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**NEW FOUNDATIONS FOR IMPERATIVE LOGIC I: Logical connectives, consistency, and quantifiers**

Peter B. M. Vranas University of Wisconsin-Madison Talk at the University of Warsaw, 14 May 2012

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INTRODUCTION There is little current or recent work on imperative logic. This is probably because earlier on some well-known philosophers: have tried but failed to develop an adequate imperative logic (Rescher 1966, Sosa 1967); have argued that imperative logic is impossible (Williams 1963); have argued that imperative logic is iso-morphic to standard logic (Castañeda 1975). I hope to resurrect imperative logic. Today you will assist at the birth of a new branch of logic, namely imperative logic. By calling imperative logic a new branch of logic I don’t mean to suggest that no previous work on imperative logic exists. On the contrary, a lot of previous work exists. But most of this work is quite old, having been published at the latest in the 1970s. Nowadays, few people work on imperative logic. Indeed, in the massive, eighteen-volume second edition of the Handbook of philosophical logic, there is no chapter on imperative logic. What explains this lack of current or recent work on imperative logic? I think that people have given up on the subject because most of the previous work has been negative, in three ways. First, some well-known philosophers, most notably Nicholas Rescher and his former student Ernest Sosa, have tried but have by their own admission failed to develop an adequate imperative logic. Second, some other well-known philosophers, most notably Bernard Williams, have denied the very possibility of imperative logic. And third, some other well-known philosophers, most notably Hector-Neri Castañeda, have argued that imperative logic is isomorphic to standard declarative logic, and this has led many people to conclude that imperative logic is uninteresting. Today I hope to show that this pessimism about imperative logic is unwarranted. I plan to argue that imperative logic is both possible and interesting. And I hope that my work will inspire a renaissance of the subject.

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**WHAT IS IMPERATIVE LOGIC?**

Distinguish imperative sentences from what they typically express, namely prescriptions: com-mands, requests, instructions, suggestions, … English and French imperative sentences can express the same prescription. Declarative sentences can express prescriptions (“You will open the door”). Imperative sentences can express propositions (“Marry in haste and repent at leisure”). Imperative logic is the logic of prescriptions. To start with, let’s get clear on what we are talking about. What is imperative logic? Obviously, it’s the logic of imperatives. But what is an imperative? It’s important to distinguish imperative sentences, like “open the door”, from what such sentences typically express, namely commands, requests, instructions, suggestions, and so on. It’s helpful to have a generic term for all these entities that imperative sentences typically express, so I will call them prescriptions. You can also call then imperatives if you like, but this might lead one to confuse them with imperative sentences. The reasons for distinguishing imperative sentences from prescriptions are familiar. As you know, an English declarative sentence and its French translation normally express the same proposition; similarly, an English imperative sentence and its French translation normally express the same prescription. Moreover, a declarative sentence can express a prescription; for example, an officer who says to a soldier “you will open the door” is normally issuing an order. Conversely, an imperative sentence can express a proposition; for example, the imperative sentence “marry in haste and repent at leisure” normally expresses a proposition, to the effect that if you marry in haste you will repent at leisure. So I hope that the distinction is clear. I take declarative logic to be primarily concerned with propositions, and only secondarily with declarative sentences. Similarly, I take imperative logic to be primarily concerned with prescriptions, and only secondarily with imperative sentences. I realize that some people may disagree, but the disagreement would be inessential for my purposes; although I will formulate my results in terms of prescriptions, they can be easily reformulated in terms of imperative sentences. To my mind, however, this would result in a loss of precision.

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**OVERVIEW Part 1: A MODEL OF PRESCRIPTIONS Part 2: LOGICAL CONNECTIVES**

(Negation, conjunction, disjunction, conditional, and biconditional) Part 3: CONSISTENCY AND QUANTIFIERS At this stage we have a preliminary answer to the question of what imperative logic is: it’s the logic of prescriptions, and prescriptions are what imperative sentences typically express, namely commands, requests, instructions, suggestions, and so on. But this answer by itself does not get us very far. We need to say more on what prescriptions are. Indeed, the main conceptual innovation of my work consists in proposing a simple, powerful, and novel model of prescriptions. I propose this model in the first main part of my talk. In the second part I propose and defend definitions of logical connectives for imperative logic: the negation of a prescription, the conjunction of two prescriptions, and so on. In the third part I examine the consistency of a set of prescriptions and I also deal with quantified prescriptions.

