## A logic and semantics for imperativesIan Williams Goddard
1. Introduction
1.1 The problem Formal semantics defines criteria for evaluating the truth of declarative statements with respect to domains of discourse. However, imperative statements like “Shut the door!” are not obviously true or false in any domain and therefore fall outside the realm of truth-valuable statements. Without truth values imperative arguments cannot be shown to be truth preserving, or semantically valid, even if they are intuitively valid. As such, it is held that imperative arguments, which form a large body of everyday arguments, fall outside the scope of formal logical reasoning. This paper proposes to bring them in scope.
1.2 The path followed Given their lack of truth values, proposed imperative logics often define alternatives to truth. An important example is Anthony Kenny's substitution of being Kenny's Translating imperatives into declaratives to express their meaning is not new. For example, H. G. Bohnert proposed: “There exists a set of grammatically declarative sentences which can be put in one-to-one correspondence with commands.” He posited such a mapping wherein any command A can be translated into a declarative of the form
2 The Kenny's premise that an imperative expresses an imperator's wish for a goal state is my premise too. By this premise, what an imperator It should be noted that Hare has observed that imperatives can be translated into declaratives denoting what an imperator
To flash forward briefly, the semantic structure in Figure 1 shall be explicitly implemented with modal operators that denote Wishes and possible Changes above. Given a set of conceivable states of affairs S = { P(S´S), whose members are sets of state pairs. So for example, suppose for simplicity that Wishes(a_{1}) = {(s, u)}, then the set of wishes of agent a_{1} contains one wish (s, u) that means: in state s agent a_{1} wants state u. Changes shares the same structure except that if Changes(a_{1}) = {(s, u)}, then (s, u) means: in state s agent a_{1} can cause state u. This semantic structure is illustrated below in Figure 2 showing a subset tree of P(S´S).
The objection might be raised that (
The language wants and cause. is similar in construction to epistemic logics. For example, Fagin Let al.^{(8)} define knowledge operators indexed to intelligent agents such that for n agents there are K_{1}, … , K operators where each _{n}K means “Agent _{i}i knows” and so Kφ means “Agent _{i}i knows φ ” where φ is a proposition variable. We'll also define modal wants and cause operators that are specific to agents. Let us then begin with a generative grammar for .LDEFINITION 1 ( vocabulary = á P, N, U, B, M, A ñ composed of six sets of atomic propositions P = { p, p', p, …}, ''names N = { n_{1}_{}, … , n }, _{n}unary connectives U = { Ø }, binary connectives B = { ® , Ù , Ú }, modalities M = { [ω], áωñ, [c], ácñ }, and auxiliary symbols { (, ) } the formulae of form the smallest set F such that:L
The syntactic structure for the proxy-imperatives we'll define appears in 1.4 above. The ω modality means
Other translations are possible. Instead of 'must have' in mode 1 above we could say 'requires'. As for mode 2 above, 'accepts' may in some cases be replaced with 'likes', and so 'loves' might replace 'must have' in mode 1 since Both
So for example, á•ñ
2.2 Proxy-imperative schemata From the modes of
These are the statement schemata we'll use for proxy-imperatives. It's essential to note that they do Schema 1 denotes conditions underlying the strongest imperatives, n, n' Î N, and any φ Î F we accept as true:
In the case of
2.3 Proxy-imperative behavior Now we'll compare proxy-imperatives with real imperatives. First, observe that because the minimal mode of wanting áωñ denotes what is Ø[ω] That equivalence implies that the negation of “Shut the door!” is not the contrary command “Don't shut the door!” but the contrary permit: “You may leave the door open.” So according to the proxy-imperatives of repeals the command and permits contrary behavior. This in fact matches natural commands of which public laws are canonical. Take for example the military draft. What happens when we repeal a command by a leader that any man, let's say Jon, must enlist? Let's see (the proposition p that's commanded to be made true is 'Jon is enlisted.')
