1 The interpretation of conditional imperatives

There are a number of different ways in which, at the purely informal level, a conditional imperative might be understood. Some of them make a difference to the formal semantics and to the logic, while others merely make a difference to how the formal semantics should itself be construed. It will be helpful to make clear where I stand on these various interpretations, and, although my focus is here on the case of imperatives, it should be noted that the same distinctions (and some others) will also apply in the deontic case.

2 Syntax

We describe the language for the logic of imperatives that we shall be adopting. The primitive vocabulary consists of the following symbols:

• indicative atoms ${\mathrm{s}}_1$ , ${\mathrm{s}}_2,\dots$

• imperative atoms ${\mathrm{p}}_1$ , ${\mathrm{p}}_2,\dots$

• the verum constants $\top$ and $\top !,$

• the connectives $\neg, \vee, \wedge$ , and $\to,$

• the parentheses (and).

The indicative atoms ${\mathrm{s}}_1,{\mathrm{s}}_2,\dots$ are meant to signify situations, i.e., to have situations as their truth- and falsity-makers, while the imperative atoms ${\mathrm{p}}_1$ , ${\mathrm{p}}_2$ , … are meant to signify actions or plans, i.e., to have actions or plans as their compliance- and contravention-makers.

The language contains two verum constants: $\top$ for the indicative made true by the null situation and $\top !$ for the imperative with which the null action is in compliance. We use $\perp$ (the falsum constant) as an abbreviation for $\neg \top$ and $\perp !$ as an abbreviation for $\neg \top !$ .

Indicative and imperative formulas are defined by means of the following inductive clauses:

• IN1. $\top$ and any indicative atom is an indicative formula.

• IN2. If $\mathrm{S}$ is an indicative formula, then so is $\neg \mathrm{S}$ .

• IN3. If $\mathrm{S}$ and $\mathrm{T}$ are indicative formulas, then so are $(\mathrm{S}\vee \mathrm{T})$ and $(\mathrm{S}\wedge \mathrm{T})$ .

• IM1. $\top !$ and any imperative atom is an imperative formula.

• IM2. If $\mathrm{P}$ is an imperative formula, then so is $\neg \mathrm{P}$ .

• IM3. If $\mathrm{P}$ and $\mathrm{Q}$ are imperative formulas, then so are $(\mathrm{P}\vee \mathrm{Q})$ and $(\mathrm{P}\wedge \mathrm{Q})$ .

• IM4. If $\mathrm{S}$ is an indicative formula and $\mathrm{P}$ an imperative formula, then $(\mathrm{S}\to \mathrm{P})$ is an imperative formula.

A formula is any indicative or imperative formula. We use $\mathrm{S},\mathrm{T},$ etc. for indicative formulas and $\mathrm{P},\mathrm{Q},$ etc. for imperative formulas, and we allow ourselves freely to put ‘!’ at the end of an imperative formula as a way of making clear that it is an imperative rather than an indicative formula.

Note that clause IM4 is the only source of ‘hybrids’, in which indicative and imperative formulas come together. It permits an indicative to appear to the left of a conditional, though not to the right, and it permits an imperative to appear to the right of a conditional, though not to the left.

We shall have occasion in part IV to distinguish an infinite subset ${\mathrm{a}}_1!$ , ${\mathrm{a}}_2!$ , … of the imperative atoms and a corresponding infinite subset of indicative atoms ${\mathrm{a}}_1$ , ${\mathrm{a}}_2$ , …. We call them A-atoms and suppose that ${\mathrm{a}}_1!$ , ${\mathrm{a}}_2!$ , … form a co-infinite subset of all the imperative atoms and that ${\mathrm{a}}_1$ , ${\mathrm{a}}_2$ , … form a co-infinite subset of all the indicative atoms. These co-infinite subsets behave in exactly the same way, syntactically speaking, as the regular imperative and indicative atoms, but the intention is that the imperative A-atoms should designate actions (rather than plans) and that the indicative A-atoms should designate corresponding states (to the effect that the given action is performed). It is through the coordination of imperative and indicative atoms that we are able to treat an action as itself part of a situation that may serve as the condition for other actions to be performed.

3 Background

I give a brief overview of standard truthmaker semantics for a propositional language (the reader might like to consult [Reference Fine, Hale, Wright and Miller7] for a general introduction to the topic).

A partial order (p.o.) on a set $S$ is a reflexive, transitive, and anti-symmetric relation on $S$ . Given the p.o. $\sqsubseteq$ on $S$ , we say:

• $s$ is null if $s$ $\sqsubseteq$ $s^{\prime }$ for all $s^{\prime }$ $\in$ $S$ and otherwise is non-null;

• $s$ is an upper bound of $T$ if $t$ $\sqsubseteq$ $s$ for each $t$ $\in$ $T$ ;

• $s$ is a lower bound of $T$ if $s$ $\sqsubseteq t$ for each $t$ $\in$ $T$ ;

• $s$ is a least upper bound (lub) of $T$ if $s$ is an upper bound of $T$ and $s\sqsubseteq s^{\prime }$ for any upper bound $s^{\prime }$ of $T$ ;

• $s$ is a greatest lower bound (glb) of $T$ if s is a lower bound of $T$ and $s\sqsubseteq s^{\prime }$ for any lower bound $s^{\prime }$ of $T$ ;

• $s\sqsubset t$ ( $s$ is a proper part of $t$ ) if $s\sqsubseteq t$ but not $t\sqsubseteq s$ ;

• $s$ overlaps $t$ if for some non-null $u$ , $u\sqsubseteq s$ and $u\sqsubseteq t$ ;

• $s$ is disjoint from $t$ if $s$ does not overlap $t$ .

The least upper bound of $T\subseteq S$ if it exists is unique. We denote it by $\bigsqcup T$ and call it the fusion of $T$ (or of the members of $T$ ). When $T=\{{t}_1,{t}_2,\dots \}$ , we shall sometimes write $\bigsqcup T$ more perspicuously as ${t}_1\sqcup {t}_2\sqcup \cdots$ . Likewise, for the greatest lower bound $\sqcap T$ .

A state space $\boldsymbol{S}$ is a pair $(S,\sqsubseteq )$ , where $S$ (states) is a non-empty set and $\sqsubseteq$ (part) is a partial order on $S$ which is complete, i.e., any subset $T$ of $S$ has a least upper bound $\bigsqcup T$ . We use $\square$ (the null state) for the least state $\bigsqcup \varnothing$ of $\boldsymbol{S}$ and $\blacksquare$ (the full state) for the greatest state $\bigsqcup S$ of $\boldsymbol{S}$ . In a state space, any subset $T$ of $S$ will also have a greatest lower bound $\sqcap T$ . For now we are thinking of states in a generic way, which will be compatible with thinking of them as either situations or actions or plans.

A state space $\boldsymbol{S}^{\prime}=(S^{\prime },\sqsubseteq^{\prime})$ is said to be a subspace of the state space $\boldsymbol{S}=(S,\sqsubseteq )$ if $S^{\prime }$ is a subset of $S$ and $\sqsubseteq^{\prime }$ is the restriction $\sqsubseteq \upharpoonleft S^{\prime }$ of $\sqsubseteq$ to $S^{\prime }$ . If $\boldsymbol{S}=(S,\sqsubseteq )$ and $\boldsymbol{T}=(T,\sqsubseteq^{\prime})$ are two state-spaces, then their product $\boldsymbol{S}\times \boldsymbol{T}$ is the structure $(S\times T,{\sqsubseteq}_{\times})$ , where

$$\begin{align*}(s,s^{\prime})\;{\sqsubseteq}_{\times}\;(t,t^{\prime})\ \mathrm{iff}\ s\sqsubseteq t\ \mathrm{and}\ s^{\prime}\sqsubseteq t^{\prime } .\end{align*}$$

It is readily verified that $\boldsymbol{S}\times \boldsymbol{T}$ is also a state space.

Let us now suppose that the formulas of our language are constructed from certain atoms ${\mathrm{x}}_1$ , ${\mathrm{x}}_2$ , … and the verum and falsum constants $\top$ and $\perp$ by means of the usual connectives, $\vee, \wedge,$ and $\neg$ (for now we do not allow the conditional). Proceeding in this way allows us to remain neutral as to the exact identity of the atoms. However, we shall, in this and the next section, use $\mathrm{X},\mathrm{Y},\mathrm{Z}$ … for formulas of the present neutral language rather than for the imperative formulas of the previous papers.

A state model $\boldsymbol{M}$ over the state space $\boldsymbol{S}=(S,\sqsubseteq )$ is a triple $(S,\sqsubseteq, | \cdot |)$ , where $| \cdot |$ (the valuation) is a function taking each atom $\mathrm{x}$ into a pair of non-empty subsets ${|\mathrm{x}|}^{+}$ and ${|\mathrm{x}|}^{-}$ of $S$ (the respective truth- and falsity-makers of $\mathrm{x}$ ). Such a valuation is said to be a bilateral valuation over $S$ on the given atoms. It is to be contrasted with a unilateral valuation which simply assigns a subset of states to each atom. We also allow a model, both here and elsewhere, to be defined over a subset of the relevant atoms.

The atom $\mathrm{x}$ is said to be definite in a model if ${|\mathrm{x}|}^{+}$ is singleton, to be bi-definite if both ${|\mathrm{x}|}^{+}$ and ${|\mathrm{x}|}^{-}$ are singleton, and to be non-vacuous if neither ${|\mathrm{x}|}^{+}$ nor ${|\mathrm{x}|}^{-}$ is empty. The model $\boldsymbol{M}$ itself is said to be definite (bi-definite, non-vacuous) if each atom is definite (bi-definite, non-vacuous) in the model.

Given any model $\boldsymbol{M}=(S,\sqsubseteq, | \cdot |)$ , there is a standard way of extending the valuation $| \cdot |$ to all other formulas. For subsets $T$ and $U$ of $S$ , let $T\sqcup U=\{t\sqcup u:t\in T\; and\;u\in U\}$ . We then define $| \cdot |$ on all formulas by means of the following clauses:

• ${|\top |}^{+}=\{\square \}$ .

• ${|\top |}^{-}\kern0.48em =\varnothing$ .

• ${|\neg \mathrm{X}|}^{+}={|\mathrm{X}|}^{-}.$

• ${|\neg \mathrm{X}|}^{-}={|\mathrm{X}|}^{+}$ .

• ${|\mathrm{X}\wedge \mathrm{Y}|}^{+}={|\mathrm{X}|}^{+}\sqcup {|\mathrm{Y}|}^{+}.$

• ${|\mathrm{X}\wedge \mathrm{Y}|}^{-}={|\mathrm{X}|}^{-}\cup {|\mathrm{Y}|}^{-}.$

• ${|\mathrm{X}\vee \mathrm{Y}|}^{+}={|\mathrm{X}|}^{+}\cup {|\mathrm{Y}|}^{+}$ .

• ${|\mathrm{X}\vee \mathrm{Y}|}^{-}={|\mathrm{X}|}^{-}\sqcup {|\mathrm{Y}|}^{-}$ .

The negative clause for the verum constant is somewhat different from that in [Reference Fine9]. We there took ${|\top |}^{-}$ to be $|\blacksquare|$ rather than $\varnothing$ . Each clause has its advantages and disadvantages, but, in the case of conditionals, it will be convenient to adopt the present clause. Note that we have required ${|\mathrm{x}|}^{+}$ and ${|\mathrm{x}|}^{-}$ , for any atom $\mathrm{x}$ , to be non-empty, whereas ${|\top |}^{-}$ is empty. This means that we cannot think of an atom as standing in for $\top$ .

The above clauses can be stated in more orthodox fashion by specifying when a state $s$ is a truth-maker ( $s\ ||\!\text{-}\ \mathrm{X}$ ) or a falsity-maker $s$ $\text{-}\!||\ \mathrm{X}$ for $\mathrm{X}$ . Thus the clauses for $\mathrm{X}\wedge \mathrm{Y}$ and $\neg \mathrm{X}$ now take the following form:

• $s\ ||\!\text{-}\ \mathrm{X}\wedge \mathrm{Y}$ iff for some $t$ and $u$ , $t\ ||\!\text{-}\ \mathrm{X}$ , $u\ ||\!\text{-}\ \mathrm{Y}$ and $s=t\sqcup u$ .

• $s\ \text{-}\!||\ \mathrm{X}\wedge \mathrm{Y}$ iff $s\ \text{-}\!||\ \mathrm{X}$ or $s\ \text{-}\!||\ \mathrm{Y}$ .

• $s\ ||\!\text{-}\ \neg \mathrm{X}$ iff $s\ \text{-}\!||\ \mathrm{X}$ .

• $s\ \text{-}\!||\ \neg \mathrm{X}$ iff $s\ ||\!\text{-}\ \mathrm{X}$ .

The two kinds of clauses will then be in conformity with one another, in that $| \mathrm{X}|^+=\{s{:}\ s\ ||\!\text{-}\ \mathrm{X}\}$ and $| \mathrm{X}|^- =\{s{:}\ s\ \text{-}\!||\ \mathrm{X}\}$ .