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**A MODEL OF PRESCRIPTIONS**

Any prescription (“Run”) has a satisfaction proposition (“You run”) and a violation proposition (“It is not the case that you run”). “If it rains, run” is (1) satisfied if it rains and you run, (2) violated if it rains and it is not the case that you run, (3) avoided if it doesn’t rain. So the violation proposition is not always the negation of the satisfaction proposition, and in general we need two propositions to model a prescription: A prescription is an ordered pair of logically incompatible propositions. So let’s start with the first part, namely my model of prescriptions. My starting point is the observation that prescriptions can be satisfied or violated. For example, the prescription that I express by addressing to you the imperative sentence “run” is satisfied if you run and is violated if it is not the case that you run. Call the proposition that you run the satisfaction proposition of the prescription, and call the proposition that it is not the case that you run the violation proposition of the prescription. More generally, to any given prescription correspond two incompatible propositions: its satisfaction proposition, which is true exactly if the prescription is satisfied, and its violation proposition, which is true exactly if the prescription is violated. The two propositions are incompatible because it is impossible for a prescription to be both satisfied and violated. In the example I just gave, the violation proposition (it is not the case that you run) is the negation of the satisfaction proposition. But this is not always so. Take the conditional prescription expressed by “if it rains, run”. If it rains and you run, this prescription is satisfied. If it rains and it is not the case that you run, the prescription is violated. But if it doesn’t rain the prescription is neither satisfied nor violated; we can say that it is then avoided. So in general we need two propositions to model a prescription; one proposition is not enough, since the violation proposition is not always the negation of the satisfaction proposition. Here is then my model of prescriptions: a prescription is an ordered pair of logically incompatible propositions, namely the satisfaction proposition and the violation proposition of the prescription.

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AN ANALOGY Descartes identified points in the plane with ordered pairs of numbers. This enabled one to do geometry by using tools from algebra. I identify prescriptions with ordered pairs of propositions. This enables one to do imperative logic by using tools from declarative logic. Descartes did not claim that points are identical with pairs of numbers. Similarly, I am not claiming that prescriptions are identical with pairs of propositions. My model of prescriptions is so simple that it can take a while for its importance to sink in. It may help to consider an analogy: my identification of prescriptions with ordered pairs of propositions is analogous to Descartes’s identification of points in the plane with ordered pairs of numbers. As you know, Descartes’s idea enabled one to do geometry by using tools from algebra, and so it marked the birth of analytic geometry. Similarly, my idea enables one to do imperative logic by using tools from declarative logic, and so I hope it will mark the birth, or if you will the rebirth, of imperative logic. Descartes did not claim that points in the plane are identical with pairs of numbers; similarly, I am not claiming that prescriptions are identical with pairs of propositions. I am rather claiming that for the purposes of logic prescriptions can be identified with, or modeled as, pairs of propositions. So I am not primarily addressing the question of what prescriptions really are, what their nature is. I am rather primarily addressing the question of what prescriptions are like, what their structure is. To give another analogy, I am proceeding like those mathematicians who identify the number zero with the empty set without thereby committing themselves to the claim that the number zero is identical with the empty set.

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**CONDITIONAL PRESCRIPTIONS**

