is again the standard interpretation of the quantifier) if $p(w)$ necessarily holds for all $w$.
Thus, this gives one syntactic formalization of the informal meaning of “necessity” and “possibility”. The natural semantics for these base change operations is a generalization of the simple traditional possible worlds semantics of propositional necessity and possibility modalities. (There are, however, more complicated possible worlds semantics.)
Moreover, with this formalization, the modal operator $\underset{W}{\lozenge}$ is left adjoint to $\underset{W}{\Box}$ and hence both form an adjoint modality. As discussed there, this is a formalization of opposite concepts, which reflects well the opposition of necessity and possibility in their informal meaning.
Observe how the corresponding unit and counit maps properly reflect the intended logic of necessity, actuality and possibility:
Some technical remarks:
In general, $\underset{W}{\lozenge}$ and $\underset{W}{\Box}$ as defined above are, while being a monad and comonad, respectively, not an idempotent monad and idempotent comonad if generalized from first-order hyperdoctrines to more general dependent type theories. But this just reflects the usual issues with propositions as types, see there for more discussion.
While base change-adjunctions are essentially unique and not free to choose, there is a genuine choice in the above given by the choice of context $W$. This is reflected in the subscripts of $\underset{W}{\lozenge}$ and $\underset{W}{\Box}$ above. It is the choice of this $W$ that gives different kinds of possibility and necessity. More generally there is in fact not just a choice of a context, but of a morphism of contexts, reflecting what is often called “accessibility of possible worlds”. This we come to below.
There is also an adjoint pair on the other side, $\mathbf{H}_{/\ast}$, of the base change maps, in which the left adjoint is given by context extension back to $\mathbf{H}_{/W}$ followed by $\exists_W$, and dually the right adjoint is given by $W^\ast$ followed by $\forall_W$. The former is the coreader comonad, whereas the latter is the reader monad (aka function monad):
If we think of the types $P_\bullet \,\in\, Type_B$ in the given context – now called “$B$” – as actual types (2) then the types down in the base context $P \,\in\, Type$ should be thought of as potential types that fail to be actual in that they are missing information about their $B$-dependency (see also at nondeterministic computation, here):
The definiteness-counit $\star P \rightarrow P$ witnesses their inhabitation as following from that of a definite type after forgetting which definite world $b \colon B$ was used to inhabit it.
More in detail, since $(p_B)^\ast$ is a conservative functor, the monadicity theorem (see this example) says
that the types $P \,\in\, Type$ are equivalently
those actual types $P_\bullet$ which carry a possibility-monad action $\lozenge_B P_\bullet \longrightarrow P_\bullet$
those actual types $P_\bullet$ which carry a necessity-co-monad co-action $P_\bullet \longrightarrow \Box_B P_\bullet$
In words this says that for such potential propositions the standard implications
may be reversed. And indeed, if a kind of proposition holds as soon as it possibly holds, then it makes sense to say that it holds potentially.
With this axiomatization via base change, it is immediate to consider the relative case where instead of base change to a unit type $W \to \ast$ one considers base change
along any morphism
and set
If here $\omega$ is an effective epimorphism (a 1-epimorphism) then it exibits an equivalence relation on $W$, where $w_1\sim w_2$ is given by $\omega(w_1) = \omega(w_2)$. In traditional possible worlds semantics such equivalence relation is called an “accessibility relation between possible worlds”. Now
$\underset{\omega}{\lozenge} p$ is true/inhabited at $w\in W$ iff it is true/inhabited at at least one $\tilde w$ in the same equivalence class of $w$;
$\underset{\omega}{\Box} p$ is true/inhabited at $w\in W$ iff it is true/inhabited at all $\tilde w$ in the same equivalence classes of $w$.
Historically, one informal example whose formalization in modal logic has been controversially discussed (see for instance IEP “Rudolf Carnap: Modal Logic”) is the pair of informal assertions
“9 is necessarily greater than 7.”
“The number of planets in the solar system is 9.”
(Remark. It maybe adds to the joy of modal logic to notice that the second sentence, which was regarded as true in our world at the time the above example was brought up, actually no longer is. Of course this only highlights that indeed this statement is not to be expected to be “necessarily true” in any sense.)
The issue with this example is that if one does not fix a decent formalization of these statements, then naively they seem to imply as correct the statement “The number of planets in the solar system is necessarily greater than 7.”, which however sounds like it ought to be wrong.
We now formalize and then analyze this example with the above prescription.
So let $W$ be any type, to be thought of as the type of possible worlds. Write
for the $W$-dependent type that is constant on the ordinary type of natural numbers, i.e.$\mathbb{N}(w) = \mathbb{N}$ for all $w\colon W$.
The terms of $\mathbb{N}$, i.e. the natural numbers, canonically extend to $W$-dependent terms of this dependent type
namely to the constant terms, which take the same value (here: 9) for all $w \colon W$.
In contrast to this, assume now that $W$ is such there is at least one non-constant function $W \to \mathbb{N}$. In fact, in the spirit of the informal problem at hand, we require a surjective function. This gives a non-constant term of the constantly $W$-dependent type of natural numbers, which we may just as well call
That is the formalization of the above example we consider, and it should be evident enough. Now we may step back and see what the above formalization produce from this.