We may define various logical notions. Let $\Delta$ be a well-ordered sequence ${\mathrm{X}}_1,{\mathrm{X}}_2,\dots$ of formulas. Given a model $\boldsymbol{M}$ , we say that:

• $\Delta$ exactly entails the formula $\mathrm{Y}$ in $\boldsymbol{M}$ if ${s}_1\sqcup {s}_2\sqcup \dots \ ||\!\text{-}\ \mathrm{Y}$ in $\boldsymbol{M}$ whenever ${s}_1\ ||\!\text{-}\ {\mathrm{X}}_1,{s}_2\ ||\!\text{-}\ {\mathrm{X}}_2,\dots$ in $\boldsymbol{M}$ , and that:

• $\Delta$ contains $\mathrm{Y}$ in $\boldsymbol{M}$ —symbolically, $\Delta\;{>}_{\boldsymbol{M}}\mathrm{Y}$ —if (i) whenever ${s}_1\ ||\!\text{-}\ {\mathrm{X}}_1,{s}_2\ ||\!\text{-}\ {\mathrm{X}}_2,\dots$ in $\boldsymbol{M}$ then for some $t\sqsubseteq {s}_1\sqcup {s}_2\sqcup \dots, t\ ||\!\text{-}\ \mathrm{Y}$ in $\boldsymbol{M}$ , and (ii) whenever $t\ ||\!\text{-}\ \mathrm{Y}$ in $\boldsymbol{M}$ then for some ${s}_1,{s}_2,\dots$ , ${s}_1\ ||\!\text{-}\ {\mathrm{X}}_1,{s}_2\ ||\!\text{-}\ {\mathrm{X}}_2,\dots$ in $\boldsymbol{M}$ and $t\sqsubseteq {s}_1\sqcup {s}_2\sqcup \dots$ .

Given formulas $\mathrm{X}$ and $\mathrm{Y}$ , we also say that:

• $\mathrm{X}$ and $\mathrm{Y}$ are (positive) (exact) equivalents in $\boldsymbol{M}$ —in symbols, $\mathrm{X}\;{\approx}_{\boldsymbol{M}}\mathrm{Y}$ or $\mathrm{X}\;{\approx}^{+}_{\mathrm{M}}\mathrm{Y}$ —if, for any state s, s $\Vdash$ X in $\boldsymbol{M}$ iff s ╟ Y in $\boldsymbol{M}$ ;

• $\mathrm{X}$ and $\mathrm{Y}$ are negative (exact) equivalents in $\boldsymbol{M}$ —in symbols, $\mathrm{X}\approx ^{-}_{\mathrm{M}}\;\mathrm{Y}$ —if, for any state $s$ , $s\ \text{-}\!||\ \mathrm{X}$ in $\boldsymbol{M}$ iff $s\ \text{-}\!||\ \mathrm{Y}$ in $\boldsymbol{M}$ ; and

• $\mathrm{X}$ and $\mathrm{Y}$ are full (exact) equivalents in $\boldsymbol{M}$ —in symbols, $\mathrm{X}\approx ^{\pm}_{\mathrm{M}}\;\mathrm{Y}$ —if they are both positive and negative exact equivalents.

We can also define non-relative analogues of these notions in the obvious way. For example, $\mathrm{X}$ and $\mathrm{Y}$ will be (positive) exact equivalents— $\mathrm{X}\approx \mathrm{Y}$ —if $\mathrm{X}\approx ^{\pm}_{\mathrm{M}}\kern0.24em \mathrm{Y}$ for any model $\boldsymbol{M}$ . We may also sometimes restrict these non-relative analogues to a sub-class of models. (Note that this notation slightly departs from the notation of Fine [Reference Fine8], where $\mathrm{X}\approx \mathrm{Y}$ is used as an abbreviation in the object language.)

Although we shall not give a precise statement or proof, we should mention that full equivalents may be substituted for one another in any formula, preserving full equivalence, while positive equivalents may be substituted within positive contexts (in the usual sense of the term), thereby preserving positive equivalence.

4 Multi-set semantics

For reasons that will later become clear, we shall find it helpful to formulate a slightly more fine-grained form of the semantics. Suppose $s$ is the sole truth-maker for the atom $\mathrm{x}$ . Then we will want to say not merely that $s$ is a truth-maker for $\mathrm{x}\vee \mathrm{x}$ , but that it is a truth-maker twice, once via the left disjunct and once via the right disjunct. To this end, we should take the content of a formula to be a multi-set $[s,s]$ in which the truth-maker $s$ occurs twice.

I shall forgo a precise treatment of multi-sets, but the reader may find it helpful to think of them in graphical terms. [1, 2, 2, 3, 3, 3], for example, is the multi-set in which $1$ occurs once, $2$ twice, and $3$ thrice, $[1,2,3]$ is the multi-set in which $1,2$ , and $3$ occur only once, while $[]$ is the empty multi-set. Given two multi-sets $X=[{x}_1,{x}_2,\dots ]$ and $Y=[{y}_1,{y}_2,\dots ]$ , their union is the multi-set $[{x}_1,{x}_2,\dots, {y}_1,{y}_2,\dots ]$ and their fusion is the multi-set $[{x}_1\sqcup {y}_1,{x}_1\sqcup {y}_2,\dots, {x}_2\sqcup {y}_1,{x}_2\sqcup {y}_2,\dots, \dots ]$ . So, for example, is $[0,1,1,2,2,2,2,3,3,3]$ .

We shall need to make use of multi-maps. Suppose $X=[{x}_1,{x}_2,\dots ]$ is a multi-set and $Y=\{{y}_1,{y}_2,\dots \}$ is a set. Then a multi-map $f$ from $X$ into $Y$ is a multi-set of ordered pairs $[{<}{x}_1,{y}_1{>},{<}{x}_2,{y}_2{>},\dots ]$ . Thus identical members of $X$ may be associated with distinct members of $Y$ .

A multi-set model $\boldsymbol{M}=(S,\sqsubseteq, | \cdot |)$ over a state space $\boldsymbol{S}=(S,\sqsubseteq )$ is the same as before, except now the valuation $| \cdot |$ takes each atom $\mathrm{x}$ into a pair of non-empty multi-sets of states (we might call this a multi-set valuation). The valuation $| \cdot |$ can then be extended to all formulas by means of the following clauses:

• ${|\top |}^{+}=[\square ]$ .

• ${|\top |}^{-}\kern0.6em =\varnothing (=[\;])$ .

• ${|\neg \mathrm{X}|}^{+}={|\mathrm{X}|}^{-}$ .

• ${|\neg \mathrm{X}|}^{-}={|\mathrm{X}|}^{+}$ .

• .

• .

• |.

• .

Thus, the clauses are essentially the same as before, but the set-theoretic operations have been converted into corresponding operations on multi-sets. We may write ${|\mathrm{X}|}_M^{\pi }$ , for $\pi =+$ or $-$ , to make explicit the relativization of $| \mathrm{X}|$ to the model $\boldsymbol{M}$ .

Suppose $X=[{x}_1,{x}_2,\dots, {y}_1,{y}_2,\dots, {z}_1,{z}_2,\dots, \dots ]$ is a multi-set, specified in such a way that ${x}_1,{x}_2,\dots$ are the same element $x$ , ${y}_1,{y}_2,\dots$ are the same element $y$ , …., ${z}_1,{z}_2,\dots$ are the same element $z$ , …, and $x,y,z,\dots$ are pairwise distinct. We then let $\{X\}$ be the corresponding set {x, y, z, …}. Thus $\{X\}$ is like $X$ but for the fact that each element of $X$ , which may occur many times in $X$ , only occurs once in $\{X\}$ .

Given a multi-set model $\boldsymbol{M}=(S,\sqsubseteq, | \cdot |_{\boldsymbol{M}})$ , let $\{\boldsymbol{M}\}$ be the corresponding (set) model $(S,\sqsubseteq, | \cdot |_{\{\boldsymbol{M}\}})$ , where ${|\mathrm{x}|}_{\{M\}}^{\pi}=\{{|\mathrm{x}|}_M^{\pi}\}$ for each atom $\mathrm{x}$ and where $\pi =+$ or $-$ . Then $\{\boldsymbol{M}\}$ and $\boldsymbol{M}$ evaluate formulas in the same way but for a difference in the ‘count’:

We might compare the present positive clause for disjunction with the conception of intuitionistic disjunction as disjoint union, where the disjoint union $X{\cdot}\! \cup\! {\cdot} Y$ of the sets $X$ and $Y$ might be defined as $\{(x,0):x\in X\}\cup \{(y,1):y\in Y\}$ . In the intuitionistic case, we distinguish between verification on the left and right (so that $X{\cdot}\! \cup \!{\cdot} Y$ will not, in general, be the same as $Y{\cdot}\! \cup \!{\cdot} X$ ), whereas, in the present case, we make no such distinction.

The similarity in the clauses for the two kinds of models belies a fundamental difference. For whereas in the earlier case, we can restate the clauses relationally, in terms of when a state is a truth-maker or falsity-maker for a given formula, this is no longer possible in the present case, since it does not provide us with the means of stating that a formula, such as α ∨ α, will be made true more than once by any given truth-maker for $\alpha$ .

Although this is not a topic I shall pursue here, it is worth noting that the previous order of explanation can be reversed and that, instead of explaining our symbolism in terms of multi-sets, we can use our symbolism to provide a convenient way of specifying multi-sets. Suppose, for example, that we wish to specify the multi-set $[1,1]$ . The usual set-theoretic formulation $[x{:}\ x=1]$ is not then appropriate. However, if we subject the defining conditions to a multi-set semantics, then $[1,1]$ can be specified as $[x{:}\ x=1\vee x=1]$ , which will be different from $[x{:}\ x=1]=[1]$ and, similarly, if f is a function, then we might specify its multi-range as $[y{:}\ \mathrm{for}\kern0.17em \mathrm{some}\;x,f(x)=y]$ , where y now will occur as many times in the multi-set as there are arguments $x$ for which $f(x)=y$ .

Indeed, for these purposes we can adopt a purely extensional quasi-classical version of the multi-set semantics in which there is just one element $\square (=\blacksquare )$ in the state space, corresponding to the actual world. The pair of multi-sets ( $[\square, \square, \dots ],[\square, \square, \dots ]$ ) assigned to a formula will then indicate how many times the formula is true and how many times it is false. So, for example, the condition x = 1 ∨ x = 1, when $1$ is assigned to $x$ , will have ([□, □], []) as its bilateral content, thereby indicating that $1$ is to occur twice in the corresponding multi-set $[x:x=1\vee x=1]$ .

The semantics of each formula in this simple model is given, in effect, by a pair of non-negative integers, where the first indicates how many times the formula is true and the second how many times it is false. Supposing $| \mathrm{X}| =(m,n)$ and $| \mathrm{Y}| =(p,q)$ , we will then have the following clauses for the truth-functional connectives:

• $| \neg \mathrm{X}| =(n,m)$ .

• $[(\mathrm{X}\vee \mathrm{Y})]=(m+p,n\times q)$ .

• $| (\mathrm{X}\wedge \mathrm{Y})| =(m\times p,n+q)$ .

The notions of exact entailment and equivalence can be extended to the multi-set semantics in the obvious way. Thus the formula $\mathrm{X}$ will exactly entail the formula $\mathrm{Y}$ if whenever ${|\mathrm{X}|}^{+}=[{s}_1,{s}_2,\dots ]$ in a model $\boldsymbol{M}$ then ${|\mathrm{Y}|}^{+}$ is of the form $[{s}_1,{s}_2,\dots, \dots ]$ in $\boldsymbol{M}$ and $\mathrm{X}$ will be an exact equivalent of $\mathrm{Y}$ if the multi-sets ${|\mathrm{X}|}^{+}$ and | ${|\mathrm{Y}|}^{+}$ are the same. A related definition of containment could also be given, although it will not be required.

5 Logics

We briefly review some earlier findings concerning the logic of equivalence and containment. Given two formulas $\mathrm{X}$ and $\mathrm{Y}$ , we use $\mathrm{X}\equiv \mathrm{Y}$ to indicate that they are exact equivalents and use $\mathrm{X}>\mathrm{Y}$ to indicate that $\mathrm{X}$ contains $\mathrm{Y}$ .

We shall be interested in the following axioms and rules for $\equiv$ :

• E1 $\mathrm{X}\equiv \neg \neg \mathrm{X}$ .

• E2 $\mathrm{X}\wedge \mathrm{Y}\equiv \mathrm{Y}\wedge \mathrm{X}$ .

• E3 $(\mathrm{X}\wedge \mathrm{Y})\wedge \mathrm{Z}\equiv \mathrm{X}\wedge (\mathrm{Y}\wedge \mathrm{Z})$ .

• E4 $\mathrm{X}\vee \mathrm{Y}\equiv \mathrm{Y}\vee \mathrm{X}$ .