A prescription is unconditional if its satisfaction and violation propositions are contradictories, and is conditional otherwise. Why not say that “If it rains, run” is satisfied (rather than avoided) if it doesn’t rain? Because then it would be identical with “Let it be the case that if it rains you run”. But the two prescriptions are distinct. Cf. bet-ting that “If it rains, you run” is true vs betting, on the condition that it rains, that you run. One reason why I take my model of prescriptions to be important is that it allows a unified treatment of unconditional and conditional prescriptions. A prescription is unconditional exactly if its satisfaction and violation propositions are contradictories: the one is the negation of the other, so the prescription must be either satisfied or violated. On the other hand, a prescription is conditional exactly if its satisfaction and violation propositions are not contradictories. As we saw, a conditional prescription can be avoided, namely neither satisfied nor violated; for example, the prescription expressed by “if it rains, run” is avoided if it doesn’t rain. But why not say instead that this prescription is satisfied if it doesn’t rain? Because then this prescription would be identical with the unconditional prescription, expressed by “let it be the case that if it rains you run”, whose satisfaction proposition is the material conditional expressed by “if it rains, you run”. But we want to distinguish the two prescriptions. To see why, consider an analogy with conditional bets. Betting that the material conditional expressed by “if it rains, you run” is true differs from betting, on the condition that it rains, that you run. Indeed, if it doesn’t rain, then a person who bets that the material conditional is true wins, but a person who bets on the condition that it rains neither wins nor loses. A conditional order is like a conditional bet: if I order you to run if it rains and it doesn’t rain, then my order is neither obeyed nor disobeyed. Similarly for conditional prescriptions in general. These remarks have an important consequence. Since conditional prescriptions can be satisfied, violated, or avoided, prescriptions in general are three-valued. But propositions in standard logic are two-valued: they must be either true on false. So here we have a first reason why imperative logic is not isomorphic to standard logic, contrary to what, as I said, some well-known philosophers have argued.

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PREVIOUS MODELS Motivation of previous models: “Run” and “You run” have a common element. According to R. M. Hare (1952): “Run” = “Your running, please” “You run” = “Your running, yes” Common element (“your running”): phrastic Different element (“please”/ “yes”): neustic But then “If it rains, run” = “Your running if it rains, please” = “Let it be the case that if it rains you run”, so Hare’s model is inadequate. The above considerations also show that previously proposed models of prescriptions are inadequate. Previously proposed models of prescriptions are motivated by the plausible idea that for example the prescription expressed by “run” and the proposition expressed by “you run” have a common element. According to Richard Hare, the above prescription could also be expressed by “your running, please”, and the above proposition could also be expressed by “your running, yes”. Hare called the element that is shared by the prescription and the proposition, namely “your running”, the phrastic, and called the element that differentiates the prescription from the proposition, namely “please” or “yes”, the neustic (or, in later publications, the tropic). So Hare proposed that a prescription consists of a phrastic and an imperative neustic, whereas a proposition consists of a phrastic and an indicative neustic. (Strictly speaking, Hare was talking about sentences, not about prescriptions or propositions.) If it’s unclear to you what exactly a phrastic or a neustic is, that’s good: it’s also unclear to me. A lot of ink has been spilled in attempts to make these concepts precise, and several variants of Hare’s model of prescriptions have been proposed. Apart from their lack of clarity or precision, these models face the fundamental problem of being unable to properly account for conditional prescriptions. Indeed, according to Hare’s model, the conditional prescription expressed by “if it rains, run” could also be expressed by “your running if it rains, please”, and is thus identical with the unconditional prescription expressed by “let it be the case that if it rains you run”. But as we saw the two prescriptions are distinct, so Hare’s model is inadequate. Note also that my model nicely accounts for the plausible idea that the prescription expressed by “run” and the proposition expressed by “you run” have a common element. The common element is the proposition itself, which is the satisfaction proposition of the prescription, and is thus the first member of the ordered pair of propositions that constitutes the prescription.

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**HOW MANY PRESCRIPTIONS?**

Every prescription is an ordered pair of incompatible propositions. But does the converse hold? Is every ordered pair of incompatible propositions a prescription? Yes: the pair <S, V> is the prescription expressed by “If S or V is true, let S be true”. The satisfaction proposition of this prescription is: (S or V) and S. This is just S, since S and V are incompatible. The violation proposition is: (S or V) and ~S. This is just V. I hope by now you are convinced that every prescription is, or at least can be fruitfully identified with, an ordered pair of incompatible propositions. But does the converse hold? Is every ordered pair of incompatible propositions a prescription? I believe so: any pair <S, V> of incompatible propositions can be identified with the prescription expressed by “if S or V is true, let S be true”. Indeed, the satisfaction proposition of the prescription is: (S or V) and S. This is just S, since S and V are incompatible. Similarly, the violation proposition is: (S or V) and not S. This is just V. One might argue, however, that some pairs of incompatible propositions do not correspond to prescriptions. For example, some people have argued that there are no prescriptions about the past. I disagree: I take the sentence “let it be the case that my son survived yesterday’s battle” to express the unconditional prescription whose satisfaction proposition is the proposition that my son survived yesterday’s battle. But the disagreement is inessential for my purposes: if you want, you can take for example only pairs of incompatible propositions which are not about the past to be prescriptions. What matters is that every prescription is a pair of incompatible propositions; I am not committed to the converse, although as said I believe that the converse is true.