So consider the $W$-dependent identity type
By the assumption that $NumberOfPlanetsInTheSolarSystem(w)$ is not constant in $W$ it follows that the dependent product over $W$ of that dependent identity type is the empty type, and so the same is true for $\Box_W$ applied to it:
However, if we assume that there is one $w$ for which indeed $NumberOfPlanetsInTheSolarSystem(w)$ takes the value 9, then the dependent sum over the dependent identity type contains that coincidence as a term, and so $\lozenge_W$ of our dependent identity type is inhabited. Hence
In English words, these formal consequences are to be pronounced as:
“It is false that it is necessary that the number of planets in the solar system is 9.”
“It is true that it is possible that the number of planets in the solar system is 9.”
Which is just as it should be.
Similarly, the $W$-dependent type
is only going to be inhabited if indeed the value $NumberOfPlanetsInTheSolarSystem(w)$ is greater than 7 for all $w\colon W$. With our formalization assumption above this is not the case, and so one finds that
hence
under construction
We discuss^{1} the modal logic induced from base change of dependent types, as above, now applied to dependent linear types over finite sets (family of finite types) — to find a natural modal quantum logic (and in fact a natural quantum programming language for quantum circuits, see at quantum circuits via dependent linear types for more on this).
Given a base type (which we take to be finite)
write $LinType_B$ for the $B$-dependent linear types
The intended categorical semantics is obtained by interpreting $\mathscr{H}_\bullet$ as a $B$-indexed set of complex vector spaces (or rather: Hermitian inner product spaces) which the reader may want to think of as being finite dimensional (hence: as Hilbert spaces if positive definite), though this finite-dimensionality condition of $\mathscr{H}_b$ is not used in the following (as opposed to the finiteness condition on $B$, which is used).
We assume that our dependent linear type theory realizes base change of linear types along maps of non-linear base types in the form of Wirthmüller-type yoga of six operations:
Moreover, we assume that along maps with finite-fibers, such as the terminal map $B \to \ast$, the left base change agrees with the right base change (“ambidexterity”, so that both give the dependent biproduct $\otimes_{b : B}$ over $B$):
Curiously, in terms of modal logic, this is the Gell-Mann principle of quantum physics, in that it means:
Both these assumptions are verified in the linear homotopy type theory of Riley (2022), §2.4, which should have categorical semantics in (a type-theoretic model category presenting) the $(\infty,1)$-topos of parameterized $H\mathbb{C}$-module spectra, among which complex vector spaces $\mathscr{H}$ embed as their Eilenberg-MacLane module spectra $H\mathscr{H}$.
Recalling from above the quadruple of (co-)monadic modalities associated with any base change one finds that its quantum analog for dependent linear types (along finite fibers (4))
where the interpretation of the modal (co-)units is as follows:
In the context of linear types not just $(p_B)^\ast$ is monadic, but also $(p_B)_!$/$(p_B)_\ast$ (for details see the quantum reader monad), so that the factorization above enhances to the following situation:
abbreviate $\mathscr{H} \,\coloneqq\, p_! \mathscr{H}_\bullet \,\simeq\, p_\ast \mathscr{H}_\bullet$
notationally suppress any outer application of $(p_B)^\ast \colon LinType \to LinType_B$;
Thereby, the definiteness/randomness modalities in relation to necessity and possibility of classical modal logic (above) may be expressed in quantum modal logic by:
Considering the basic dependent linear type
(which is the tensor unit in $LinType_B$) we may understand
as the Hilbert space which is spanned by a measurement basis $\vert b \rangle$.
From all this we find, first of all, that
the necessity-counit exhibits the collapse of quantum states under a quantum measurement in the $B$-basis
the possibility-unit exhibits the condition quantum state preparation in this basis:
The collapse-principle of quantum measurement accurately expressed by the above linear necessity$\to$actuality$\to$possibility map was maybe first highlighted and made explicit in von Neumann (1932), §III.3, at least for the special case that (in our notation here) $\mathscr{H}_\bullet \,=\, (p_B)^\ast \mathbb{C}$ (cf. Lüders (1951)).
We see that the base type $B$ (4) of quantum measurement outcomes plays the role of both
(…)
In
is discussion of modality as one of the categories of pure thought, with sub-categories necessity, possibility and actuality (Wirklichkeit).
In
this is picked up and claimed to be refined to a dialectic system of unities of opposites.
Saul Kripke, Naming and Necessity (1980)
Stanford Encyclopedia of Philosophy, Modal Logic
Jake Chandler, Modality: Necessity and Possibility, lecture notes (pdf)
Gavin Bierman and Valeria de Paiva, On an Intuitionistic Modal Logic, Studia Logica (65):383-416, 2000 (preprint from 1992) (web)
Alex K. Simpson, The Proof Theory and Semantics of Intuitionistic Modal Logic, Ph.D. Thesis, University of Edinburgh, 1994, web.
Internet Encyclopedia of Philosophy, Rudolf Carnap: Modal Logic
Following discussion at Topological Quantum Programming in Linear Homotopy Type Theory ↩
Last revised on December 29, 2022 at 11:43:00. See the history of this page for a list of all contributions to it.