• E5 $(\mathrm{X}\vee \mathrm{Y})\vee \mathrm{Z}\equiv \mathrm{X}\vee (\mathrm{Y}\vee \mathrm{Z})$ .

• E6 $\neg (\mathrm{X}\wedge \mathrm{Y})\equiv (\neg \mathrm{X}\vee \neg \mathrm{Y})$ .

• E7 $\neg (\mathrm{X}\vee \mathrm{Y})\equiv (\neg \mathrm{X}\wedge \neg \mathrm{Y})$ .

• E8 $\mathrm{X}\wedge (\mathrm{Y}\vee \mathrm{Z})\equiv (\mathrm{X}\wedge \mathrm{Y})\vee (\mathrm{X}\wedge \mathrm{Z})$ .

• E9 $\mathrm{X}\equiv \mathrm{Y}/\mathrm{Y}\equiv \mathrm{X}$ .

• E10 $\mathrm{X}\equiv \mathrm{Y},\mathrm{Y}\equiv \mathrm{Z}/\mathrm{X}\equiv \mathrm{Z}$ .

• E11 $\mathrm{X}\equiv \mathrm{Y}/\mathrm{X}\wedge \mathrm{Z}\equiv \mathrm{Y}\wedge \mathrm{Z}$ .

• E12 $\mathrm{X}\equiv \mathrm{Y}/\mathrm{X}\vee \mathrm{Z}\equiv \mathrm{Y}\vee \mathrm{Z}$ .

• E13 $\mathrm{X}\wedge \top \equiv \mathrm{X}$ .

• E14 $\mathrm{X}\wedge \perp \equiv \perp$ .

• E15 $\mathrm{X}\vee \perp \equiv \mathrm{X}$ .

• E16 $\mathrm{X}\equiv \mathrm{X}\vee \mathrm{X}$ .

• E17 $\mathrm{x}\wedge \mathrm{x}\equiv \mathrm{x}$ $\mathrm{x}$ an atom.

• E18 $\neg \mathrm{x}\wedge \neg \mathrm{x}\equiv \neg \mathrm{x}$ $\mathrm{x}$ an atom.

The core system CS is comprised of the axioms and rules E1–E15. A sequent of the form $\mathrm{X}\equiv \mathrm{Y}$ is said to hold in a model $\boldsymbol{M}$ if $\mathrm{X}$ and $\mathrm{Y}$ are exact equivalents in $\boldsymbol{M}$ . A single sequent such as E1 is said to be valid if it is true in all models (or in all models from a pre-designated subclass of models) and a rule such as E9 is said to be valid if its conclusion holds in any model in which its premises hold. It may then be shown that CS is sound and complete for the multi-set semantics, i.e., that the provable sequents coincide with the valid sequents. CS is also sound and complete for the class of multi-set models $\boldsymbol{M}=(S,\sqsubseteq, | \cdot |)$ in which the state space $S$ is singleton.Footnote ³ This is somewhat surprising, and no similar result holds for the set-theoretic semantics. Axiom E16 corresponds to the adoption of a set-theoretic semantics, E17 to the restriction of the semantics to definite models, and E17 and E18 to the restriction of the semantics to bi-definite models. So, for example, the system CS + E16 will be sound and complete for the class of all set-theoretic models, the system CS + E17 sound and complete for the class of all definite multi-set models, and the system CS + E17 + E18 sound and complete for the class of all bi-definite multi-set models.

Let me now state the normal form theorem, which will play an important role in establishing the extended normal form theorem below. For this purpose, it will be helpful to refer to conjunctions of formulas ${\mathrm{X}}_1$ , ${\mathrm{X}}_2$ , …, ${\mathrm{X}}_n$ , where $n$ is allowed to be $0$ or $1$ . When $n=0$ , the conjunction is the verum constant $\top$ , and when $n=1$ , it is the single formula ${X}_1$ . We shall refer similarly to disjunctions of formulas ${\mathrm{X}}_1$ , ${\mathrm{X}}_2$ , …, ${\mathrm{X}}_n$ , where $n$ is again allowed to be 0 or 1. When $n=0$ , the disjunction is the falsum constant $\perp$ , and when $n=1$ , it is the single formula ${\mathrm{X}}_1$ .

A literal $\pm \mathrm{x}$ is an atom $\mathrm{x}$ or its negation $\neg \mathrm{x}$ , a state description is a conjunction (possibly empty) of literals, and a disjunctive normal form is a disjunction (possibly empty) of state descriptions. Say that the formulas $\mathrm{X}$ and $\mathrm{Y}$ are provably equivalent in a given system if $\mathrm{X}\equiv \mathrm{Y}$ is a theorem of the system. Then we can establish along the lines of Fine [Reference Fine5]:

Let us note a couple of refinements of this result. We may suppose that the atoms of our language occur in a fixed order ${\mathrm{x}}_1$ , ${\mathrm{x}}_2\dots$ . Let ${\mathrm{x}}_1,\neg {\mathrm{x}}_1,{\mathrm{x}}_2,\neg {\mathrm{x}}_2,\dots$ be a corresponding order for the literals. Then a state description is said to be standard if its literals occur in the fixed order with association from left to right and it is said to be non-repetitive if no literal occurs more than once. Clearly, a standard state description is uniquely determined by the multi-set of its literal conjuncts and a standard non-repetitive state description is uniquely determined by the set of its literal conjuncts. The standard state descriptions might also be given in a fixed order. A disjunctive normal form is then said to be in standard form if it is a disjunction of standard state descriptions that occur in the fixed order with association from left to right. It is said to be conjunctively non-repetitive if each of its state descriptions is non-repetitive, to be disjunctively non-repetitive if none of its state descriptions occurs more than once, and to be fully non-repetitive if it is both conjunctively and disjunctively non-repetitive. Clearly, a standard form is uniquely determined by the multi-set of all those multi-sets of literals that correspond to an occurrence of one of its state descriptions, a fully non-repetitive standard form by the set of all those sets of literals that correspond to one of its state descriptions, and similarly for the other cases.

We turn to the second refinement of the result. Take a realization scheme $\Sigma$ for a set of atoms $\Delta$ to be a set of formulas of the form $\neg \mathrm{x}\equiv ({\mathrm{y}}_1\vee {\mathrm{y}}_2\vee \cdots \vee {\mathrm{y}}_n)$ , one for each atom $\mathrm{x}$ of $\Delta$ , where ${\mathrm{y}}_1,{\mathrm{y}}_2,\dots, {\mathrm{y}}_n$ for $n>0$ , are distinct atoms (though not necessarily in $\Delta$ ) and say that $\Sigma$ is a realization scheme for a formula $\mathrm{X}$ if it is a realization scheme for the set of atoms that occur in $\mathrm{X}$ . Intuitively, a realization scheme provides a positive characterization of when the given atoms are false.

Say that a state description is positive if it is a conjunction of atoms and that a disjunctive normal form is positive if it is a disjunction of positive state descriptions. From Theorem 1, we easily derive:

Combining the previous results, we obtain:

I have stated these results using the notion of provable equivalence. But we should note that by Soundness corresponding results will hold for the notion of semantic equivalence. Thus in the case of Theorem 1, we will have that every formula is exactly equivalent to a disjunction of state descriptions and, in the case of Corollary 3, we will have that, given a realization scheme $\Sigma$ , every formula $\mathrm{X}$ will be $\Sigma$ -equivalent to a positive disjunctive normal form $\mathrm{Y}$ , i.e., that $| \mathrm{X}|$ and $| \mathrm{Y}|$ will be the same in any model in which the equivalences of $\Sigma$ hold. I should also remark that in the subsequent treatment we could dispense with realization schemes by taking the models to be bi-definite (and adopting axioms E17 and E18). However, the present treatment is more general in that it allows an atom to have any positive finite number of falsity-makers.

Related rules can be given for the logic of containment within the set-theoretic semantics [Reference Fine8, Section 6]. An alternative approach, which we adopt here, builds the logic of containment on top of the logic of equivalence. Using ‘ $\mathrm{X}>\mathrm{Y}$ ’ as a symbolic notation for the containment relation, this requires that the following rules be added to the previous rules E1–E16:

• $\mathrm{C}1.\ \mathrm{X}>\mathrm{Y},\mathrm{X}\equiv \mathrm{X}^{\prime },\mathrm{Y}\equiv \mathrm{Y}^{\prime }/\mathrm{X}^{\prime }>\mathrm{Y}^{\prime }$ ;

• $\mathrm{C}2.\ \mathrm{X}\wedge \mathrm{Y}>\mathrm{X}$ as long as $\mathrm{Y}$ does not contain a negative occurrence of $\top$ ;

• $\mathrm{C}3.\ \mathrm{X}>\mathrm{X}^{\prime },\mathrm{Y}>\mathrm{Y}^{\prime }/\mathrm{X}\vee \mathrm{Y}>\mathrm{X}^{\prime}\vee \mathrm{Y}^{\prime }$ .

Note the restriction on C2. This is required since $\mathrm{X}\ \wedge \perp\ >\mathrm{X}$ , for example, is not valid (for discussion, see the sub-section on partial content in [Reference Fine6]).

Using normal forms, we can also establish completeness for the various underlying systems along the lines of Fine [Reference Fine5]. Thus in the case of the system CS, we have:

Or again, in the case of the system for the logic of equivalence and containment described above, we can first use the original rules E1–E16 to put the formulas $\mathrm{X}$ and $\mathrm{Y}$ into appropriate disjunctive normal form and then use the additional rules C1–C3 to establish $\mathrm{X}>\mathrm{Y}$ when $\mathrm{X}$ semantically contains $\mathrm{Y}$ .

6 Situations and actions

The semantics for conditional imperatives will involve both situations and actions. Roughly, the idea is that a situation is a situation in which one acts—as when one takes an umbrella in the situation in which it rains. Actions may themselves be, or may constitute, situations—as when one puts on a rain coat in a situation in which one has taken an umbrella. We accordingly take a situation–action space (or S/A space) ${\boldsymbol{S}\kern-1.2pt}_\mathrm{I,A}$ to be a structure $({S}_{\mathrm{I}},{S}_{\mathrm{A}},\sqsubseteq )$ , where ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{I}}=({S}_{\mathrm{I}},\sqsubseteq )$ is a state space and ${S}_{\mathrm{A}}$ is a subset of ${S}_{\mathrm{I}}$ closed under fusion $(\bigsqcup B\in {S}_{\mathrm{A}}\;\mathrm{whenever}\;B\subseteq {S}_{\mathrm{A}})$ . Intuitively, ${S}_{\mathrm{I}}$ consists of states or situations and ${S}_{\mathrm{A}}$ of actions. We have here presupposed that each action is identical to a situation, but this constitutes no great metaphysical commitment on our part. We could just as well have taken each action to constitute the situation in which that action is performed.

We set ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{A}}=({S}_{\mathrm{A}},\sqsubseteq \upharpoonleft {S}_{\mathrm{A}})$ and call it an action (or A-) space. It should be evident that ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{A}}$ is a subspace of ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{I}}$ , and often we shall use $\sqsubseteq$ in place of $\sqsubseteq \upharpoonleft {S}_{\mathrm{A}}$ when it is clear that the relata are actions. The null action ${\square}_{\mathrm{A}}$ from ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{A}}$ is identical to the null situation ${\square}_{\mathrm{I}}$ from ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{I}}$ (to do nothing and for nothing to be is one and the same)—for which reason, we designate both as $\square$ . But although $\square_{\mathrm{A}}$ and $\square_{\mathrm{I}}$ are the same, the full action ${\blacksquare}_{\mathrm{A}}$ (the fusion of all actions) will not, in general, be the same as the full situation ${\blacksquare}_{\mathrm{I}}$ (the fusion of all situations) but merely a proper part of ${\blacksquare}_{\mathrm{I}}$ .

Given any situation $s\in S,$ there will be a greatest action ${a}_s$ that is a part of $s$ , viz. $\bigsqcup \{a\in {S}_{\mathrm{A}}:a\sqsubseteq s\}$ . Since ${S}_{\mathrm{A}}$ is closed under arbitrary fusions, ${a}_s$ is indeed an action. However, we cannot in general assume that all parts of actions will also be actions.

Call a situation pure if it contains no non-null action (one might for this reason think of it as a pure ‘state of nature’) and otherwise call it impure. We can plausibly assume:

Situation Decomposition: Any situation $s$ is the fusion of a pure situation and an action.

We might, similarly, call an action pure if it does not contain a pure situation (other than the null situation). Although Situation Decomposition has some plausibility, it is not so plausible to assume:

Action Decomposition: Any action is the fusion of a pure situation and a pure action.

Consider, for example, the action of raising my arm. This contains the situation of my arm rising, which arguably is a pure situation, i.e., one that does not contain a non-null action. However, it is far from clear that it fuses with any pure action to form the action of raising my arm since this would appear to require an analysis of the action of my raising my arm into a purely physical and a purely mental component.