So the negation of command
So according to both our proxy-imperatives and natural intuition, repealing a draft's command “Enlist!” does not mean “
2.4 A proxy-imperative semantics And now let's explore the meaning, or semantics, of that defines a frame of objects and relations between them from which domains of discourse can be built and in which, by way of an interpretation, the statements of L have their meaning. Here then is such a model for L. LDEFINITION 2 ( = á S, A, Wishes, Changes, α , V ñ whereMá S, A, Wishes, Changes ñ is a domain frame and á α, V ñ is an interpretation for :L
The wanting casts a wider net over states than alethic possibility given that one can want the impossible. For example, you could want to be as big as a mountain or to travel in time, but such conceivable states are not possible states. So in the frame the set of states S contains Lconceivable states that may be impossible but sill wantable. On the other hand, Changes does assign access relations to agents. For all a Î A and all s, s' Î S, if (s, s' ) Î Changes(a), then state s' is possible from state s, and perhaps because agent a can cause s' . Conceivable states are plausibly infinite and possible states are a proper subset of S. 2.5 defines a name-assignment function α such that for any name DEFINITION 3 ( , the Mtruth conditions in any state s Î S are (where (s) φ is read: in state s, φ is true):
By Axiom 1, in any , if (Ms) [ω]n (φ), then (s) áωñn (φ). So by Definitions 3.6 and 3.7, every agent wants at least one conceivable state. Otherwise, [ω]n (φ) can be vacuously true by Definition 3.6 when agent α(n) wants no state. This serial condition also blocks vacuous truth in alethic modal logic and applies to the cause modality such that every agent can cause at lest one state. Axiom 1 is intuitively valid as well.^{(10)} Definitions 3.6 through 3.9 are unique and implement my thesis.Definitions for the proxy-imperatives follow directly from Definitions 3.6 through 3.9. However, it's worth presenting them explicitly. They are for brevity presented in meta-logic rather than the meta-language of English used for Definitions 3.6 - 3.9. DEFINITION 3 (amendment -
Figure 3 below extends Figure 2 by articulating the mapping on
3 By proxy semantic validation of imperative argument Now we put our proxy-imperatives to work to provide semantic proofs by proxy for imperative arguments. We assume that any meaningful imperative has an imperator and thus that there is at least one agent who p = 'You see Jesse' and q = 'The police are notified '.
PROOF: By 1b we assume that in model s) p ® [ω]i [c]n (q). By Definitions 3.1 and 3.10 this means that we accept as true that if state s Î V(p), then for all s' , s'' Î S, if (s, s' ) Î Wishes(α(i)) and (s', s'' ) Î Changes(α(n)), then (s'' ) q. Now, by 2b we have it as a fact that state s Î V(p); therefore, by assumption 1b and Definition 3.10 we also have it as a fact that for all conceivable states s' and s'' , if agent α(i) wants state s' and in state s' agent α(n) must cause state s'', then in state s'' proposition q is true, which is to say by Definition 3.10 again that we have it that: (s) [ω]i [c]n (q). £ Since perhaps most imperative arguments can be expressed in
4 Conclusion The goal of this project has been to understand the semantic structure of natural imperatives and from such insight build a formal model of imperative semantics that can integrate imperatives into classical logic. So matching the behavior of natural imperatives has been both a goal and guide. Following the frequented path of defining alternatives to truth values in a new kind of logical system used only for imperatives was not an attractive option. My goal has been to facilitate semantic evaluation of imperatives within
(1) Kenny, A. J. (1966). Practical Inference. (2) Geach, P. T. (1966). Dr. Kenny on Practical Inference. (3) Gombay, A. (1967). What is imperative inference? (4) Bohnert, H. G. (1945). The Semiotic Status of Commands. (5) Hare, R. M. (1949). Imperative Sentences. (6) Most research on imperative logic was done between the 1930s and 1970s. For a short review of some recent work let me suggest: B. Žarnić's (7) Hare, R. M. (1952). (8) Fagin, R., Halpern, J.Y., Moses, Y., & Moshe Y. Vardi. (9) Definition 2.5 might seem to add excessive semantic machinery, however, it better segregates et al the number of modal operators K_{1}, … , K in the language reflects the number of agents _{n}n in the domain.^{(4)} But the number of modal operators in syntax is independent of the number of agents in the domain. This abstracts the L modalities from domains. In natural thought, Lwanting and causing, as well as knowing, are concepts we've abstracted from our domains of experience such that we can conceive of them independent of specific instances. And so in natural language, wants, cause, and knows are atomic operators rather than Adam wants, Amy wants, … and so on, per person. For these and other reasons, I feel that the extra semantic machinery better models natural semantics. (10) If you
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