Diaspora of a Mathematics of Argument

R. LouiDept of Computer Science

Washington University

St. Louis

Outline

I. Intellectual History of Process-Based Models of Reasoning

II. Some Technical Issues regarding Argument

III. Foundations:

A. Probability

B. Decision

C. Legal Reasoning

D. Belief Revision/Deontic Logic

E. Negotiation

F. Rhetoric

IV. Future Work

A. Fairness

B. Computation

CS TALKScope of My Current CS Work

cgi in gawk

book with S. Sachs

independent co-malloc for localizing dynamically allocated objects

optimal average hash chain length for gawk

malloc with a vmstat time series estimator for elective memory expansion

gnu release(s) with M. Waldvogel, M. Pachos, K. Krouse

something with FPGA's

patent & license with J. Lockwood, J. Moscola, M. Pachos

CS TALKScope of My Current CS Work (cont.)

purely probabilistic negotiating agents

model, simulations

real-time object recognition for aerial targets

hiding among non-combatants

half-baked ideas, proposal with R. Pless

AI and Law service

journal, ICAIL, JURIX, special issues, workshops, JD-PhD's, no $

WHAT IS LOGIC?What do Computer Scientists think is Logic?

Roughly: Hilbert-Russell-Whitehead tradition:

1. there is one correct logic:

it is either the predicate logic or the propositional logic or both

2. entailment (syntactic or semantic?) has something to do with mathematical proof

3. logic codifies correct ways of reasoning

4. logic has something to do with the success of hardware

WHAT IS LOGIC?What do Computer Scientists think is Logic?

(cont.)

Some more advanced members of our species:

Knowledge Representation

1. logics are like programming languages; can be chosen or designed

w/o metaphysical consequence

2. some logics are more expressive than others

3. some logics license more inferences than others

4. inferential license and expressiveness are complementary

WHAT'S NEW/DIFFERENTWhat is the Difference between Argument and

Deduction?

Diagram of an argument

<p, { < {a,b}, p >,

<{c}, a>,

<{d}, b>

} >

Diagram of a proof

<p1, p2, …, p>

Obvious

Argument Deduction

nondemonstrative demonstrative

if p then defeasibly q if p then materially q

nonmonotonic monotonic

argument proof

subargument subproof

counterargument (counterproof?)

defeat (fallibility? corrigibility?)

inconsistency-tolerating inconsistency-degenerating

Less ObviousArgument Deductionfocus is on metalanguage: focus is on object language:conflict, rebuttal, warrant and, or, not

anytime: idealwarrant w.r.t. arguments commitment at all tproduced in time t

Warranted(S,t) Thm(S), Proved(S,t)

constructive: nonconstructivep's warrant underdetermined

ampliative: nondeterminism nonampliative: conclusionsof process in the meanings of words

Less Obvious (continued)

Argument Deduction

strategy-based: constraint-based:

choices proof = constraint-propagation

dialectical unilateral

sits between objectivist and invites principle of charity:

relativist conceptions of truth

formally, 10-20 years old formally, 100-150 years old

20thC seminal in social sciences 20thC dominant in logic

Variations of Logic

Modal Logic opaque contexts

>> notation for beliefs about beliefs

Fuzzy Logic predication weakened

>> smoother control, washing machines

Multivalued Logic truth weakened

>> semantic curiosities, reductions

Relevance Logic implication weakened

>> model of limited inferential capacity

Variations of Logic (cont.)

Intuitionist Logic weak negation added>> first step toward elevation of process

Counterfactual Logic second implication added>> plausible alternative conditional

Paraconsistent Logic meaning from inconsistency>> proof-theory for consistent subsets

Belief Revision recovery from inconsistency>> model of premise adoption/retraction

Argument ties logic to computation in a fundamental way>> rewrite foundations of other fields

INTELLECUTAL HISTORY

1. where did the idea of defeasibility come from?

2. where did the idea of procedural rationality come from?

3. where did the idea of argument come from?

AI:"tweety is a bird, but tweety does not fly"

McCarthy-Hayes -- Modal Belief1973

Reiter --Closed World Databases1978Doyle -- TMS1978Kowalski -- PROLOG1974Clark -- Negation as Failure1978

Argument systems1987...