There are three kinds of models, corresponding to the three kinds of state space. A situation (or S-) model ${\boldsymbol{M}}_{\mathrm{I}}$ over the situation space ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{I}}=({S}_{\mathrm{I}},{\sqsubseteq}_{\mathrm{I}})$ is a triple $({S}_{\mathrm{I}},{\sqsubseteq}_{\mathrm{I}},{| \cdot |}_{\mathrm{I}})$ , where ${| \cdot |}_{\mathrm{I}}$ (the indicative valuation) is a function taking each indicative atom $\mathrm{s}$ into a pair of multi-sets $| \mathrm{s}|^{+}_{\mathrm{I}}$ and $| \mathrm{s}|^{-}_{\mathrm{I}}$ of situations (the respective truth- and falsity-makers of $\mathrm{s}$ ). Relative to such a model, we can then assign a multi-set of truth-makers $| \mathrm{S}|^{+}_{\mathrm{I}}$ and a multi-set $| \mathrm{S}|^{-}_{\mathrm{I}}$ of falsity-makers to each indicative sentence S in accordance with the clauses given for the multi-set semantics in Section 4.

For the purposes of the extended normal form theorem, we assume that the situation models are definite, i.e., that $| \mathrm{s}| ^{+}_{\mathrm{I}}$ for each indicative atom $\mathrm{s}$ is singleton. This means that we can take the underlying logic for indicative formulas to be given by CS + E17 and that we can assume that, for any realization scheme $\Sigma$ for an indicative formula S, S will be provably $\Sigma$ -equivalent in CS + E17 to a standard conjunctively non-repetitive normal form.

An action (A-) model ${\boldsymbol{M}}_{\mathrm{A}}$ over the action space ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{A}}=({S}_{\mathrm{A}},{\sqsubseteq}_{\mathrm{A}})$ is a triple $({S}_{\mathrm{A}},{\sqsubseteq}_{\mathrm{A}},{| \cdot |}_{\mathrm{A}})$ , where ${| \cdot |}_{\mathrm{A}}$ (the imperative valuation) is a function taking each imperative A-atom $\mathrm{a}!$ into a pair of sets $| \mathrm{a}!| ^{+}_{\mathrm{A}}$ and $| \mathrm{a}!| ^{-}_{\mathrm{A}}$ of actions (the respective actions in compliance with or in contravention to a!). Relative to such a model, we can then assign a set of compliance-makers $| \mathrm{P}| ^{+}_{\mathrm{A}}$ and a set $| \mathrm{P}| ^{-}_{\mathrm{A}}$ of contravention-makers to each $\to$ -free imperative P constructed from the imperative A-atoms in accordance with the clauses given for the standard set-theoretic semantics in Section 3. Since the semantics is set-theoretic, we can take the underlying logic to be CS + E16 and any such formula P will therefore be provably equivalent to a standard disjunctive non-repetitive normal form.

Finally, a situation-action (S/A) model ${\boldsymbol{M}}_\mathrm{I,A}$ over the situation–action space ${\boldsymbol{S}\kern-1.2pt}_{I,A}=({S}_{\mathrm{I}},{S}_{\mathrm{A}},\sqsubseteq )$ is a quintuple $({S}_{\mathrm{I}},{S}_{\mathrm{A}},\sqsubseteq, {| \cdot |}_{\mathrm{I}},{| \cdot |}_{\mathrm{A}})$ , where $({S}_{\mathrm{I}},{S}_{\mathrm{A}},\sqsubseteq, {| \cdot |}_{\mathrm{I}})$ is a situation model and $({S}_{\mathrm{I}},{S}_{\mathrm{A}},\sqsubseteq, {| \cdot |}_{\mathrm{A}})$ is an action model. Thus a situation–action model enables us, separately, to interpret indicative and imperative formulas, though not the conditional imperatives in which the two kinds of formulas come together. We shall make a critical assumption about any situation–action model. Suppose $|\mathrm{a}!|^{+}_{\mathrm{A}}=\{{a}_1,{a}_2,\dots \}$ and $| \mathrm{a}!| ^{-}_{\mathrm{A}}=\{{b}_1,{b}_2,\dots \}$ , where the ${a}_1,{a}_2,\dots$ and the ${b}_1,{b}_2,\dots$ are chosen to be pairwise distinct. Then $| \mathrm{a}| ^{+}_{\mathrm{I}}=[{a}_1,{a}_2,\dots ]$ and $| \mathrm{a}| ^{-}_{\mathrm{I}}=[{b}_1,{b}_2,\dots ]$ . In other words, the interpretation of the A-atom $\mathrm{a}!$ and the corresponding indicative atom $\mathrm{a}$ should be coordinated so that the truth-makers for indicative $\mathrm{a}$ are the actions in compliance with imperatival $\mathrm{a}!$ and the falsity makers for $\mathrm{a}$ are the actions in contravention to $\mathrm{a}!$ .

This assumption gives rise to an issue. For whereas a state that is the truth-maker for ‘you shut the door’ may well be an action in compliance with ‘(you) shut the door’, it is far from clear that any state that is a truth-maker for ‘you did not shut the door’ need be an action in compliance with ‘do not shut the door’. For suppose, upon hearing you command ‘do not shut the door’, that I race to the door with the intention of shutting it but trip and end up leaving the door open. Then it is plausible to suppose that what I do is not in compliance with the imperative ‘do not shut the door’. For this would appear to require, not merely that I fail to shut the door, but that I refrain from shutting the door.Footnote ⁴

This means that in order to ensure that the truthmakers for ‘you do not shut the door’ are the same as the ‘actions’ compliant with ‘do not shut the door’, we must either adopt a narrower view of what the truth-makers might be or a broader view of what the actions might be. I am inclined to think that for most purposes we should adopt the former view. For we may suppose that under normal circumstances accidents like the one above will not happen and that, given that a negative imperative has been invoked, the agent will only fail to perform the required action by refraining from performing that action.

7 Plans

In extending the semantics that we earlier gave for categorical imperatives in terms of compliance- and contravention-conditions to conditional imperatives, we face a basic question. What should we take to be the nature of the items or ‘states’ in compliance with or in contravention to a conditional imperative? What, for example, should we take to be in compliance with the imperative ‘if it rains then take an umbrella’?

One natural answer is that we should take it to be an action, as before, but one whose compliance with the imperative is relative to a situation. Thus relative to the situation in which it rains, the action of taking an umbrella will be in compliance with the imperative, while, relative to a situation in which it is not rainy, the null action will be in compliance with the imperative. The clauses for the various connectives will then also be relativized to a situation. Thus relative to any given situation, we may take the actions in compliance with a conjunctive imperative to be the fusions of those actions, which, relative to the given situation, are in compliance with its respective conjuncts.

Unfortunately, this will not in general give us what we want. Consider the following two imperatives:

1. (1) (if it rains take an umbrella and if it is sunny take a parasol) or (if it rains wear a coat and if it is sunny wear a hat) ( $(\mathrm{R}\to \mathrm{U}!)\wedge (\mathrm{S}\to \mathrm{P}!)\vee ((\mathrm{R}\to \mathrm{C}!)\wedge (\mathrm{S}\to \mathrm{H}!)$ );

2. (2) (if it rains take an umbrella and if it is sunny wear a hat) or (if it rains wear a coat and if it is sunny take a parasol) ( $(\mathrm{R}\to \mathrm{U}!)\wedge (\mathrm{S}\to \mathrm{H}!)\vee ((\mathrm{R}\to \mathrm{C}!)\wedge (\mathrm{S}\to \mathrm{P}!)$ ).

The two imperatives are clearly different in their import. In preparing for the first, one will have an umbrella and a parasol or a coat and a hat at hand, while, in preparing for the second, one will have an umbrella and a hat or a coat and a parasol at hand. But in any situation, the same actions will be in compliance with each of the two imperatives. Thus in the situation in which it rains, the actions in compliance with either imperative will be taking an umbrella or wearing a coat and, in the situation in which it is sunny, the actions in compliance with either imperative will be taking a parasol or wearing a hat. What the present semantics ignores is an interdependence that may exist between what is required in different situations. Thus in the first case, taking an umbrella in the situation in which it rains is to be paired with taking a parasol in the situation in which it is sunny, while, in the second case, it is to be paired with wearing a hat.

We may avoid this difficulty by packaging the situation s and the action $a$ into a single item, something which we might represent by the ordered pair $(s,a)$ . We might think of $(s,a)$ as a conditional action, the action of doing $a$ conditional upon $s$ , where the performance of this action requires nothing in case $s$ does not obtain and requires the performance of $a$ (or better: the performance of $a$ in the situation $s$ ) in case $s$ does obtain.Footnote ⁵ These conditional actions can then be fused in a natural way. Thus the fusion of $(s,a)$ and $(t,b)$ , for $s\ne t$ , will be a complex action, which we can represent as $\{(s,a),(t,b)\}$ and which will consist in doing $a$ in situation $s$ and $b$ in situation $t$ . The previous problem then disappears for, using an obvious notation, $\{(r,u),(s,p)\}$ , for example, will be in compliance with the first imperative but not the second.

A pleasing simplification to the present approach is available to us. For presumably, we can identify the fusion of the states $(s,{a}_1),(s,{a}_2),\dots$ , where the first component $s$ is always the same, with $(s,({a}_1\sqcup {a}_2\sqcup \cdots ))$ , and presumably, we can identify the case in which, relative to a situation $s$ , nothing has been required of us, with the conditional action $(s,\square )$ (where □, recall, is the null action). But this means that the items in compliance with a conditional imperative or with a truth-functional compound of conditional imperatives can be taken to be total plans, i.e., with functions which associate each situation with a single action. We can, moreover, identify any action a in compliance with a categorical imperative with the conditional action $(\square, a)$ of doing $a$ in the null situation. This then means that we can treat total plans, telling us what is to be done in each situation, as the compliance- and contravention-makers for all imperatives.

This plan-based semantics is suggestive of a richer, more subjective, understanding of imperatives. It was perhaps true all along that we should think of the action in compliance with an imperative, such as ‘shut the door’, not simply as the action of shutting the door but as the action of shutting the door by way of conforming to the imperative (or what it prescribes). But we can now think of an imperative as telling us, not simply what to do, but what to plan on doing. Thus the item in conformity with the conditional imperative ‘take an umbrella if it rains’ will now be the adoption of a plan to take an umbrella if it rains (or perhaps we should say the effective adoption, so that the plan is indeed acted upon should the occasion arise). Similarly, the item in conformity with the positive categorical imperative ‘shut the door’ will be an (effective) decision to shut the door and the item in conformity with the negative categorical imperative ‘do not shut the door’ will be an effective decision not to shut the door. Thus from this perspective, what an imperative most immediately requires is an internal change in our readiness to act rather than an external change in how we actually do act. I am not saying that this understanding of imperatives is forced upon us, but it is one that naturally arises when conditional imperatives are in play and when the agent is indeed capable of making decisions or of planning for future contingencies. In such a case, the imperative, if successful, will serve to ‘prime’ the agent to do what the imperative requires.

Let us now make the previous informal discussion more precise. Suppose ${\boldsymbol{S}\kern-1.2pt}_\mathrm{I,A}=({S}_{\mathrm{I}},{S}_{\mathrm{A}},\sqsubseteq )$ is an S/A space. A plan $p$ in ${\boldsymbol{S}\kern-1.2pt}_\mathrm{I,A}$ is then a (total) function from ${S}_{\mathrm{I}}$ into ${S}_{\mathrm{A}}$ . Intuitively, a plan tells the agent what to do in each situation $s\in {S}_{\mathrm{I}}$ . We use $S^{+}_{\mathrm{P}}$ to denote the set of all plans in ${\boldsymbol{S}\kern-1.2pt}_\mathrm{I,A}$ (the ‘ $+$ ’ here indicates fullness not positivity), and, given $p,q\in S^{+}_{\mathrm{P}}$ , we define the relation $\sqsubseteq ^{+}_{\mathrm{P}}$ of part on plans in usual point-wise fashion:

$$\begin{align*}p\sqsubseteq ^{+}_{\mathrm{P}}\kern0.24em q\;\mathrm{iff}\;p[s]\sqsubseteq q[s]\;\mathrm{for}\kern0.17em \mathrm{each}\;s\in {S}_{\mathrm{I}}.\end{align*}$$

Thus one plan $p$ is said to be part of another if what it enjoins in any situation is part of what the other plan enjoins in that situation.

There are two variants on this notion of a plan that we should mention even though they will not be pursued. Under the first, a plan $p$ is a partial function from the set of situations to the set of actions. With each partial plan $p$ may be associated the corresponding total plan ${p}^{+}$ , where:

• ${p}^{+}[s]=p[s]\;\mathrm{when}\;p[s]\;\mathrm{is}\kern0.17em \mathrm{defined}$ ,

• $= \square\ \mathrm{otherwise}$ .

However, one cannot, conversely, recover $p$ from ${p}^{+}$ since it may be that ${p}^{+}={q}^{+}$ even though $p$ , say, takes the value $\square$ for a specific situation $s$ for which $q$ takes no value, and, on a certain highly fine-grained conceptions of content, this difference in the situations on which a plan is defined may be regarded as relevant to the meaning of an imperative.