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TERMINOLOGY I will talk interchangeably of e.g. satisfaction propositions and satisfaction sets. Context (C = S V): union of satisfaction and violation sets. (Context of “If it rains, run”: “It rains”.) Avoidance set: complement of context. (“It doesn’t rain”.) The satisfaction, violation, and avoidance sets partition logical space, so to specify a prescrip-tion it suffices to specify any two of them: the third is the complement of the union of the two. I would like now to introduce some useful terminology. From now on I will understand propositions as sets; for example, sets of possible words. So I will talk interchangeably about for example the satisfaction proposition and the satisfaction set of a prescription. Those who object to identifying propositions with sets can just translate what I will say from a language of sets to a language of propositions. Let the context of a prescription be the union of its satisfaction and violation sets; in other words, the disjunction of its satisfaction and violation propositions. Let the avoidance set of a prescription be the complement, in other words the negation, of its context. If a prescription is unconditional, then its context is necessary and its avoidance proposition is impossible. For example, the context of the unconditional prescription expressed by “run” is the necessary proposition that either you run or it is not the case that you run. On the other hand, the context of the conditional prescription expressed by “if it rains, run” is the proposition that it rains, and the avoidance proposition of this prescription is the proposition that it doesn’t rain. Note that the satisfaction, violation, and avoidance sets of a prescription partition logical space: they are mutually exclusive and collectively exhaustive. So to specify a prescription it is enough to specify any two of the three sets: the third is the complement of the union of the two.

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**(Negation, conjunction, disjunction, conditional, and biconditional)**

PART 2 Part 1: A MODEL OF PRESCRIPTIONS Part 2: LOGICAL CONNECTIVES (Negation, conjunction, disjunction, conditional, and biconditional) Part 3: CONSISTENCY AND QUANTIFIERS This completes the first main part of my talk. I turn now to the second part, in which I define logical connectives for prescriptions.

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**Definition 1 (Negation): ~<S, V> = <V, S>**

Negation of “Run”: “Don't run”. Negation of “If it rains, run”: “If it rains, don't run”. Satisfaction table for negation: Definition 1 (Negation): ~<S, V> = <V, S> I and ~I have the same context. Double nega-tion holds: ~(~<S, V>) = ~(<V, S>) = <S, V>. I = <S, V> Sat. Av. Viol. ~I = <V, S> Let’s start with negation. The negation of “run” is clearly “don’t run”. This simple example already enables us to partially fill in the satisfaction table for negation A satisfaction table is analogous to a truth table: for each of the three possible satisfaction values of a negated prescription I, namely satisfaction, avoidance, and violation, the table gives us the satisfaction value of the negation, tilde I. Suppose first that the negated prescription, “run”, is satisfied. Then you run. But then the negation, which is “don’t run”, is violated. Suppose next that the negated prescription is violated. Then you don’t run. But then the negation, which is “don’t run”, is satisfied. Now what if the negated prescription is avoided? Here our example is of no help, since the prescription expressed by “run” is unconditional and thus cannot be avoided. So take another example: the conditional prescription expressed by “if it rains, run”. In other words, “run if it rains”. Its negation is “don’t run if it rains”. In other words: “if it rains, don’t run”. Suppose the negated prescription is avoided. Then it doesn’t rain. But then the negation is also avoided. So this is the complete satisfaction table for negation. And this is the formal definition: if a prescription I is the ordered pair of a satisfaction set S and a violation set V, then its negation, tilde I, is the ordered pair with satisfaction set V and violation set S. In other words, to negate you just exchange S and V. This concept of negation has the nice property of satisfying the law of double negation: the negation of the negation of a given prescription is the given prescription. Here is a half-line proof.

