Question 3 Consider the general sentence “All ravens are black.” What could be more intuitive than to say that this sentence is scientifically confirmed just in case existing ravens we have observed so far are, in fact, black. Let H = ∀x(Rx → Bx) be the paraphrase of the scientific sentence or hypothesis “All ravens are black”. Call sentence E = (Ra ∧ Ba) confirmatory evidence for H if E is true. Nicod’s criterion of confirmation says, roughly, that E confirms H if E is true. (a) Using models, explain why the following argument is valid. 1. (Ra∧¬Ba) 2. ¬∀x(Rx → Bx) Since (a) is a valid argument, one way to think of Nicod’s criterion for confirmation is that a claim like “All ravens are black” is confirmed by its positive instances, i.e., terms that satisfy both the antecedent and consequent of H because the only time “All ravens are black” is disconfirmed is when some raven is non-black. But things are not quite so simple! To see why, do the following exercises. (b) Using models, show that ∀x(Rx → Bx) is equivalent to ∀x(¬Bx → ¬Rx) in every model. To do this show that every model that makes ∀x(Rx → Bx) true also makes ∀x(¬Bx → ¬Rx) true and every model that makes ∀x(¬Bx → ¬Rx) true also makes ∀x(Rx → Bx) true. Now consider Hempel’s Equivalence Condition which says that if E confirms H and H′ is equivalent to H, then E confirms H′. The Equivalence Condition should be satisfied by any logic of confirmation because, otherwise, the establishment of a confirmation relation would depend on the mode of formulation or presentation of the hypothesis, which would contradict the goal of finding a logic of inductive inference, which should be entirely formal. (c) Explain why (¬Ba ∧ ¬Ra) confirms ∀x(¬Bx → ¬Rx)? Hint: Is (¬Ba ∧ ¬Ra) a positive instance of ∀x(¬Bx → ¬Rx)? (d) Explain why (¬Ba ∧ ¬Ra) also confirms ∀x(Rx → Bx)? Hint: Use Hempel’s Equivalence Condition. (e) Explain why you can read the predicate ¬Bx as "x is white" and the predicate ¬Rx as "x is a piece of chalk"? (f) Do you find it paradoxical that evidence of a white piece of chalk confirms the scientific sentence that "All ravens are black"? Why or why not? Where do you think the paradox comes in?
Question 3
Consider the general sentence “All ravens are black.” What could be more intuitive than to say that this sentence is scientifically confirmed just in case existing ravens we have observed so far are, in fact, black.
Let H = ∀x(Rx → Bx) be the paraphrase of the scientific sentence or hypothesis “All ravens are black”. Call sentence E = (Ra ∧ Ba) confirmatory evidence for H if E is true. Nicod’s criterion of confirmation says, roughly, that E confirms H if E is true.
(a) Using models, explain why the following argument is valid.
1. (Ra∧¬Ba)
2. ¬∀x(Rx → Bx)
Since (a) is a valid argument, one way to think of Nicod’s criterion for confirmation is that a claim like “All ravens are black” is confirmed by its positive instances, i.e., terms that satisfy both the antecedent and consequent of H because the only time “All ravens are black” is disconfirmed is when some raven is non-black.
But things are not quite so simple! To see why, do the following exercises.
(b) Using models, show that ∀x(Rx → Bx) is equivalent to ∀x(¬Bx → ¬Rx) in every model. To do this show that every model that makes ∀x(Rx → Bx) true also makes ∀x(¬Bx → ¬Rx) true and every model that makes ∀x(¬Bx → ¬Rx) true also makes ∀x(Rx → Bx) true.
Now consider Hempel’s Equivalence Condition which says that if E confirms H and H′ is equivalent to H, then E confirms H′. The Equivalence Condition should be satisfied by any logic of confirmation because, otherwise, the establishment of a confirmation relation would depend on the mode of formulation or presentation of the hypothesis, which would contradict the goal of finding a logic of inductive inference, which should be entirely formal.
(c) Explain why (¬Ba ∧ ¬Ra) confirms ∀x(¬Bx → ¬Rx)? Hint: Is (¬Ba ∧ ¬Ra) a positive instance of ∀x(¬Bx → ¬Rx)?
(d) Explain why (¬Ba ∧ ¬Ra) also confirms ∀x(Rx → Bx)? Hint: Use Hempel’s Equivalence Condition.
(e) Explain why you can read the predicate ¬Bx as "x is white" and the predicate ¬Rx as "x is a piece of chalk"?
(f) Do you find it paradoxical that evidence of a white piece of chalk confirms the scientific sentence that "All ravens are black"? Why or why not? Where do you think the paradox comes in?
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