NEW FOUNDATIONS FOR IMPERATIVE LOGIC I: Logical connectives, consistency, and quantifiers Peter B....

NEW FOUNDATIONSFOR 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

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.

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.

OVERVIEW

Part 1:A MODEL OF PRESCRIPTIONS

Part 2:LOGICAL CONNECTIVES

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

Part 3:CONSISTENCY AND QUANTIFIERS

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.

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.

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.

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.

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.

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.

PART 2

Part 1:A MODEL OF PRESCRIPTIONS

Part 2:LOGICAL CONNECTIVES

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

Part 3:CONSISTENCY AND QUANTIFIERS

NEGATION

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> Viol. Av. Sat.

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.

Sat. Sat. Sat. Viol.

Av. Sat. Av. Viol.

Viol. Viol. Viol. Viol.

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.

Sat. Sat. Sat. Viol.

Av. Sat. Av. Viol.

Viol. Viol. Viol. Viol.

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.

Sat. Sat. Sat. Sat.

Av. Sat. Av. Viol.

Viol. Sat. Viol. Viol.

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 Sat. Av. Viol.

False Av. Av. Av.

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:P I Sat. Av. Viol.

True Sat. Av. Viol.

False Viol. Av. Sat.

PART 3

Part 1:A MODEL OF PRESCRIPTIONS

Part 2:LOGICAL CONNECTIVES

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

Part 3:CONSISTENCY AND QUANTIFIERS

(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 2 + 2 = 5” is unsatisfiable but not self-contradictory.

(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.

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>.

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.