The other variant is to allow a plan $p$ to be indeterminate so that its value $p[s]$ for a given situation $s$ can be a non-empty set of actions (or also the empty set if partiality is also allowed). However, it may be argued that any indeterminate plan can just as well be viewed as a disjunction of determinate plans (for which the value $p[s]$ is always a single action). Thus adopting the plan of doing $a$ or $b$ in situation $s$ and $c$ or $d$ in situation $t$ can be viewed as adopting the plan to do $a$ in $s$ and $c$ in $t$ or the plan to do $a$ in $s$ and $d$ in $t$ or the plan to do $b$ in $s$ and $c$ in $t$ or the plan to do $b$ in $s$ and $d$ in $t$ . There would therefore appear to be no need to appeal to indeterminate plans in developing a semantics for conditional imperatives.

However, there may well be a need to consider indeterminate transitions in the case of counterfactuals and other conditional constructions. Thus even though the imperative ‘if it rains then take an umbrella or wear a coat’ can be taken to be equivalent to ‘if it rains then take an umbrella or if it rains then wear a coat’, it is not so clear that the counterfactual ‘if it were to rain then you would take an umbrella or wear a coat’ should be taken to be equivalent to ‘if it were to rain then you would take an umbrella or if it were to rain then you would wear a coat’. Thus in a more general theory of conditionals we should allow for the route from antecedent to consequent to correspond to an indeterminate transition.

With each S/A space ${\boldsymbol{S}\kern-1.2pt}_{I,A}=({S}_{\mathrm{I}},{S}_{\mathrm{A}},\sqsubseteq )$ may be associated the full planning space ${\boldsymbol{S}\kern-1.2pt}_\mathrm{I,A,P}^{+}=({S}_{\mathrm{I}},{S}_{\mathrm{A}},S^{+}_{\mathrm{P}},\sqsubseteq )$ and the full plan space $\boldsymbol{S}^{+}_{\mathrm{P}}=(S^{+}_{\mathrm{P}},\sqsubseteq ^{+}_{\mathrm{P}})$ . $\boldsymbol{S}^{+}_{\mathrm{P}}$ is, of course, the ‘functional’ space from the situation space ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{I}}$ into the action space ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{A}}$ . It is readily verified that $\boldsymbol{S}^{+}_{\mathrm{P}}$ is indeed a state space, whose bottom element ${\square}_{\mathrm{P}}$ is the function $p$ on ${S}_{\mathrm{I}}$ whose value $p(s)$ is always ${\square}_{\mathrm{A}}=\square$ and whose top element ${\blacksquare}_{\mathrm{P}}$ is the function $p$ on ${S}_{\mathrm{I}}$ whose value $p(s)$ is always ${\blacksquare}_{\mathrm{A}}$ .

However, our interest will often not be in all of the plans of $S^{+}_{\mathrm{P}}$ but only in a subset of ‘admissible’ plans that are subject to certain constraints. Accordingly, a (closed) planning space is taken to be a structure ${\boldsymbol{S}\kern-1.2pt}_\mathrm{I,A,P}$ of the form $({S}_{\mathrm{I}},{S}_{\mathrm{A}},{S}_{\mathrm{P}},{\sqsubseteq}_{\mathrm{P}})$ , where the plan space ${\boldsymbol{S}\kern-1.5pt}_{\mathrm{P}}=({S}_{\mathrm{P}},{\sqsubseteq}_{\mathrm{P}})$ is a subspace of the full plan space $\boldsymbol{S}^{+}_{\mathrm{P}}=(S^{+}_{\mathrm{P}},\sqsubseteq ^{+}_{\mathrm{P}})$ subject to the condition that for each plan $p$ in $S^{+}_{\mathrm{P}}$ there is a plan $q\sqsupseteq ^{+}_{\mathrm{P}}\;p$ in ${S}_{\mathrm{P}}$ . Thus, according to this condition, any possible plan can always be extended to an admissible plan. Where it is evident from the context, we may drop the subscript ‘ $\mathrm{P}$ ’ or ‘ $^{+}_{\mathrm{P}}$ ’ from $\sqsubseteq$ .

Since the plan space ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{P}}$ is complete, it follows that there will be a least plan in ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{P}}$ containing any given plan $p$ of $S^{+}_{\mathrm{P}}$ , which we dub the closure ${p}^{* }$ of $p$ . The operation: $p\to {p}^{* }$ has the standard features of a closure operation with respect to $\sqsubseteq_{\mathrm{P}}$ :

• $p\ {\sqsubseteq}_{\mathrm{P}}\ {p}^{\ast}$

• $p\ {\sqsubseteq}_{\mathrm{P}}\ q\Rightarrow {p}^{* }{\sqsubseteq}_{\mathrm{P}}\ {q}^{\ast}$

• ${p}^{\ast \ast }\ {\sqsubseteq}_{\mathrm{P}}\ {p}^{* }$

appeal to which will often be implicit in what follows. A plan $p$ of $S^{+}_{\mathrm{P}}$ is itself said to be closed if $p={p}^{* }.$ Clearly, the closed plans of $S^{+}_{\mathrm{P}}$ are simply the plans of $S_{\mathrm{P}}$ .

We should note that the fusion of plans ${p}_1,{p}_2,\dots$ in ${\boldsymbol{S}\kern-1.4pt}_{\mathrm{P}}$ will be ${p}^{* }$ , where $p={p}_1\sqcup {p}_2\sqcup \cdots$ is the fusion of ${p}_1,{p}_2,\dots$ in $\boldsymbol{S}^{+}_{\mathrm{P}}.$ In other words, taking $\bigsqcup^{*}$ to be the fusion operation for the closed subspace, ${\bigsqcup}^*{p}_k= (\bigsqcup {p}_k)^*$ , and there is no reason why ${\bigsqcup }^* {p}_k$ in general should be identical to $\bigsqcup {p}_k$ .

The closure operation: $p\to {p}^{* }$ on plans could, in principle, obliterate all distinctions between plans by making ${p}^{* }$ always equal to ${\blacksquare}_{\mathrm{P}}$ .Footnote ⁶ We shall therefore sometimes find it helpful to suppose that there is some constraint in how far ${p}^{* }$ can go in extending a given plan $p$ . Set ${p}^{\oplus}[s]=({a}_s\sqcup $ $\bigsqcup Rg(p))$ . Then the operation $p\to {p}^{* }$ is said to be bound if ${p}^{\ast}\sqsubseteq {p}^{\oplus }$ . We might think of this condition as telling us that, in defining ${p}^{* }(s)\sqsupseteq p(s)$ , the only actions we can add to $p(s)$ are those already in $s$ or in the range of $p$ .

8 Conditions on plans

There are a number of natural conditions one might wish to impose on a plan space. They will not make a difference to the logics considered below. However, they will make a significant difference to the logics considered in part IV, once we allow for the coordination $a\iff a!$ between indicative and imperative atoms, so that the indicative atom ${a}_k$ corresponding to an imperative atom ${a}_k!$ is allowed to occur in the antecedent of a conditional imperative.

One condition often introduced in the context of a functional space is

$$\begin{align*}\mathbf{Monotonicity}\mathbf{:}\ s\sqsubseteq t\Rightarrow p[s]\sqsubseteq p[t].\end{align*}$$

From a technical point of view, this is a very desirable condition. However, under our semantics, it would require us to accept Antecedent Strengthening (so that from ‘if it rains then take an umbrella’ we can infer ‘if it rains and there is a strong wind then take an umbrella’), and since we not wish in general to accept Antecedent Strengthening, nor can we accept Monotonicity.

One condition I have previously suggested is

$$\begin{align*}\mathbf{Semi}\text{-}\mathbf{Inclusion}\mathbf{:}\ \mathrm{if} a\sqsubseteq s\ \mathrm{then}\ a\sqsubseteq p[s].\end{align*}$$

In other words, when the situation on which a plan operates contains an action then the plan can be taken to require that action. Since ${a}_s$ is the maximal action to be contained in the situation s, we can state the condition more simply in the form

$$\begin{align*}\kern-51pt\mathbf{Semi}\text{-}\mathbf{Inclusion}\mathbf{:}\ {a}_s\sqsubseteq p[s].\end{align*}$$

When $s$ is an action $a$ then ${a}_s=a$ and, so in this special case, we get the more familiar

$$\begin{align*}\kern-56pt\mathbf{Action\ Inclusion}\mathbf{:}\ a\sqsubseteq p[a].\end{align*}$$

However, we cannot in general assume:

$$\begin{align*}s\sqsubseteq p\left[s\right]\end{align*}$$

since $p[s]$ will be an action of which the situation $s$ might not be a part.

Another condition we wish to accept is

$$\begin{align*}\mathbf{Follow}\text{-}\mathbf{Through}\mathbf{:}\ p[s]\sqsupseteq t\Rightarrow p[s]=p[s\sqcup t].\end{align*}$$

This says that what we are to do under a given plan should be the same as what we are to do under partial compliance with the plan. Thus we cannot evade or enhance the demands of a plan through partial compliance. This condition will later be of critical importance in establishing the validity of Imperative Detachment.

We may divide the condition into two parts

$$\begin{align*}&\mathbf{Follow}\text{-}\mathbf{Up}\mathbf{:}\ p[s]\sqsupseteq t\Rightarrow p[s]\sqsupseteq p[s\sqcup t]\\&\mathbf{Follow}\text{-}\mathbf{Down}\mathbf{:}\ p[s]\sqsupseteq t\Rightarrow p[s\sqcup t]\sqsupseteq p[s].\end{align*}$$

The first says that what we are to do under a given plan should contain (looking up) what we are to do under partial compliance with the plan, while the second says that what we are to do under partial compliance should contain (looking down) what we are to do under the given plan.

In the special case in which $t=p[s]$ , we get:

$$\begin{align*}p\left[p\left[s\right]\sqcup s\right]=p\left[s\right].\end{align*}$$

What we are to do under a given plan should be the same as what we are to do under complete compliance with the plan. When $s$ is an action $a$ , this becomes:

$$\begin{align*}p[p[a]\sqcup a]=p[a].\end{align*}$$

Given Action Inclusion, $a\sqsubseteq p[a]$ and so this becomes the familiar closure condition:

$$\begin{align*}p[p[a]]=p[a]\end{align*}$$

and so, in particular,

$$\begin{align*}p[p[\square]]=p[\square].\end{align*}$$

To illustrate: perhaps in the null situation, I am (according to the plan) to turn on the stove and, in a situation in which I turn on the stove, I am to light the stove; it then follows, according to the condition, that in the null situation, I am (according to the plan) to turn on the stove and light it. However, we do not in general have

$$\begin{align*}p[p[s]]=p[s]\end{align*}$$

since $p[s]$ may not contain $s$ .

I wish now to introduce a general method for getting from a plan that may not satisfy some of the above conditions to one that does. To this end, it will be helpful to introduce some familiar material on fixed points, which will be presupposed in what follows. (The less technically inclined reader may omit the rest of this section and the related material in Section 11).

Suppose $F$ is an inclusive operation on plans, i.e., $p\sqsubseteq F(p)$ for any plan $p$ . Given a plan $p$ , define a (transfinite) sequence of plans ${p}_0,{p}_1,\dots$ by

• ${p}_0\kern1.0em =p$ ,

• ${p}_{\beta +1}=F\!({p}_{\beta}),\mathrm{and}$

• ${p}_{\lambda}\kern1.0em =\bigsqcup \{{p}_{\zeta }{:}\ \zeta <\lambda \}$ .

We may then establish by induction that ${p}_{\alpha}\sqsubseteq {p}_{\beta}$ whenever $\alpha <\beta$ ; for ${p}_{\beta}\sqsubseteq F({p}_{\beta})={p}_{\beta +1}$ , given that $F$ is inclusive, and, for $\zeta <\lambda$ , ${p}_{\zeta}\sqsubseteq$ $\bigsqcup \{{p}_{\zeta }:\varsigma <\lambda \}={p}_{\lambda }.$ It follows, by considerations of cardinality, that there will be a least ordinal $\alpha$ for which ${p}_{\alpha}={p}_{\alpha +1}=F({p}_{\alpha})$ . We dub ${p}_{\alpha}$ the inclusive fixed point of $F$ and, where $\alpha$ is this least ordinal, we set ${p}^F={p}_{\alpha }$ .