Pollock -- Defeasible Reasoning1968-1974-1986-1987

Nute -- Defeasible PROLOG1985

Kyburg -- system for probability based on defeat1961-1974

Epistemology:"it seems red therefore it is red" is defeasible

Belzer -- Defeasible Reasoning

1986Swain -- Epistemic Defeasibility1978

Pollock -- Knowledge & Justification1974

Sosa -- Conceptions of Knowledge1970Lehrer-Paxson -- Knowlege1969Firth -- Coherence1964

Chisholm -- Perceiving1957-1964

Ladd -- Structure of a Moral Code

1957

Reasoning (Qualitative Decision Theory):"doing a achieves the goal, therefore do a" is defeasible

Nozick -- Practical Reason/Explanations

1981

Searle -- Prima Facie Obligations/Practical Reason

1978-1985

Raz -- Practical Reason/Norms

1970

Gauthier -- Practical Reasoning

1963

Ethical Reasoning:"a person has a prima facie obligation or responsibility"

Glover -- Responsibility1970Nozick -- Moral Structures1968

Feinberg -- Action and Responsibility1965

Wellman -- Language of Ethics1961

Brandt -- Blameworthiness and obligation1958Melden -- Action/Rights1956-1959Mackie -- Responsibility and Language1955

Hare -- Language of Morals1952

(note: Stevenson 1938 and Ross 1930)

Political Justification:

Barry -- Political Argument

1965

Rawls -- Pure Procedural Justice

1958-1974

Dialectics/Rhetoric:

Rescher -- Dialectics

1977

Perelman -- Justice and Argument

1963

Toulmin -- Uses of Argument

1958

Origins:

Ladd

Raz

Gauthier

Wellman

Brandt/Melden/Hare

Barry

Rawls

Perelman/Toulmin

Hart

Origins (continued):

Hart -- Ascription of Responsibility1948

Wisdom -- Gods1945Waismann -- Verifiability1951Austin -- Speech Acts1947?

Wittgenstein -- Remarks on Foundations of Mathematics1935?

Keynes -- Treatise on Probability1908 thesis begins “Part of our knowledge we obtain direct; and part by argument.”Bentham -- Principles of Morals and Legislation185?

Defeasibility:

When the student has learnt that in English law here are positive conditions required for the existence of a valid contract, - his understanding of the legal concept of a contract is still incomplete, ... For ... he has still to learn what can defeat a claim that there is a valid contract, even though all these conditions are satisfied. The student has still to learn what can follow on the word "unless", which should accompany the statement of these conditions. This characteristic of legal concepts is one for which no word exists in ordinary English ... but the law has a word which with some hesitation I borrow and extend: this is the word “defeasible”...

(Hart vs. Aristoteliean Society, 1951, p. 152)

Process:

... [Principia Mathematica] gives rise to questions about the relation in which ordinary reasoning stands to this ordered system, and in particular, as to the precise connection between the process of inference, in which the older logicians were principally interested, but which [Russell] ignores, and the relation of implication, on which his scheme depends.

The gradual perfection of the formal treatment ... had been to empty [logic] of content and to reduce it more and more to mere dry bones, until finally it seemed to exclude ... most of the principles, usually deemed logical, of reasonable thought.

(Keynes vs. Russell, Whitehead, Ramsey, 1908/1921/1973, p. 118, 1972, p. 243)

Formal Inconsistency:

WITTGENSTEIN: Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone-- Because the thing works like this: if a man says 'I am lying' we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? ... It does not matter. ... it is just a useless language-game, and why should anyone be excited?

TURING: ... one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong.

WITTGENSTEIN: Yes -- and more: nothing has been done wrong. ... where will the harm come?

(Wittgenstein vs. Turing, 1939, Hodges, p. 154)

RECENT TECHNICAL QUESTIONS

0. knee-jerk reaction (deductivists)

Q. isn't

p > r,

p & q > ~r

reducible to

p & ~q --> r

p & q --> ~r ?

A. no.

what can be concluded with just p? r.

does that imply ~q? no.

TECHNICAL

I. old issues (Touretzky, Horty, Thomason, 1987)Q. skeptical vs. credulous

A. skepticalQ. ambiguity-propagating vs. blocking

A. depends whether undercut or rebutQ. syntactic specificity

1. (strict specificity)p > r p & q > ~r

2. (shortcut specificity)p > q; q > r p > ~r

3. (defeasible specificity)p > r q > ~r

p > q

TECHNICAL (continued)

A. keep it simple (Nute, 1990)

A. appeal to convention (Simari-Loui, 1992)

A. give explicit ordering (Lin-Shoham, 1987, Vreeswijk, 1991)

A. provide for meta-argument about defeat

e.g., context-dependent defeat (Prakken-Sartor, 1995)

[r1] p > q

[r2] r > ~q

s > [r1>r2]

TECHNICAL (continued)

II. principles versus rules (Loui-Norman, Hage, Verheij, 1995-2001)

Q. Is there a formal difference between:

1. no vehicles in the park

2. parks should be peaceful

A. Rationales: can recall rationales during dispute

A. Principles can be weighed: free speech versus privacy

TECHNICAL (continued)

III. rules for fair dialectic (Loui, Gordon, Vreeswijk, Lodder, 1992-2001)

Q. What is the exact procedural burden?

1. pro: argument 1 for p

2. con: argument 2 for ~p, and

argument 2 defeats argument 1.

a. should con have the burden of showing

argument 2 defeats argument 1?

or

b. should the claim be presumable and subject to dispute?

