1. In what conditions is the meaning of conditionals perceived as equivalent to the meaning of a corresponding disjunction? Alessandra Tasso - Università della Valle d’Aosta Andrea Manfrinati - Università della Valle d’Aosta Gianmarco Altoè. - Università di Padova Paolo Cherubini - Università di Milano Bicocca Pierdaniele Giaretta - Università di Verona Birkbeck - University of London The 4 th London Reasoning Workshop London, 27 – 28 July, 2009

2. in a recent paper on conditionals, Ruth Byrne and Phil Johnson-Laird write that ‘if’ is a puzzle (Byrne & Johnson-Laird, 2009 – ‘If’ and the problems of conditional reasoning) this is because even if we use conditionals in everyday life, we understand conditionals, we draw inferences from conditionals and we know that there are different kinds of conditionals either in philosophy and in psychology there is no agreement on the logical and psychological properties of conditionals “if p then q”

3. according to Edgington (2001) there are three different theoretical views on conditionals 1)conditionals are truth – functional (material implications) a conditional “if p then q” is false only when its antecedent is true and its consequent is false. Otherwise the conditional is true. (Jackson, 1991;Ggrice, 1989; Johnson-Laird & Byrne, 1991, 2002) 2)conditionals are non truth - functional when its antecedent is false, a conditional “if p then q” is not determined (it is impossible to establish a truth value of the conditional) (Stalnaker, 1968; Rips & Marcus, 1977; Braine & O’Brien, 1991)

4. 3)conditionals have assertability conditions a conditional “if p then q” is assertable when it is possible to suppose its antecedent is true. (Adams, 1975; Edgington, 1995; Evans & Over, 2004) according to Quine (1952) when you say “if p then q” when the antecedent turns out to be false, it is as if ithe whole conditional had never been asserted

5. experimental results are mixed and contradictory if there is a black triangle, then there is a white square true false ? ? reasoners show a defective truth table (Wason, 1966; Evans & Newstead, 1977) In everyday language, there are several counter – examples to the defective truth table (Manfrinati, 2007) if Paolo is not in Milan, then he is in Padova If I don’t finish my work, we will stay at home

6. these conditionals are very easily understood and asserted even when theirs antecedents turn out to be false Paolo is not in MilanI finish my work Giaretta et al., (2007) show that also abstract conditionals if there is not an Ace, then there is a King are often evaluated as true when compared to a hand of cards in which the antecedent doesn’t hold there is an Ace expecially when the conditional is preceded by the equivalent explicit disjunction there is a King or there is an Ace or both

7. are some conditionals evaluated as true and assertable when they can be interpreted as the corresponding disjunctions? “if p then q” equivalent to “not p or q” “if not p then q” equivalent to “p or q” according to the propositional logic these propositions have the same meaning they have the same truth table they are true and false in the same situations

8. the experiment investigation on the psychological relation between conditionals and equivalent disjunctions contentreasoning is easier with concrete material typebiconditionals are psychologically easier than conditionals polaritydoes polarity affect meaning equivalence? orderdoes order affect meaning equivalence? (Richardson & Ormerod (1997; Ormerod & Richardson, 2003) Giaretta et al., (2007)

9. indipendent variables contentconcrete vs. abstract typeconditionals vs. biconditionals polaritynegative (negated antecedent) vs. affirmative (affirmative antecedent) orderconditionaldisjunctive vs. disjunctiveconditional experimental design2 (content) X 2 (type) X 2 (polarity) X 2 (order) X 4 (sentences) X 21 participants 1344 observations

10. methodparticipants:21 students materials:48 pairs of premises 32 pairs of fillers proceduresentences were presented on the screen one by one participants‘ task was to evaluate whether conditionals and disjunctives have the same meaning on a 5-points Likert like scale: 1 = different meanings5 = same meaning participants give their answers using the computer keyboard RTs from the presentation of the 5-points scale (3 rd slide) were measured

11. examplesAnna says:if it is raining the ground will be wet Luca says:it is not raining or the ground will be wet Franco says:there is a square or there is a circle, not both Rita says:if and only if there is not a square there is a circle first analysis RTs as dependent variable were analyzed by a mixed effect model in which content, order, polarity, type of conditional, & responses were entered as fixed factors subject and item were entered as random effects fixed factors: under experimental control random factors: not under experimental control but affecting the data (ex. variability between & within subjects)

12. differences in RTs between and within 21 participants

13. resultsRTs are faster for responses 1 and 5 than for responses 2, 3, 4, for negative polarity type X polarity interaction was also significant

14. frequency distribution of the 5 different response categories responses 1 (different meaning) & 5 (same meaning) are more frequent than responses 2, 3, 4

15. frequency distribution of the 5 different response categories for each participant

16. second analysis participant’s judgements are classified in two groups: “correct”response 5(from a logical point of view) “wrong”responses 1, 2, 3, 4 RTs, type of conditional, order, polarity, & content were entered as fixed effects subject and item were entered as random effects results RTs faster for “correct” than for “wrong” responses “correct” responses are more frequent with biconditionals than with conditionals with negative polarity

17. the difference between conditionals and biconditionals is greater with concrete material and negative polarity type X polarity X content interaction was also significant

18. third analysis participant’s judgements are classified in three categories: “correct”response 5according to logic “wrong”responses 1 “don’t know”responses 2, 3, 4 no differences in RTs between “wrong” and “correct” “wrong” responses are less frequent with negated antecedent type X polarity X content interaction was also significant comparison between “wrong” and “correct”

19. RTs are slower for“don’t know” than for “correct” responses “don’t know” responses are less frequent with negated antecedents and with biconditionals comparison between “don’t know” and “correct”

20. conclusionperceived meaning equivalence is affected by concrete content (not as a main effect) type (biconditionals vs. conditionals) polarity (negated antecedent vs. affirmatvie antecedents) no order effect RTs are affected by response (1 and 5 vs 2, 3, 4) polarity (negated antecedent vs. affirmatvie antecedents)

21. Negative polarity If not king, ace MMnot-kingace king … Affirmative polarity If king, ace MMkingace … king or ace MMking ace models integrated without fleshing-out not king or ace MMnot-king ace fleshing-out necessary for integration

22. examples affirmative conditional & negated disjunction if you study hard you’ll get a good mark you don’t study hard or you’ll get a good mark negated conditional & affirmative disjunction if you don’t study hard you’ll get a bad mark you study hard or you’ll get a bad mark affirmative biconditional & negated disjunction if you cut the grass I’ll give you 10 £ you don’t cut the grass or I’ll give you 10 £ negated biconditional & affirmative disjunction if Paolo is not in Milan then he is in Padova Paolo is in Milan or he is in Padova