A probabilistic analysis of argument cogency DAVID GODDEN

A probabilistic analysis of argument cogency DAVID GODDEN Michigan State FRANK ZENKER Lund/Konstanz/Bratislava Bochum, 2 DEC 2016

§ Pascalian probabilistic treatment of the conditions for cogent argument in informal logic: relevance, sufficiency, acceptability (RSA). § Argument-as-product (vs. process) § Aim: to specify content features of defeasible argument on which the RSA conditions (should) depend. § Why care? Making the RSA conditions more precise shows how formal and informal approaches (can) align.

Overview § Terms & definitions § The impact term, i § Interpret sensitivity and selectivity § Strongest vs. weakest reason § Specify RSA conditions § Open question: update on weak reasons § Upshot

PPC-view: premise, premise; ergo conclusion RRC-view: reason, reason; ergo claim

Terms & Definitions § Pascalian P. : 0 P( )=1 P( ) 1 § ARG: R= R 1, …, Rn 1, Rn ; ergo C § P(R 1), P(C|R 1): commitment in R 1, C, C given R 1 [not belief!] § Pf(C)=P(C|R 1, …, Rn 1, Rn) __ P(C) § ts, ta: threshold(-value) § Pf(C)=P(C|R)__ ts__P(C) § >, <, = : “R (sufficiently) supports, undermines, or is irrelevant to, C. ”

Basic Idea (Receiving) the reason, R, does or does not make a difference to one’s commitment in the claim, C. ‘makes a difference’ = ‘has impact’

The “oomph” § Pf(C)=P(C|R)=P(C) i [i =“impact of reason”]* § i=P(R|C)/P(R) [likelihood; 0 L< ] § P(R)=P(C) P(R|C)+P( C) P(R| C) § Pf(C)=P(C|R)=P(R|C) P(C) _______ P(R|C) P(C)+P(R| C) P( C) § P(R|C): sensitivity of the reason to the claim. § P( R| C)=1−P(R| C): specificity of R to C. * Carnap (1962: 466) calls i the relevance quotient, or the probability ratio; Strevens (2012: 30) calls it the Bayes multiplier (see Joyce, 2009: 5).

Sensitivity & specificity are readily meaningful for long run frequencies (e. g. , medical test) But: folks do argue for claims about single events such as “Oswald shot Kennedy. ” Hence: subjective interpretation of probability

Interpretation proposal § Reason R is sensitive to claim C to the extent that R supports C more than R supports any other claim C*, that itself entails ~C, i. e. , P(C|R)>. 5>P(~C|R). § R is specific to C to the extent that R rather than any other reason R*, itself entailing ~R, supports C, i. e. , P(C|~R)<. 5<P(~C|~R).

Drawing this together, … § …the support that R generates for C thus depends: § on the extent to which the Csupporting-reason R fails to support ~C, on one hand, and § on the extent to which argumentative support for C cannot be generated by reasons besides R, on the other.

Hence, … § …in the extremal cases P(C|R)=1 and P(C|R)=0, support is § strongest where R is an exclusive and decisive supporting reasonfor-C, and § weakest where R is a common and indecisive supporting reason -for-C.

Exploit the i term… …to characterize: relevance, sufficiency (and acceptability) Pf(C)=P(C|R)=P(C) i

Relevance § Pf(C)=P(C|R)=P(C) i If … § i > 1, R is positively relevant to C § i = 1, R is irrelevant to C § i < 1, R is negatively relevant to C § Compare: Relevance as probability raising, or rather probability-change.

Sufficiency § Pf(C)=P(C|R)=P(C) i § Pf(C)=P(C|R)≥ts>P(C) [ts: s. -threshold] § Inferential sufficiency entails that i 1 § Note: a necessary reasons is a special case of an insufficient R (cf. Spohn, 2012)

Acceptability § Pf(C)=P(C|R)=P(C) i § Pf(R)≥ta [ta: acceptability threshold]

Open question § Those who understand offering arguments as the issuing of “invitations to inference” (e. g. , Pinto, 2001) can interpret the sufficiency criterion as prohibiting any inferential use of reasons failing the threshold. § Sufficiency condition would thus act as an “inference gate, ” asking you to “ignore” weak reasons.

Problem case § R={R 1, R 2, R 3, R 4}; P(R 1)=P(R 2)=P(R 3)=P(R 4) § P(Rn|C)=. 25 [“R is weakly sensitive to C”] § P(Rn|~C)=. 15 [“R is weakly selective for C”] § It follows that P(Rn)=. 167 § Now set P(C)=. 17 [“weak reason”] [“hardly supported C”] § But: when successively updating P(C) on R 1 to R 4, using Bayes’ theorem, P(C|R 1)=. 1755; P(CR 1|R 2)=. 3625; P(CR 1, 2|R 3)=. 4833; and P(CR 1, 2, 3|R 4)=. 6093. Compare ts=0. 5

Upshot § RSA conditions depend on: § Change in the acceptability of R § R’s sensitivity and selectivity to C § One’s prior commitment to C § Contextually determined thresholds for reasons and claims § Particularly: Inferential update may be obligatory rather than permissive. § New Q: study linked vs. convergent ARG; compare to probability-model

In sum § We suggest a specific understanding of the RSA criteria concerning their conceptual (in)dependence, their function as update-thresholds, and their status as obligatory rather than permissive norms, which shows how these formal and informal normative approaches (can) in fact align.

Forthcoming with Synthese

Thank you! also on behalf of David frank. zenker@fil. lu. se