TECHNICAL (continued)

Q. are normal default rules fair?

p : q / q

1. pro: argument 1 for p

2. con: a. proof of ~q

b. (it does not suffice to argue ~q?)

Q. what is the penalty of failed attempts to rebut?

1. pro: argument 1 for p based on b and c, etc.

2. con: ~b and ~c.

3. pro: why ~b?

4-15. con: ... pro: ... con loses the subdispute over ~b.

16. con: nevertheless, ~c.

A. rhetorical costs: HIGH. logical costs: NONE?

TECHNICAL (continued)

IV. rules extracted from casesQ. What is the structure of a precedent case?

A. (Raz, 1970) a b c d e f / q

A. (Rissland-Ashley, 1985-1990)a+ b+ c+ d- e- f- / q+

A. (Loui-Norman, 1992)argument 1(a,(b,c); q)argument 2(a,d,e; ~q)argument 3(e,f; ~d)argument 4(c; ~f)_______________

q

TECHNICAL (continued)

Q. What is the rule of the case?

A. (Loui-Norman, 1995)

sufficient premises of arguments in dialectical subtree with leafs that are pro arguments

a b c d e f > q

but no false specificity:

a b c f > q

A. (Prakken-Sartor, Bench-Capon, 1992-2001)

{Argument1, Argument3} > {Argument2, Argument4}

TECHNICAL (continued)

V. criteria for theory-formation when theories are defeasible? (Peczenik, McCarty, 1997-2001)

Q. Given a set of cases:case 1 a b d e f j qcase 2 a b d ~q...case n b d ~r

what is the "best-fitting" set of defeasible rules?

1. all cases predicted by rules2. no error

(so far this is a learning problem with no simplicity measure)3. sets of rules restricted or justified by principles?

FOUNDATIONS: I. Probability

A probability calculation is an argument.A statistical argument is an argument.

Reference Class:

Reichenbach (1949): "use the narrowest reference class for which there are adequate statistics"

Kyburg (1961,1974,1983): maximum in a partial order? dominance ~= defeat. Each "inference structure" permits an argument from a different sample class.

Prob(A | B C D)? Sample from BCD: 5 A's/9 Sample from BC: 14 A's/20

what is the logic of combinatorial significance tests?

FOUNDATIONS: I. Probability (cont.)

<S1,[p1,q1]> and <S2,[p2,q2]> disagree

iff

not([p1,q1] in [p2,q2]) and

not([p2,q2] in [p1,q1])

<S1,[p1,q1]> defeats <S2,[p2,q2]>

iff

they disagree and

S1 strict subset of S2

also Pollock (1985, 1990) who uses defeat explicitly

FOUNDATIONS: I. Probability (cont.)

Protocols

Shafer on Monte-Hall type probability "paradoxes" (1985)

the probability argument

is improved through knowledge of the protocol

Neyman-Pearson tradition of crucial tests

two crucial tests

produce two competing statistical arguments?

FOUNDATIONS: II. Decision

Problem of Small Worlds

Savage (1954, 1967):

considering fresh and stale

should not change the calculation

based on good and rotten.

But of course it does.

So: a grand world which contains all detail.

pseudomicrocosm vs. real microcosm.

Shafer-Tversky (1988):

framing problems

constructive decision theory

FOUNDATIONS: II. Decision (cont.)

Loui (1990):

u(s) given T(P1,s) m(P1) = 5

u(s) given T(P2,s) m(P2) = -4

u(s) given T(P1 & P2, s) m(P1 & P2) = 6; u(s) = 5-4+6

("defeasibility" of linearity)

Holds(P1&P2, s) > u(s) = 7

defeats

Holds(P1,s) > u(s) = 5

(defeasibility of valuation)

FOUNDATIONS: II. Decision (cont.)

But s is a lottery: s = {r/p; t;(1-p)}Prob(p) = .5; u(r) = 10; u(t) = 0; so expected u(s) = .5(10) +.5(0) = 5

(defeasibility of outcome)

Simon (1955-1967)substantive vs. procedural rationality, yes,

but more importantly:decision is more like chess than constraint-propagation:heuristic valuation changes as search/computation proceeds

a defeasible independence/substitution axiom?

paradoxes of certainty, menu-dependence, framing effects based on description

FOUNDATIONS: III. Legal Reasoning

FOUNDATIONS: IV. Belief Revision/Deontic Logic

contrary-to-duty imperatives (Chisholm, 1963, Nute etc., 1996)

1. O(p-->q), O(p-->r) and O(~p) are consistent.

(there can be two expiations)

2. O(~p) entails O(p --> x) for any x.

(all expiations are obliged)

von Wright (1982):

"It only means that, if the prohibition is violated, the coordinated Contrary-to-Duty imperatives require, for their satisfaction, that both q and that r come true. ... If ... the conjunction of the two ... is a logical impossibility ..., the legislator would presumably take steps to remove

the conflict."