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**CONJUNCTION “Run” & “Smile” = “Run and smile”.**

“If it rains, run” & “If it rains, smile” = “If it rains, run and smile”. “If it rains, run” & “If it doesn't rain, run” = “Run (whether or not it rains)”. Satisfaction table for conjunction: I & I' Sat. Av. Viol. Let’s turn now to conjunction. The conjunction of “run” with “smile” is clearly “run and smile”. This simple example already enables us to partially fill in the satisfaction table for conjunction. Suppose first that both conjuncts are satisfied. Then you run, and you smile. But then the conjunction, which is “run and smile”, is satisfied. Suppose next that at least one conjunct is violated. For example, you don’t run. Then you don’t both run and smile, so the conjunction is violated. Now what if at least one conjunct is avoided? Here out example is of no help, since both conjuncts are unconditional and thus cannot be avoided. So take another example: the conjunction of “if it rains, run” with “if it rains, smile” is clearly “if it rains, run and smile”. Suppose that both conjuncts are avoided. Then it doesn’t rain. But then the conjunction, which is of the form “if it rains…”, is also avoided. Finally, for the remaining cells of the table we need yet another example. The conjunction of “if it rains, run” with “if it doesn’t rain, run” is clearly “run whether or not it rains”; in other words, just “run”. (By the way, this is an important example: it shows that the conjunction of two conditional prescriptions can be unconditional, contrary to what has been maintained in the literature.) Suppose the first conjunct is avoided, so it doesn’t rain, and the second conjunct is satisfied, so it doesn’t rain and you run. Then you run, so the conjunction is satisfied. Now suppose the first conjunct is avoided, so it doesn’t rain, and the second conjunct is violated, so it doesn’t rain and you don’t run. Then you don’t run, so the conjunction is violated. So this is the complete satisfaction table for conjunction. But now you might ask: would we have reached the same table if we had used different examples? Indeed, we would have; I don’t have the time to defend this claim now, but I defend it in the paper on which this talk is based, by examining a variety of additional examples.

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**CONJUNCTION II AVI&I' = AVIAVI'. CI&I' = CICI'. VI&I' = VIVI'.**

Definition 2 (Conjunction): <S, V> & <S', V'> = <(CC')-(VV'), VV'>. Rescher (1966): <SS', (CC')-(SS')>. Storer (1946): <SS', VV'> (cf. Łukasievicz 1920). I & I' Sat. Av. Viol. We can use the above satisfaction table, which I reproduce here, to get a formal definition of conjunction. Note first that the conjunction is avoided exactly if both conjuncts are avoided. So the avoidance set of the conjunction is the intersection of the avoidance sets of the conjuncts. Now recall that the context is the complement of the avoidance set; so, by one of de Morgan’s laws, we get that the context of the conjunction is the union of the contexts of the conjuncts. Note next that the conjunction is violated exactly if at least one of the conjuncts is violated. So the violation set of the conjunction is the union of the violation sets of the conjuncts. We are done, since the satisfaction set is the context minus the violation set. So here is the formal definition: the conjunction of two prescriptions with satisfaction sets S and S and with violation sets V and V respectively is the prescription with violation set the union of V with V and with satisfaction set the union of the contexts minus the union of the violation sets. This definition can be readily generalized to arbitrarily, and even infinitely, many conjuncts: as you may recall, unions are defined for arbitrary collections of sets. Now I would like to contrast my definition of conjunction with two previously proposed definitions, due to Rescher and to Storer, which I reproduce here in my notation. Contrary to my definition, both of these definitions have the consequence that the conjunction is not satisfied if one conjunct is satisfied and the other conjunct is avoided. But this gives the wrong result: as we saw, the conjunction of “if it rains, run” with “if it doesn’t rain, run” is just “run”, and thus is satisfied if for example the first conjunct is avoided, so it doesn’t rain, and the second conjunct is satisfied, so it doesn’t rain and you run. It follows that these previously proposed definitions are inadequate. These remarks have an important consequence. As we saw, imperative logic is not isomorphic to standard logic because prescriptions are three-valued whereas propositions in standard logic are two-valued. But there are also three-valued declarative logics, so one might think that imperative logic is isomorphic to such a three-valued logic. However, typical definitions of conjunction in three-valued declarative logics turn out to be isomorphic to Storer’s definition of imperative conjunction; so the inadequacy of Storer’s definition shows that imperative logic is not isomorphic to typical three-valued declarative logics either.

