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 argu-ment 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 condit-ions 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=R1, …, Rn1, Rn; ergo C P(R1), P(C), P(C|R1): commitment in R1, C, C given R1 [not belief!] Pf(C)=P(C|R1, …, 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.”

‘makes a difference’ = ‘has impact’ 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).

Hence: subjective interpretation of probability 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 C-supporting-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 reason-for-C, and weakest where R is a common and indecisive supporting reason-for-C.

…to characterize: relevance, sufficiency (and acceptability) 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)

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={R1, R2, R3, R4}; P(R1)=P(R2)=P(R3)=P(R4) 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 [“weak reason”] Now set P(C)=.17 [“hardly supported C”] But: when successively updating P(C) on R1 to R4, using Bayes’ theorem, P(C|R1)=.1755; P(CR1|R2)=.3625; P(CR1,2|R3)=.4833; and P(CR1,2,3|R4)=.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 understand-ing of the RSA criteria concerning their conceptual (in)dependence, their function as update-thresh-olds, and their status as obligatory rather than permissive norms, which shows how these formal and informal normative approa-ches (can) in fact align.

Forthcoming with Synthese

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