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NEW FOUNDATIONS FOR IMPERATIVE LOGIC I: Logical connectives, consistency, and quantifiers Peter B. M. Vranas vranas@wisc.edu 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: l have tried but failed to develop an adequate imperative logic (Rescher 1966, Sosa 1967); l have argued that imperative logic is impossible (Williams 1963); l have argued that imperative logic is iso- morphic to standard logic (Castañeda 1975). I hope to resurrect imperative logic.

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WHAT IS IMPERATIVE LOGIC? Distinguish imperative sentences from what they typically express, namely prescriptions: com- mands, requests, instructions, suggestions, … l English and French imperative sentences can express the same prescription. l Declarative sentences can express prescriptions (“You will open the door”). l Imperative sentences can express propositions (“Marry in haste and repent at leisure”). Imperative logic is the logic of prescriptions.

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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

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A MODEL OF PRESCRIPTIONS l Any prescription (“Run”) has a satisfaction proposition (“You run”) and a violation proposition (“It is not the case that you run”). l “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. l 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.

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AN ANALOGY l Descartes identified points in the plane with ordered pairs of numbers. This enabled one to do geometry by using tools from algebra. l I identify prescriptions with ordered pairs of propositions. This enables one to do imperative logic by using tools from declarative logic. l 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.

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CONDITIONAL PRESCRIPTIONS l A prescription is unconditional if its satisfaction and violation propositions are contradictories, and is conditional otherwise. l Why not say that “If it rains, run” is satisfied (rather than avoided) if it doesn’t rain? l Because then it would be identical with “Let it be the case that if it rains you run”. l 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.

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PREVIOUS MODELS l Motivation of previous models: “Run” and “You run” have a common element. l 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 l 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.

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HOW MANY PRESCRIPTIONS? l Every prescription is an ordered pair of incompatible propositions. But does the converse hold? Is every ordered pair of incompatible propositions a prescription? l Yes: the pair is the prescription expressed by “If S or V is true, let S be true”. l 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.

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TERMINOLOGY l I will talk interchangeably of e.g. satisfaction propositions and satisfaction sets. l 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”.) l 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.

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PART 2 Part 1: A MODEL OF PRESCRIPTIONS Part 2: LOGICAL CONNECTIVES (Negation, conjunction, disjunction, conditional, and biconditional) Part 3: CONSISTENCY AND QUANTIFIERS

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NEGATION l Negation of “Run”: “Don't run”. Negation of “If it rains, run”: “If it rains, don't run”. l Satisfaction table for negation: Definition 1 (Negation): ~ = l I and ~I have the same context. Double nega- tion holds: ~(~ ) = ~( ) =. I = Sat.Av.Viol. ~I = Viol.Av.Sat.

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CONJUNCTION Ê “Run” & “Smile” = “Run and smile”. Ë “If it rains, run” & “If it rains, smile” = “If it rains, run and smile”. Ì “If it rains, run” & “If it doesn't rain, run” = “Run (whether or not it rains)”. l Satisfaction table for conjunction: I & I'Sat.Av.Viol. Sat. Viol. Av.Sat.Av.Viol.

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CONJUNCTION II AV I&I' = AV I AV I'. C I&I' = C I C I'. V I&I' = V I V I'. Definition 2 (Conjunction): & =. Rescher (1966):. Storer (1946): (cf. Łukasievicz 1920). I & I'Sat.Av.Viol. Sat. Viol. Av.Sat.Av.Viol.

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DISJUNCTION Definition 3 (Disjunction): =. De Morgan’s laws hold: ~(I & I') = ~I ~I' ~(I I')= ~ I & ~I' I I' Sat.Av.Viol. Sat. Av.Sat.Av.Viol. Sat.Viol.

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CONDITIONAL Ê “It rains” “Run” = “If it rains, run”. Ë “It rains” (“It snows” “Run”) = “If it rains and snows, run”. l Truth-satisfaction table for conditional: Definition 4 (Conditional): P =. P I Sat.Av.Viol. TrueSat.Av.Viol. FalseAv.

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BICONDITIONAL l “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). l Truth-satisfaction table for biconditional: P IP I Sat.Av.Viol. TrueSat.Av.Viol. FalseViol.Av.Sat.

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PART 3 Part 1: A MODEL OF PRESCRIPTIONS Part 2: LOGICAL CONNECTIVES (Negation, conjunction, disjunction, conditional, and biconditional) Part 3: CONSISTENCY AND QUANTIFIERS

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(IN)CONSISTENCY l Is consistency just joint satisfiability? No: “If it rains, run” and “If it doesn’t rain, don’t run” are consistent but jointly unsatisfiable. l I propose instead: A set of prescriptions is inconsistent iff the conjunction of its members is self-contradictory. l What is it for a prescription to be self- contradictory? Is it to be unsatisfiable? l No: “If you run, prove that 2 + 2 = 5” is unsatisfiable but not self-contradictory.

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(IN)CONSISTENCY II l 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). l “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.

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QUANTIFIERS l “Push every button” =. l So to formalize a quantified prescription we don’t need special quantifiers: we can use standard quantifiers to formalize its satisfaction and violation propositions. l Still, imperative quantifiers are useful. Definition 7 (Quantifiers): x = ; x =.

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FUTURE PLANS l New foundations for imperative logic II: Pure imperative inference. l New foundations for imperative logic III: A general definition of argument validity. l New foundations for deontic logic I: Unconditional deontic propositions. l New foundations for deontic logic II: Conditional deontic propositions. l Imperative and deontic logic: New foundations.

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