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**DISJUNCTION Definition 3 (Disjunction):**

<S, V> <S', V'> = <(CC')-(SS'), SS'>. De Morgan’s laws hold: ~(I & I') = ~I ~I' ~(I I')= ~ I & ~I' I I' Sat. Av. Viol. By using a process similar to the one that I followed for conjunction, one can construct the satisfaction table and extract the definition of disjunction. I will spare you the details, but I note that my three definitions so far have the nice property of satisfying de Morgan’s laws: the negation of the conjunction of two prescriptions is the disjunction of the negations of the two prescriptions, and the negation of the disjunction is the conjunction of the negations. It’s easy to prove.

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**Definition 4 (Conditional):**

“It rains” “Run” = “If it rains, run”. “It rains” (“It snows” “Run”) = “If it rains and snows, run”. Truth-satisfaction table for conditional: Definition 4 (Conditional): P <S, V> = <P S, P V>. P I Sat. Av. Viol. True False Consider now the conditional whose antecedent is the proposition that it rains and whose consequent is the prescription expressed by “run”. This conditional is the conditional prescription expressed by “if it rains, run”. Suppose first that the antecedent is true, so it rains, and the consequent is satisfied, so you run. Then the conditional is satisfied. Similarly, if the antecedent is true and the consequent is violated, then the conditional is violated. Suppose next that the antecedent is false. Then it doesn’t rain, so the conditional is avoided. Now what if the consequent is avoided? Here our example is of no help, so take another example. Suppose I say: ”if it rains, then if it snows, run”. This clearly expresses the conditional prescription expressed by “if it rains and snows, run”. Suppose that the consequent is avoided: it doesn’t snow. Then it doesn’t both rain and snow, so the conditional is avoided. So here is the complete truth-satisfaction table for the imperative conditional. And this is the formal definition. By the way, the fact that we need a truth-satisfaction table for the imperative conditional provides another reason why imperative logic is not isomorphic to declarative logic: imperative logic mixes propositions with prescriptions.

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**Definition 5 (Biconditional):**

“Run if and only if it rains” = “Run if it rains” & “Run only if it rains” = “If it rains, run” & “If it doesn't rain, don't run”. So: Definition 5 (Biconditional): P I = (P I) & (~P ~I). Truth-satisfaction table for biconditional: Finally, I don’t have much to say about the biconditional. It’s easy: it’s the conjunction of two conditionals. Take, for example, the prescription expressed by “run if and only if it rains”. This is the conjunction of “run if it rains” with “run only if it rains”. And this is in turn the conjunction of “if it rains, run” with “if it doesn’t rain, don’t run”. So in general, the biconditional P double arrow I, where P is a proposition and I is a prescription (or imperative), is the conditional P arrow I conjoined with the conditional tilde P arrow tilde I. Note an ambiguity here: I use the tilde both for propositional and for imperative negation. But I trust that the context disambiguates. Finally, this is the truth-satisfaction table for the biconditional. I will not go over it. P I Sat. Av. Viol. True False

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**CONSISTENCY AND QUANTIFIERS**

PART 3 Part 1: A MODEL OF PRESCRIPTIONS Part 2: LOGICAL CONNECTIVES (Negation, conjunction, disjunction, conditional, and biconditional) Part 3: CONSISTENCY AND QUANTIFIERS I turn now to the third and final main part of my talk, in which I examine consistency and quantifiers.

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(IN)CONSISTENCY Is consistency just joint satisfiability? No: “If it rains, run” and “If it doesn’t rain, don’t run” are consistent but jointly unsatisfiable. I propose instead: A set of prescriptions is inconsistent iff the conjunction of its members is self-contradictory. What is it for a prescription to be self-contradictory? Is it to be unsatisfiable? No: “If you run, prove that = 5” is unsatisfiable but not self-contradictory. It is commonly suggested in the literature that two or more prescriptions are logically consistent exactly if they are jointly satisfiable; in other words, exactly if the intersection of their satisfaction sets is non-empty. Here is a counterexample to this common suggestion. The prescriptions expressed by “if it rains, run” and “if it doesn’t rain, don’t run” are clearly consistent: as we saw, their conjunction is the biconditional “run if and only if it rains”. And yet these prescriptions are not jointly satisfiable: their contexts are disjoint, so the intersection of their satisfaction sets is empty. So joint satisfiability will not do as a definition of consistency. I propose a different definition: a set of prescriptions, just like a set of propositions, is inconsistent exactly if the conjunction of its members is self-contradictory. But what is it for a prescription to be self-contradictory? One might think that it is to be unsatisfiable: to have an empty satisfaction set. But this will not do. The prescription expressed by “if you run, prove that 2+2=5” is unsatisfiable but not self-contradictory: it is possible that you don’t run, and thus that the prescription is not violated.