When the operation $F$ is also monotonic, then ${p}^F$ is the least fixed point of $F$ to contain $p$ . For suppose $q$ is a fixed point containing $p$ , i.e., $F\kern-0.8pt(q){\kern-1pt}={\kern-1pt}q$ and $q{\kern-1pt}\sqsupseteq{\kern-1pt} p$ . We may show by induction on $\alpha$ that ${p}_{\alpha}\sqsubseteq q$ . For: when $\alpha =0$ , ${p}_0=p\sqsubseteq q$ by supposition; when $\alpha =\beta +1$ , then ${p}_{\beta}\sqsubseteq q$ by IH, and so ${p}_{\beta +1}=F\kern-0.8pt({p}_{\beta})\sqsubseteq F\kern-0.8pt({q}_{\beta})=q$ by the monotonicity of $F$ ; and when $\alpha =\lambda$ , ${p}_{\zeta}\sqsubseteq q$ for each $\zeta <\lambda$ by IH and so ${p}_{\lambda}=$ $\bigsqcup \{{p}_{\zeta }:\zeta <\lambda \}.$ Moreover, when $F$ is monotonic, the operation $p\to {p}^F$ is also monotonic. For suppose $p\sqsubseteq q$ . Then ${p}^F$ is the least fixed point of $F$ to contain $p$ . But ${q}^F$ is a fixed point of $F$ that contains $p$ and so ${p}^F\sqsubseteq {q}^F.$

We shall later find it useful to consider a generalization of the above construction. Suppose that ${<}{\boldsymbol{F}}_{\alpha }{>}$ is a sequence of inclusive operations on plans (the ${\boldsymbol{F}}_{\alpha }$ ’s may repeat and, for simplicity, we have taken ${<}{\boldsymbol{F}}_{\alpha }{>}$ to be defined on all ordinals $\alpha$ ). Given a plan $p$ , we define a corresponding sequence of plans ${<}{p}_{\alpha }{>}$ by

• ${p}_0\kern1.0em =p,$

• ${p}_{\beta +1}={\boldsymbol{F}}_{\beta}({p}_{\beta}),\mathrm{and}$

• ${p}_{\lambda}\kern1.0em =\bigsqcup \{{p}_{\zeta }{:}\ \zeta <\lambda \}$ .

The only difference from the previous definition is that the operation applied at the successor stage $\beta +1$ is allowed to vary with $\beta$ . Again, by considerations of cardinality, there will be a least ordinal $\alpha$ for which ${p}_{\alpha}={p}_{\alpha +\zeta }$ for all ordinals $\zeta$ . For this least ordinal $\alpha$ , we may then set ${p}^{\boldsymbol{F}}={p}_{\alpha }.$ And again, we may show, in the same way as before, that when each of the operations $p\to {\boldsymbol{F}\!}_{\alpha }(p)$ is monotonic then ${p}^{\boldsymbol{F}}$ is the least fixed point containing $p$ for all of the ${\boldsymbol{F}\!}_{\alpha }$ and that the operation: $p\to {p}^{\boldsymbol{F}}$ is monotonic.

I now wish to use the above constructions to show how any plan can be minimally extended to a plan subject to the previous constraints.

Lemma 6 (Follow-Up)

For any plan $p$ there is a least plan ${p}^{\uparrow}\sqsupseteq p$ conforming to the condition:

$$\begin{align*}{p}^{\uparrow}[s]\sqsupseteq t\Rightarrow {p}^{\uparrow}[s]\sqsupseteq {p}^{\uparrow}[s\sqcup t]\end{align*}$$

where the operation $p\to {p}^{\uparrow }$ is monotonic.

Proof. Define the operation $p\to p\uparrow$ on plans (to be distinguished from $p\to {p}^{\uparrow }$ ) by:

$$\begin{align*}p\uparrow \left[s\right]=\bigsqcup \left\{p\left[s\sqcup t\right]:t\sqsubseteq p\left[s\right]\right\}\;.\end{align*}$$

1. (1) The operation: $p\to p\uparrow$ is inclusive.

Pf: Since $p[s]\sqsupseteq \square$ , $p[s]=p[s\sqcup \square ]\sqsubseteq \bigsqcup \{p[s\sqcup t]:t\sqsubseteq p[s]\}=p\uparrow [s]$ .

Define p^↑ as the inclusive fixed point (with respect to the operation: $p\to p\uparrow$ ).

1. (2) Suppose that $p\sqsubseteq q$ and that $q[s]\sqsupseteq q[s\sqcup t]$ whenever $q[s]\sqsupseteq t$ . Then ${p}^{\uparrow}\sqsubseteq q$ .

Pf: We show by induction on $\alpha$ that ${p}_{\alpha}\sqsubseteq q,$ where the sequence of the ${p}_{\alpha }$ is defined as above by reference to the operation $p\to p\uparrow$ . Since ${p}^{\uparrow }$ is of the form ${p}_{\alpha }$ for some $\alpha$ , it will follow that ${p}^{\uparrow}\sqsubseteq q$ .

1. (i) $\alpha =0$ . By supposition.

2. (ii) $\alpha =\beta +1$ . Then ${p}_{\alpha}={p}_{\beta}\uparrow$ , where ${p}_{\beta}\uparrow [s]=\bigsqcup \{{p}_{\beta}[s\sqcup t]:t\sqsubseteq {p}_{\beta}[s]\}$ . So we need to show ${p}_{\beta}[s\sqcup t]\sqsubseteq q[s]$ whenever $t\sqsubseteq {p}_{\beta}[s]$ . But ${p}_{\beta}\sqsubseteq q$ by IH. Hence ${p}_{\beta}[s]\sqsubseteq q[s]$ , and so, given $t\sqsubseteq {p}_{\beta}[s]$ , $t\sqsubseteq q[s]$ . So by the condition imposed upon $q$ in (2), $q[s\sqcup t]\sqsubseteq q[s]$ . But by IH, ${p}_{\beta}[s\sqcup t]\sqsubseteq q[s\sqcup t]$ , and so ${p}_{\beta}[s\sqcup t]\sqsubseteq q[s]$ .

3. (iii) $\alpha =\lambda$ . Then ${p}_{\alpha}=\bigsqcup \{{p}_{\zeta }:\zeta <\alpha \}$ . By IH, ${p}_{\zeta}\sqsubseteq q$ for each $\zeta <\alpha$ , and so ${p}_{\alpha}=$ $\bigsqcup \{{p}_{\zeta }:\zeta <\alpha \}\sqsubseteq q$ .

1. (3) ${p}^{\uparrow}[s]\sqsupseteq t\Rightarrow {p}^{\uparrow}[s]\sqsupseteq {p}^{\uparrow}[s\sqcup t]$ .

Pf: Suppose ${p}^{\uparrow}[s]\sqsupseteq t$ . Then ${p}^{\uparrow}[s]={p}^{\uparrow}\uparrow [s]\sqsupseteq {p}^{\uparrow}[s\sqcup t]$ by definition of ${p}^{\uparrow }$ and $\uparrow$ .

1. (4) The operation: $p\to p\uparrow$ is monotonic.

Pf: Suppose $p\sqsubseteq q$ . Then:

• $p\uparrow =\bigsqcup \{p[s\sqcup t]:t\sqsubseteq p[s]\}$

• $\sqsubseteq {\kern-1pt}\bigsqcup \{q[s\sqcup t]{\kern-1pt}:{\kern-1pt}t\sqsubseteq q[s]\}$ $(\mathrm{since}\kern0.17em \mathrm{if}\;t\sqsubseteq p[s]\;\mathrm{then}\;t{\kern-1pt}\sqsubseteq{\kern-1pt} q[s]\;\mathrm{and}\;p[s{\kern-1pt}\sqcup{\kern-1pt} t]\sqsubseteq{\kern-1pt} q[s{\kern-1pt}\sqcup{\kern-1pt} t])$ $=q\uparrow.$

From (2) and (3) it follows that ${p}^{\uparrow }$ is the least plan $q\sqsupseteq p$ conforming to Follow-Up and from (4) it follows, from our earlier observation, that the operation $p\to p^{\uparrow}$ is also monotonic.

Lemma 7 (Follow-Down)

For any plan $p$ there is a least plan ${p}^{\downarrow}\sqsupseteq p$ conforming to the condition

$$\begin{align*}{p}^{\downarrow}\left[s\right]\sqsupseteq t\Rightarrow {p}^{\downarrow}\left[s\sqcup t\right]\sqsupseteq {p}^{\downarrow}\left[s\right],\end{align*}$$

where the operation $p\to {p}^{\downarrow }$ is monotonic.

Pf: Note that the condition imposed on ${p}^{\downarrow }$ is equivalent to

$$\begin{align*}u=s\sqcup t\&{p}^{\downarrow}\left[s\right]\sqsupseteq t\Rightarrow {p}^{\downarrow}\left[u\right]\sqsupseteq {p}^{\downarrow}\left[s\right].\end{align*}$$

So let us define the operation $p\to p\downarrow$ on plans by

$$\begin{align*}p\downarrow \left[u\right]=\bigsqcup \left\{p\left[s\right]:u=s\sqcup t\;\mathrm{and}\;p\left[s\right]\sqsupseteq t\right\}.\end{align*}$$

1. (1) The operation $p\to p\downarrow$ is inclusive.

Pf: We set $u=s$ and $t=\square$   in the definition of $p\downarrow [u]$ . Since $u=u\sqcup \square$ and $p[u]\sqsupseteq \square$ , it follows that $p[u]=\bigsqcup \{p[s]:u=s\sqcup \square$ and $p[s]\sqsupseteq \square \}\sqsubseteq \bigsqcup \{p[s]:u=s\sqcup t$ and $p[s]\sqsupseteq t\}=p\downarrow [u]$ .

Define ${p}^{\downarrow }$ as the inclusive fixed point (with respect to the operation: $p\to p\downarrow$ ).

1. (2) Suppose that $p\sqsubseteq q$ and that $q[u]\sqsupseteq q[s]$ whenever $u=s\sqcup t$ and $q[s]\sqsupseteq t$ . Then ${p}^{\downarrow}\sqsubseteq q$ .

Pf: It suffices to show ${p}_{\alpha}\sqsubseteq q$ , where the sequence of ${p}_{\alpha }$ ’s is defined as before, but by reference to the operation $p\to p\downarrow$ . The proof is by induction on $\alpha$ . The cases in which $\alpha =0$ or $\alpha =\lambda$ are straightforward, and so let us focus on the case in which $\alpha =\beta +1$ . Then ${p}_{\alpha}={p}_{\beta}{\downarrow}$ , where ${p}_{\beta}\downarrow [u]=\bigsqcup \{{p}_{\beta}[s]:u=s\sqcup t$ and ${p}_{\beta}[s]\sqsupseteq t\}$ . So we need to show that $q[u]\sqsupseteq {p}_{\beta}[s]$ whenever $u=s\sqcup t$ and ${p}_{\beta}[s]\sqsupseteq t$ , i.e., that $q[s\sqcup t]\sqsupseteq {p}_{\beta}[s]$ whenever ${p}_{\beta}[s]\sqsupseteq t$ . But ${p}_{\beta}\sqsubseteq q$ by IH. Hence ${p}_{\beta}[s]\sqsubseteq q[s]$ , and so, given $t\sqsubseteq {p}_{\beta}[s]$ , $t\sqsubseteq q[s].$ By the condition imposed in (2) upon $q$ , $q[s]\sqsubseteq q[s\sqcup t]$ , and so ${p}_{\beta}[s]\sqsubseteq q[s\sqcup t]$ .

1. (3) ${p}^{\downarrow}[s]\sqsupseteq t\Rightarrow {p}^{\downarrow}[s\sqcup t]\sqsupseteq {p}^{\downarrow}[s]$ .

Pf: Suppose ${p}^{\downarrow}[s]\sqsupseteq t$ . Then ${p}^{\downarrow}[s\sqcup t]={p}^{\downarrow}\downarrow [s\sqcup t]\sqsupseteq {p}^{\downarrow}[s]$ by definition of ${}^{\downarrow }$ and $\downarrow$ .

1. (4) The operation $p\to p\downarrow$ is monotonic.

Pf: Suppose $p\sqsubseteq q$ . Then

$$\begin{align*}p\downarrow \left[u\right]&=\bigsqcup \left\{p\left[s\right]:u=s\sqcup t\;\mathrm{and}\;p\left[s\right]\sqsupseteq t\right\}\\&\sqsubseteq \bigsqcup \{q[s]:u=s\sqcup t\;\mathrm{and}\;q[s]\sqsupseteq t\}\\& \quad\ (\mathrm{since\ if}\ p[s]\sqsupseteq t\ \mathrm{then}\ q[s]\sqsupseteq p[s]\sqsupseteq t)\\& =q\downarrow [u].\end{align*}$$

From (2) and (3) it follows that ${p}^{\downarrow }$ is the least plan $q\sqsupseteq p$ conforming to Follow-Down and from (4) it follows that the operation $p\to {p}^{\downarrow }$ is also monotonic.

Lemma 8 (Semi-Inclusion)

For any plan $p$ , there is a least plan ${p}^{\to}\sqsupseteq p$ satisfying the condition

$$\begin{align*}{a}_s\sqsubseteq {p}^{\to}\left[s\right],\end{align*}$$

where the operation $p\to {p}^{\to}$ is monotonic.

