Modals, conditionals, and probabilistic generative models Topic 2: - PowerPoint PPT Presentation

Modals, conditionals, and probabilistic generative models Topic 2: Indicative conditionals – Separating semantics from reasoning Dan Lassiter, Stanford Linguistics Université de Paris VII, 2/12/19

Overall plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals 3. causal models & counterfactuals 4. lazy reasoning about impossibilia

Today: Indicative conditionals • Finish up sampling demos • Probabilities of indicative conditionals • Major theories of indicative conditionals • The trivalent semantics • Avoiding triviality proofs • Conditional restriction

Probabilities of conditionals: The data

The lottery Mary can choose whether to buy a ticket in a fair lottery with 100 tickets. What is the probability of (1)? If Mary buys a ticket, she will win.

Under (un)likely Mary can choose whether to buy a ticket in a fair lottery with 100 tickets. How likely is it that, if Mary buys a ticket, she will win? How likely is it that Mary will win if she buys a ticket? It’s unlikely that Mary will win if she buys a ticket. If Mary buys a ticket it’s unlikely that she will win.

Across speakers Mary can choose whether to buy a ticket in a fair lottery with 100 tickets. Person A: If Mary buys a ticket, she will win. Person B: – That is unlikely. – What you said is probably wrong. • ‘ … but there’s a slight possibility you’re right’ – There’s only a 1% chance that you’re right. (Makes trouble for the restrictor gambit discussed later)

Varying tense Since will may be a modal, check past tense too: Mary had to choose whether to buy a ticket; we don’t know if she did. Person A: If Mary bought a ticket, she won. Person B: – That is unlikely. – What you said is probably wrong. • ‘ … but there’s a slight possibility you’re right’ – There’s only a 1% chance that you’re right.

Stalnaker’s thesis (1970) P(If A, C) = P(C | A) ‘The English statement of a conditional probability sounds exactly like that of the probability of a conditional. What is the probability that I throw a six if I throw an even number, if not the probability that: If I throw an even number it will be a six?’ (van Fraassen 1976) Many experimental studies confirm.

Stalnaker vs Adams • ‘Adams’ Thesis’ is widely discussed but crucially different – about assertibility/ acceptability, not probability (Adams ‘65, ’75) • Douven & Verbrugge (‘The Adams Family’, Cognition, 2010) : – Adams’ thesis is empirically incorrect – Stalnaker’s thesis holds up

Theories of indicative conditionals & what they predict

<latexit sha1_base64="f7lpExFIZWfTdo49gxibnsoWicw=">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</latexit> The material conditional A ⇒ C = A ⊃ C = ¬ A ∨ C = ¬ ( A ∧ ¬ C ) • Bad predictions about probabilities – P(1) depends on how likely she is to buy a ticket • Lots of other problems

<latexit sha1_base64="lciPO0B8AK4E+vug9m4Jw/dDQFA=">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</latexit> <latexit sha1_base64="Wm0MU+/QlC7spP0W3K9MHgiLdh0=">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</latexit> <latexit sha1_base64="bLkXb4KwDVZNjl+D84bw/Gr/TQ=">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</latexit> Strict conditional A ⇒ C = ∀ w ∈ E : A ( w ) ⊃ C ( w ) Assuming P(E) = 1, this entails If P ( C | A ) < 1 , then P ( A ⇒ C ) = 0 P ( Buy ⇒ Win ) = P ( Buy ⇒ ¬ Win ) = 0!! Definite description theories make similar predictions (e.g., Schlenker ‘04)

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