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**Definition 6 (Inconsistency):**

(IN)CONSISTENCY II A prescription is self-contradictory iff it is omniviolable: necessarily violated (i.e., both unsatisfiable and unconditional). Definition 6 (Inconsistency): A set of prescriptions is inconsistent iff the conjunction of its members is omniviolable (and is consistent otherwise). “If you run, smile” and “If you run, don't smile" are consistent: their conjunction, “If you run, smile and don't smile” is avoidable. I submit instead that a prescription is self-contradictory exactly if it is necessarily violated; to coin a term, exactly if it is omniviolable. Equivalently, exactly if it is both unsatisfiable and unconditional, like the prescription expressed by “Run and don’t run”. So here is my definition: a set of prescriptions is inconsistent exactly if the conjunction of its members is omniviolable, and is consistent otherwise. One consequence of my definition of consistency is that joint satisfiability is sufficient for consistencyalthough, as we saw with the example in the previous slide, it is not necessary. So the popular idea that consistency amounts to joint satisfiability is half-true, so to speak. Another consequence of my definition is that a conditional prescription and its negation are always consistent. And this seems right. For example, the conjunction of “if you run, smile” with its negation, “if you run, don’t smile” is the prescription expressed by “if you run, smile and don’t smile”. And this is not omniviolable: it is not violated if you don’t run.

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**QUANTIFIERS “Push every button” = <x(Bx Px), ~x(Bx Px)>.**

So to formalize a quantified prescription we don’t need special quantifiers: we can use standard quantifiers to formalize its satisfaction and violation propositions. Still, imperative quantifiers are useful. Definition 7 (Quantifiers): x<Sx, Vx> = <xSx & ~xVx, xVx>; x<Sx, Vx> = <xSx, xVx & ~xSx>. I turn finally to quantifiers. Take a prescription that intuitively involves quantification: “push every button”. Its satisfaction proposition is the proposition that you push every button: for all x, if x is a button, you push x. The violation proposition is just the negation of this. So to formalize this prescription we don’t need to define some special kind of imperative quantifier: it’s enough to use standard quantifiers to formalize the satisfaction and the violation proposition of the prescription. This point generalizes to every prescription, no matter how complicated. For example, take the prescription expressed by “if it rains, close at least two windows on every floor”. Its satisfaction proposition is the proposition that it rains and you close at least two windows on every floor, and its violation proposition is the proposition that it rains and it is not the case that you close at least two windows on every floor. Both propositions can be formalized by using standard quantifiers. We have thus here an illustration of a point I made earlier on: my identification of prescriptions with pairs of propositions enables one to do imperative logic by using tools from declarative logic. Nevertheless, for some purposes it is useful to define imperative quantifiers as generalizations of imperative conjunction and disjunction. I will not go over the definitions, because they are rather technical.

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FUTURE PLANS New foundations for imperative logic II: Pure imperative inference. New foundations for imperative logic III: A general definition of argument validity. New foundations for deontic logic I: Unconditional deontic propositions. New foundations for deontic logic II: Conditional deontic propositions. Imperative and deontic logic: New foundations. To conclude I would like to emphasize that what I have presented today is only the tip of the iceberg, only the beginning of a long-term research program. The most exciting parts are still to come. I am currently working on two papers, New foundations for imperative logic II and III, in which I examine respectively pure and mixed imperative inference, namely how to define the validity of pure imperative arguments, which consist only of prescriptions, and of mixed imperative arguments, which consist of both prescriptions and propositions. I am also working on two further papers, New foundations for deontic logic I and II, in which I propose a way to solve the major foundational problems that plague deontic logic by grounding deontic logic on imperative logic. These sequences of papers are planned to result in a book, “Imperative and deontic logic: New foundations”, which will probably come out with Oxford Universal Press. I can give details in the discussion period if you are interested.

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2009/9 1 Logic and Proofs §1.1 Introduction §1.2 Propositional Equivalences §1.3 Predicates and Quantifiers §1.4 Nested Quantifiers §1.5~7 Methods of Proofs.

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