"deontic obligation" is different from "technical obligation"

FOUNDATIONS: IV. Belief Revision/Deontic Logic

3. O(A|B), O(~A|C) entails ~(B&C)

Alchourron (1993):

"a set of conditional general norms entails ... a non-tautological sentence ... iff it follows in the logic for normative propositions that

the authority has inconsistently normed some action for some circumstance."

If norms are defeasible rules, no such problems:

1. two different arguments for expiation.

2. material conditionals are NOT deontic conditionals.

3. the entailment is not a result for defeasible conditionals

FOUNDATIONS: IV. Belief Revision/Deontic Logic AGM belief revision: choice and refinement, ampliativity

if p then defeasibly q

if p and r then defeasibly ~q

p --> q in K0

p & ~r --> q in K0[p&r-->~q][r]

Alchourron (1993):

"It seems to me unquestionable that the main [conditions] are the formal representation of the revisions effectively performed by an

agent and of his dispositions to revise."

"The particular details of the revisions (and the choice functions) are never analyzed by a logician (as a logician)..." Yes, defeasible

conditionals would inform choice functions, but invoke "possible confusion of logic and revision," hiding "conceptually weaker

conclusions" in "quiet darkness."

FOUNDATIONS: V. Negotiation

Acceptability can be argued:

Fisher-Ury-Patton (1981): Principled negotiation gives arguments for proposals.

“why not open the window?” “I’m cold” “I have a sweater”

Sycara (1988-1995), Loui-Moore (1993-1997), Parsons-Jennings etc. (1998-2001): case-based arguments from precedent settlements

“that raise was acceptable to you last year”

Utility can be argued:

search can lift utilities at the proposed settlements

FOUNDATIONS: V. Negotiation

Utility can be argued:

Instead of strategic form {a1, …, a4} X {b1, …, b5} with

utilities Ua and Ub,

suppose

OPTa(x,y) and OPTb(x,y), a hard optimization problem for

each agent with parameters determined by the agreement

Ua(x,y) is a’s current best solution for OPTa(x,y)

Ua and Ub are lifted at <p,q> which is the focus of dialogue

or when joint-problem-solving

Sunk cost-of-search arguments lead to settlement

FOUNDATIONS: VI. Rhetoric/"Informal Logic"

Future Work: Fairness (Procedural)Claim 1. Fairness depends on the computational abilities of agents (the

known subspace S x T of the possible strategies S* x T*;

thus, rules are changed when a strategy s+ is discovered for which all known responses t are inadequate.

Claim 2. To be fair (just), the procedure must construct its output upon the right inputs, with adequate monotonicity and invariance properties;

thus, the justification of social procedure resembles the proof of program correctness.

(concatenation) ex-post asymmetry of position should be the result of fair (just) ex-ante asymmetry adjusted only by the procedure’s effects on elective inputs (strategic choices).

Future Work: Fairness (cont.)Obsv (political scientists).

the purposes of the procedure can limit the degree of stochastics, the maximum variation of outcome, and the permissible input types.

Claim 3.procedures should be non-dictatorial (for every important different kind of outcome, e.g., victory/defeat, there is a strategy pair that would reach this outcome). (dominance is more interesting…)

Obsvrule symmetry and equivalent initial position are prima facie fair(but sometimes there are good reasons for bias, e.g. plaintiff)

Claim 4.Fairness can be inherited from class relationships among procedural types.

Future Work: Computation

Observations. Social procedures which regulate/distribute/construct distributions are games. Social programming is like distributed programming (quantify over strategy tuples). Building societies is like inventing algorithms for distributed decision-making. Argument games. Welfare distributions. Elections. Tournaments.

Objective. I want computer science to be at the foundation of the study of social procedures.

Obstacle. Game-playing is not considered computation (yet).

Claim. Two people playing chess compute the outcome of the game.

Why do you have trouble with this claim? But not modems or chess tournaments for charities.

Future Work: Computation

Paradigmatic computation:

on a machine: but, long division by hand?

causally connected: but, two people doing long division?

deterministic: but, probabilistic algorithms?

locus of control: but, distributed algorithms?

fully specified: but, pseudo-code? uncompiled code?

algorithmic: but, protocols? interactions?

(control systems?)

non-elective: but, frequently arbitrary choices

(e.g., search algorithms)

Protocol-design is a kind of programming.

Future Work: Computation

Broad computation: intentional and teleological rule-following upon symbols.

The existence of a program is the test for computation,

not the existence of an algorithm.

Not just any rule-following is computation, since the objects must be symbolic and the rule-following purposive and non-accidental.