Proof. Given a plan $p$ , define its expansion ${p}^{\to }$ by:

$$\begin{align*}{p}^{\to}\left[s\right]=p\left[s\right]\sqcup {a}_s.\end{align*}$$

Clearly, ${p}^{\to}\sqsupseteq p$ and satisfies Semi-Inclusion. It is also the least plan containing p to satisfy Semi-Inclusion. For if $q\sqsupseteq p$ and satisfies Semi-Inclusion, then $p[s]\sqsubseteq q[s]$ and ${a}_s\sqsubseteq q[s]$ and so ${p}^{\to}[s]=p[s]\sqcup {a}_s\sqsubseteq q[s]$ .

Also, if $p\sqsubseteq q$ then ${p}^{\to}[s]=p[s]\sqcup {a}_s\sqsubseteq q[s]\sqcup {a}_s={q}^{\to}[s]$ .

Lemma 9 (Monotonicity)

For any plan $p$ there is a least plan $p\,\check{}\ \sqsupseteq p$ satisfying the condition

$$\begin{align*}s\sqsubseteq t\Rightarrow p\,\check{}\ \left[s\right]\sqsubseteq p\,\check{}\ \left[s\right],\end{align*}$$

where the operation $p\to p\,\check{}$ is monotonic.

We are now in a position to combine our various results. Let us call the various conditions mentioned above the standard conditions. Then:

This result means that if we define the admissible plans ${S}_{\mathrm{P}}$ of a plan space ${\boldsymbol{S}\kern-1.4pt}_{\mathrm{P}}=({S}_{\mathrm{P}},\sqsubseteq ^{+}_{\mathrm{P}}\upharpoonleft {S}_{\mathrm{P}})$ to be those conforming to a combination of standard conditions, then ${\boldsymbol{S}\kern-1.4pt}_{\mathrm{P}}$ will be a state space in which the restricted relation $\sqsubseteq ^{+}_{\mathrm{P}}\upharpoonleft {S}_{\mathrm{P}}$ of part-whole is indeed complete. For given any set $\{{p}_1,{p}_2,\dots \}$ of admissible plans, there will be a least plan conforming to the conditions that contains ${p}_1\sqcup {p}_2\sqcup \cdots$ . Let us also note that, by examining how the standard operations $p\to {p}^{* }$ have been defined, we may easily verify that each of them is bound, i.e., that ${p}^{\ast}\sqsubseteq {p}^{\oplus }$ .

9 Plans, partial plans, and actions

In what follows, I assume given a situation–action space ${\boldsymbol{S}\kern-1.2pt}_{I,A}=({S}_{\mathrm{I}},{S}_{\mathrm{A}},\sqsubseteq )$ and an associated plan space ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{P}}=({S}_{\mathrm{P}},\sqsubseteq )$ . Recall that, for each plan $p$ in $S^{+}_{\mathrm{P}}$ , there will be a smallest plan ${p}^{\ast}\sqsupseteq p$ in ${\boldsymbol{S}\kern-1.4pt}_{\mathrm{P}}$ .

As already noted, there is a natural extension—the completion— ${p}^{+}$ of a partial plan $p$ , which is a function from ${S}_{\mathrm{I}}$ into ${S}_{\mathrm{A}}$ , where, for any state $s$ ,

• ${p}^{+}[s]=p[s]$ when $s\in Dm(p)$ , and

• ${p}^{+}[s]=\square$ otherwise.

Thus ${p}^{+}$ takes the null value $\square$ when $p$ would otherwise be undefined. We may then let the closure ${p}^{* }$ of a partial plan $p$ be ${p}^{+\ast }$ . This definition agrees, of course, with the original definition of ${p}^{* }$ when $p$ is itself a total function.

Given a situation $s$ and an action $a$ , let $s\hookrightarrow a$ be the plan ${\{{<}s,a{>}\}}^{+}$ and let $s\to a$ be the plan ${(s\hookrightarrow a)}^{\ast }$ . Thus $s\hookrightarrow a$ is the plan $p$ such that

• $p[t]=a$ when $t=s$ , and

• $=\square$ otherwise

and $s\to a$ is its closure ${p}^{* }$ . We should think of $(s\to a)$ as the singular plan from $s$ to $a$ . It embodies the instruction to do $a$ in situation $s$ , but subject to closure.

Each closed plan is the fusion of its component singular plans:

We may obtain a natural embedding of actions into plans by letting ${p}_a$ , for each action $a$ , be $\square \hookrightarrow a$ and by letting ${p}_a^{\ast}$ be ${({p}_a\!)}^{\ast }$ ( $=(\square \to a)={(\square \hookrightarrow a)}^{\ast }$ ). Thus each action is identified with the plan of doing that action in the null situation. We might think of ${p}_a^{\ast }$ as being reached in three steps: first, by identifying $a$ with the partial function $\{{<}\square, a{>}\}$ ; then, by identifying the partial function with the total function $(\square \hookrightarrow a)={\{{<}\square, a{>}\}}^{+}$ ; and, finally, by taking the closure $(\square \to a)={(\square \hookrightarrow a)}^{\ast }$ of $(\square \hookrightarrow a)$ .

The map: $a\to {p}_a^{\ast }$ will normally constitute an embedding of the action space ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{A}}$ into the plan space ${\boldsymbol{S}\kern-1.2pt}_{\mathrm{P}}$ :

This result tells us that we may identify an action $a$ with the corresponding plan $p^{*}_{\mathrm{a}}$ and this will enable us to present the semantics for categorical and conditional imperatives exclusively in terms of plans.

10 Core conditionalization

We introduce the key operation of conditionalization on plans—dealing with core non-closed plans in the present section and with closed plans in the following section.

Given a situation $s$ and a plan $p$ , we let the (core) conditionalization $(s\hookrightarrow p)$ of $p$ on $s$ be the plan

$$\begin{align*}\left(s\hookrightarrow p\right)=\bigsqcup \left\{\left(s\sqcup t\right)\hookrightarrow p\left[t\right]:t\in {S}_{\mathrm{I}}\right\}.\end{align*}$$

Thus the conditional plan $s\hookrightarrow p$ is the fusion of the singular plans that send the state $s\sqcup t$ to the action $p[t]$ . We might think of this definition in terms of resetting the value of the default situation from $\square$ to $s$ , so that $(s\hookrightarrow p)$ will take the same value when $s$ is the default as $p$ takes when $\square$ is the default.

Say $t\equiv u\;(\mathit{\operatorname{mod}}\;s)$ if $t\sqcup s=u\sqcup s$ . Clearly $\equiv (\mathit{\operatorname{mod}}\;s)$ is an equivalence relation. We have the following alternative characterizations and elementary properties of $(s\hookrightarrow p)$ :

Use of these results, and of (i) in particular, will often be implicit.

These results shows that the conditional plan $s\hookrightarrow p$ only ‘kicks in’ within a situation containing the antecedent situation $s$ and that, when the plan does kick in, it assumes a cumulative interpretation. The value of $s\hookrightarrow p$ at $s\sqcup t$ in such a case is not merely $p[t]$ but the result of adding to $p[t]$ any other values $p[t^{\prime}]$ for which $s\sqcup t=s\sqcup t^{\prime }$ . We might justify the cumulative interpretation on the grounds that there is in general no way to privilege any particular $t^{\prime }$ for which $s\sqcup t=s\sqcup t^{\prime }$ , and so all such $t^{\prime }$ should be included.

The operation $\hookrightarrow$ has been used in the context $s\hookrightarrow a$ , for $a$ an action, and also in the context $s\hookrightarrow p$ , for $p$ a plan. We may show that the two uses are in conformity with one another:

Proof.

$$\begin{align*}(s\hookrightarrow {p}_a)[t]&=(s\hookrightarrow (\square \hookrightarrow a))[t]\hspace{48pt} \mathrm{by\ definition\ of}\ {p}_a\\ &=\bigsqcup \{(\square \hookrightarrow a)[t^{\prime}]{:}\ s\sqcup t^{\prime}=t\}\ \mathrm{by\ lemma}\ 13(\mathrm{i}).\end{align*}$$

For $t=s$ :

$$\begin{align*}\left(s\hookrightarrow {p}_a\right)\left[s\right]&=\bigsqcup \left\{\left(\square \hookrightarrow a\right)\left[t^{\prime}\right]\!{:}\ s\sqcup t^{\prime}=s\right\}\\&=\bigsqcup \left\{\left(\square \hookrightarrow a\right)\left[t^{\prime}\right]\!{:}\ t^{\prime}\sqsubseteq s\right\}\\&=\left(\square \hookrightarrow a\right)\left[\square \right]\sqcup \bigsqcup \left\{\left(\square \hookrightarrow a\right)\left[t^{\prime}\right]\!{:}\ \square \sqsubset t^{\prime}\sqsubseteq s\right\}\\&=a\sqcup \bigsqcup \left\{\square {:}\ \square \sqsubset t^{\prime}\sqsubseteq s\right\}\\&=a.\end{align*}$$

For $t\ne s$ :

$$\begin{align*}\left(s\hookrightarrow {p}_a\right)\left[t\right]=\bigsqcup \left\{\left(\square \hookrightarrow a\right)\left[t^{\prime}\right]\!{:}\ s\sqcup t^{\prime}=t\right\}.\end{align*}$$

But then the $t^{\prime}\ne \square$ and so $(\square \hookrightarrow a)[t^{\prime}]=\square$ . Hence:

$$\begin{align*}\left(s\hookrightarrow {p}_a\right)\left[t\right]&=\bigsqcup \left\{\square {:}\ s\sqcup t^{\prime}=t\right\}\\&=\square .\end{align*}$$

Thus, in either case $(s\hookrightarrow {p}_a)[t]=(s\hookrightarrow a)[t]$ and hence $s\hookrightarrow {p}_a=s\hookrightarrow a$ .

We establish some basic algebraic properties of conditionalization:

1. (i) $(\square \hookrightarrow p)=p$ .

2. (ii) ${s}_1\hookrightarrow ({s}_2\hookrightarrow p)=({s}_1\sqcup {s}_2)\hookrightarrow p$ .

3. (iii) $s\hookrightarrow ({p}_1\sqcup {p}_2\sqcup \cdots )=(s\hookrightarrow {p}_1)\sqcup (s\hookrightarrow {p}_2)\sqcup \cdots$ .

4. (iv) $s\hookrightarrow {\square}_{\mathrm{P}}={\square}_{\mathrm{P}}$ .

11 Closed conditionalization

We wish to extend the previous results for core conditionalization to closed conditionalization. Recall that, given a closure operation $p\to {p}^{* }$ on plans, we define closed conditionalization by:

$$\begin{align*}\left(s\to p\right){=}_{df}\;{\left(s\hookrightarrow p\right)}^{\ast}\;.\end{align*}$$

Our aim is to extend the results on $\hookrightarrow$ in Lemmas 14 and 15 to $\to$ . To this end, we assume that the closure operation $\ast$ satisfies the following conditionFootnote ⁷

$$\begin{align*}\mathbf{Regularity}\mathbf{:}\ (s\hookrightarrow {p}^{\ast})\sqsubseteq {(s\hookrightarrow p)}^{\ast }.\end{align*}$$

In other words, the external closure of a conditional will contain the internal closure of the conditional, in which we close on the consequent of the conditional rather than on the conditional itself. (We shall later show that the standard closure operations on plans conform to Regularity.)

We first note the analogue of Lemma 14:

Lemma 16. Suppose that $\ast$ is a closure operation conforming to Regularity. Then for any situation $s$ and action $a$ ,

$$\begin{align*}(s\to p^{*}_{\mathrm{a}})=(s\to a).\end{align*}$$

We also have the analogue of Lemma 15, with $\hookrightarrow$ replaced by $\to$ and with the operation $\sqcup $ on plans replaced by the operation $\sqcup^{\ast }$ on closed plans:

1. (i) $(\square \to p)=p$ ;

2. (ii) ${s}_1\to ({s}_2\to p)=({s}_1\sqcup {s}_2)\to p$ ;

3. (iii) $s\to ({p}_1\sqcup^{\ast }{p}_2\sqcup^{\ast}\cdots )=(s\to {p}_1)\sqcup^{\ast}(s\to {p}_2)\sqcup^{\ast}\cdots$ ;

4. (iv) $s\to \square ^{*}_{\mathrm{P}}=\square ^{*}_{\mathrm{P}}$ .

Proof.

1. (i) $(\square \to p)=(\square \hookrightarrow p)$ *

   $\hspace{30pt}=p$ * by Lemma 15(i)

   $\hspace{30pt}=p$ given that $p$ is closed.

2. (ii) For the one direction, observe that:

   $$\begin{align*}({s}_1\sqcup {s}_2)\hookrightarrow p&={s}_1\hookrightarrow ({s}_2\hookrightarrow p)\quad \mathrm{by\ Lemma}\ {15}(\mathrm{ii})\\&\sqsubseteq {s}_1\hookrightarrow ({s}_2\hookrightarrow p)\quad \mathrm{by\ Right\ Monotonicity\ of}\ \hookrightarrow.\end{align*}$$

Hence:

$$\begin{align*}({s}_1\sqcup {s}_2)\to p&=(({s}_1\sqcup {s}_2)\hookrightarrow p)^*\hspace{14pt} \mathrm{by\ the\ definition\ of}\ \to\\&\sqsubseteq ({s}_1\hookrightarrow ({s}_2\hookrightarrow p)^*)^*\ \ \mathrm{by\ the\ previous\ observation\ and\ the}\\&\qquad\qquad \qquad\qquad \qquad \mathrm{Monotonicity\ of}\ \sqcup\\& ={s}_1\to ({s}_2\to p)\hspace{20pt} \ \hspace{2pt} \mathrm{by\ the\ definition\ of}\ \to\end{align*}$$

For the other direction, observe that:

$$\begin{align*}({s}_1\hookrightarrow {({s}_2\hookrightarrow p)}^{\ast})&\sqsubseteq {({s}_1\hookrightarrow ({s}_2\hookrightarrow p))}^{\ast}\quad \mathrm{by\ Regularity}\\&\sqsubseteq {(({s}_1\sqcup {s}_2)\hookrightarrow p)}^{\ast }\ \ \quad \mathrm{by\ Lemma}\ {15}(\mathrm{ii})\end{align*}$$

Hence:

$$\begin{align*} ({s}_1\to ({s}_2\to p))&={({s}_1\hookrightarrow {({s}_2\hookrightarrow p)}^{\ast})}^{\ast }\quad \mathrm{by\ definition}\\&\sqsubseteq {(({s}_1\sqcup {s}_2)\hookrightarrow p)}^{\ast \ast }\quad \ \ \, \mathrm{by\ the\ previous\ observation\ and\ the}\\ &\qquad\qquad \qquad\qquad \qquad\ \ \mathrm{Monotonicity\ of}\ \ast\\&={(({s}_1\sqcup {s}_2)\hookrightarrow p)}^{\ast }\\&= ({s}_1\sqcup {s}_2)\to p\qquad\quad\ \ \mathrm{by\ definition}. \end{align*}$$

1. (iii) For the one direction, observe that:

   $$\begin{align*}s\hookrightarrow {({p}_1\sqcup {p}_2\sqcup \cdots )}^{\ast}&\sqsubseteq {(s\hookrightarrow ({p}_1\sqcup {p}_2\sqcup \cdots ))}^{\ast }\qquad\quad \ \ \ \, \mathrm{by\ Regularity}\\&={((s\hookrightarrow {p}_1)\sqcup (s\hookrightarrow {p}_2)\sqcup \cdots )}^{\ast }\quad \mathrm{by\ Lemma}\ {15}(\mathrm{iii})\ \&\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \mathrm{Monotonicity}\\&\sqsubseteq {({(s\hookrightarrow {p}_1)}^{\ast}\sqcup {(s\hookrightarrow {p}_2)}^{\ast}\sqcup \cdots )}^{\ast }\\&=(s\to {p}_1)\sqcup^{\ast}(s\to {p}_2)\sqcup \ast \cdots .\end{align*}$$

Hence:

$$\begin{align*}s\to ({p}_1\sqcup^{\ast}{p}_2\sqcup^{\ast}\cdots )&={(s\hookrightarrow ({p}_1\sqcup^{\ast }{p}_2\sqcup^{\ast}\cdots ))}^{\ast }\\&={(s\hookrightarrow ({p}_1\sqcup {p}_2\sqcup \cdots ))}^{\ast}){}^{\ast }\\&\sqsubseteq {((s\to {p}_1)\sqcup^{\ast}(s\to {p}_2)\sqcup^{\ast}\cdots )}^{\ast }\quad \mathrm{by\ the\ previous\ observation}\\&=((s\to {p}_1)\sqcup^{\ast}(s\to {p}_2)\sqcup^{\ast}\cdots ).\end{align*}$$

For the other direction, observe that:

$$\begin{align*}s\hookrightarrow {p}_k&\sqsubseteq s\hookrightarrow ({p}_1\sqcup {p}_2\sqcup \cdots )\quad \mathrm{by\ Right\ Monotonicity\ of}\ \hookrightarrow\\&\sqsubseteq s\hookrightarrow ({p}_1\sqcup^{\ast }{p}_2\sqcup^{\ast}\cdots )\end{align*}$$

and so ${(s\hookrightarrow {p}_k)}^{\ast}\sqsubseteq {(s\hookrightarrow ({p}_1\sqcup^{\ast }{p}_2\sqcup^{\ast}\cdots ))}^{\ast }$ .

Hence:

$$\begin{align*}(s\to {p}_1)\sqcup^{\ast}(s\to {p}_2)\sqcup^{\ast}\cdots &={(s\hookrightarrow {p}_1)}^{\ast }\sqcup^{\ast }{(s\to {p}_2)}^{\ast }\sqcup^{\ast}\cdots\ \mathrm{by\ definition}\\&\sqsubseteq {(s\hookrightarrow ({p}_1\sqcup^{\ast }{p}_2\;\sqcup^{\ast}\cdots ))}^{\ast}\quad\quad\ \ \mathrm{by\ previous\ observation}\\&=s\to ({p}_1\sqcup^{\ast }{p}_2\sqcup^{\ast}\cdots ).\end{align*}$$

1. (iv) For the one direction,

   $$\begin{align*}(s\to \square ^{*}_{\mathrm{P}})&={(s\hookrightarrow {(\square_{\mathrm{P}})}^{\ast})}^{\ast }\\&\sqsubseteq {(s\hookrightarrow \square_{\mathrm{P}})}^{\ast \ast }\quad \mathrm{by\ Regularity}\\&={(s\hookrightarrow \square_{\mathrm{P}})}^{\ast }\\& =\square ^{*}_{\mathrm{P}}\qquad\qquad\ \ \ \mathrm{by\ Lemma}\ {15}(\mathrm{iv}).\end{align*}$$

For the other direction, note that $\square ^{*}_{\mathrm{P}}\sqsubseteq {(s\hookrightarrow \square ^{*}_{\mathrm{P}})}^{\ast}=(s\to \square ^{*}_{\mathrm{P}})$ since $\square ^{*}_{\mathrm{P}}\sqsubseteq {q}^{\ast}$ for any plan $q$ .

We now develop a method by which we can show that an operation on plans satisfies Regularity.

Lemma 18. Let ${<}{\boldsymbol{F}}_{\alpha }{>}$ be a sequence of inclusive monotonic operations on plans and, relative to any given plan $p$ , ${<}{p}_{\alpha }{>}$ the corresponding sequence of plans defined by reference to ${<}{\boldsymbol{F}}_{\alpha }{>}$ (as in Section 8). Suppose each ${\boldsymbol{F}\!}_{\alpha}$ conforms to Regularity. Then for each ordinal $\alpha$ :

$$\begin{align*}\left(s\hookrightarrow {p}_{\alpha}\right)\sqsubseteq {\left(s\hookrightarrow p\right)}_{\alpha }. \end{align*}$$

Proof. By induction on $\alpha$ .

$$\begin{align*}&{\boldsymbol{\alpha = 0{:}}}\ (s\hookrightarrow {p}_0)=(s\hookrightarrow p)={(s\hookrightarrow p)}_0.\\&\boldsymbol{\alpha = \beta+1{:}}\ (s\hookrightarrow {p}_{\beta +1})=(s\hookrightarrow {\boldsymbol{F}\!}_{\beta}({p}_{\beta}))\\&\kern100pt\sqsubseteq {\boldsymbol{F}\!}_{\beta}(s\hookrightarrow {p}_{\beta})\quad\ \mathrm{since}\ {\boldsymbol{F}\!}_{\beta }\ \mathrm{conforms\ to\ Regularity}\\&\kern100pt\sqsubseteq {\boldsymbol{F}\!}_{\beta}({(s\hookrightarrow p)}_{\beta})\ \ \ \mathrm{by\ IH\ given\ that}\ {\boldsymbol{F}\!}_{\beta }\ \mathrm{is\ monotonic}\\&\kern100pt={(s\hookrightarrow p)}_{\beta +1}\\&\boldsymbol{\alpha=\lambda{:}}\ (s\hookrightarrow {p}_{\lambda})=(s\hookrightarrow \bigsqcup \{{p}_{\zeta }{:}\ \zeta <\lambda \})\\&\kern72pt=\bigsqcup (s\hookrightarrow {p}_{\zeta}){:}\ \zeta <\lambda \}\quad \mathrm{by\ lemma}\ {15}(\mathrm{iii})\\&\kern72pt\sqsubseteq \bigsqcup \{{(s\hookrightarrow p)}_{\zeta }{:}\ \zeta <\lambda \}\ \ \ \mathrm{since,\ by\ IH,}\ (s\hookrightarrow {p}_{\zeta})\sqsubseteq {(s\hookrightarrow p)}_{\zeta }\\&\kern72pt={(s\hookrightarrow p)}_{\lambda }\\[-34pt]\end{align*}$$

Proof. We compute $(s\hookrightarrow ({p} \downarrow))[t]$ and $((s\hookrightarrow p)\downarrow )[t]$ .

$$\begin{align*}s\hookrightarrow ({p} \downarrow))[t]&=\bigsqcup \{({p} \downarrow)[u]{:}\ s\sqcup u=t\}\\&=\bigsqcup \{\;\bigsqcup \{p[v]{:}\ u=v\sqcup w\;\mathrm{and}\;w\sqsubseteq p[v]\}{:}\ s\sqcup u=t\}\\&=\bigsqcup \{p[v]{:}\ u=v\sqcup w,w\sqsubseteq p[v]\;\mathrm{and}\;s\sqcup u=t\}\\&=\bigsqcup \{p[v]{:}\ s\sqcup v\sqcup w=t\;\mathrm{and}\;w\sqsubseteq p[v]\}.\\({(s\hookrightarrow p)} \downarrow)[t]\kern0.36em &=\bigsqcup \{(s\hookrightarrow p)[u]{:}\ t=u\sqcup w\;\mathrm{and}\;w\sqsubseteq (s\hookrightarrow p)[u]\}\\&=\bigsqcup \{\;\bigsqcup \{p[v]{:}\ s\sqcup v=u\}{:}\ t=u\sqcup w\;\mathrm{and}\;w\sqsubseteq (s\hookrightarrow p)[u]\}\\&=\bigsqcup \{p[v]{:}\ s\sqcup v=u,t=u\sqcup w\;\mathrm{and}\;w\sqsubseteq (s\hookrightarrow p)[u]\}\\&=\bigsqcup \{p[v]{:}\ t=s\sqcup v\sqcup w\;\mathrm{and}\;w\sqsubseteq (s\hookrightarrow p)[s\sqcup v]\}\\&=\bigsqcup \{p[v]{:}\ t=s\sqcup v\sqcup w\;\mathrm{and}\;w\sqsubseteq \bigsqcup \{p[x]{:}\ s\sqcup x=s\sqcup v\}\}.\end{align*}$$

But it is then clear that $(s\hookrightarrow ({p} \downarrow))[t]\sqsubseteq ({(s\hookrightarrow p)} \downarrow)[t]$ . For $p[v]\sqsubseteq \bigsqcup \{p[x]{:}\ s\sqcup x=s\sqcup v\}$ and so, given that $w\sqsubseteq p[v]$ , $w\sqsubseteq \bigsqcup \{p[x]{:}\ s\sqcup x=s\sqcup v\}\}$ .

Proof.

$$\begin{align*}(s\hookrightarrow {p}^{\to})[t]&=\bigsqcup \{{p}^{\to}[u]{:}\ s\sqcup u=t\}\\&=\bigsqcup \{p[u]\sqcup {a}_u{:}\ s\sqcup u=t\}\\&=\bigsqcup \{p[u]{:}\ s\sqcup u=t\}\sqcup \bigsqcup \{{a}_u{:}\ s\sqcup u=t\}\\&=(s\hookrightarrow p)[t]\sqcup \bigsqcup \{{a}_u{:}\ s\sqcup u=t\},\end{align*}$$

while ${(s\hookrightarrow p)}^{\to}[t]=(s\hookrightarrow p)[t]\sqcup {a}_t$ .

But $\bigsqcup \{{a}_u{:}\ s\sqcup u=t\}=\square$ when and $\bigsqcup \{{a}_u{:}\ s\sqcup u=t\}={a}_t$ when $t\sqsupseteq s$ . Thus in either case,

$$\begin{align*}\left(s\hookrightarrow p\right)\left[t\right]\sqcup \bigsqcup \left\{{a}_u{:}\ s\sqcup u=t\right\}\sqsubseteq \left(s\hookrightarrow p\right)\left[t\right]\sqcup {a}_t.\\[-34pt] \end{align*}$$

The case for Monotonicity is somewhat more straightforward since we can then show:

We have so far distinguished between $\sqcup $ and $\sqcup^{\ast}$ and between $\square ^{*}_{\mathrm{P}}$ and $\square_{\mathrm{P}}$ , but from now on, when only closed plans are under consideration, we shall use $\sqcup $ in place of $\sqcup^{\ast}$ and $\square ^{*}_{\mathrm{P}}$ in place of $\square_{\mathrm{P}}$ .