Question 5 In this problem we will seek to understand Goodman’s Riddle of Induction. Another way of understanding the scope of Hume’s problem of induction is that it is the problem of providing a justification for good rules of inductive inference. Goodman’s Riddle of Induction shows that good rules of induction cannot be entirely formal in the same way that good rules of inference in deductive logic are formal. The key assumption in the following examples is that on a purely formal account of confirmation the meaning or interpretation of the predicate symbols in the formal system does not matter. All that matters are which transformation rules involving the predicate symbols in the formulas are allowed and which rules are truth preserving. (a) So we can come up with a colorful (pun intended!) predicate symbol B∗x ≡ (Bx∨Wx) for the predicate "x is blite" that is true of x if: · x is observed anytime before January 1st 2100 and is found to be black (Bx); or · x is observed anytime after January 1st 2100 and is found to be white (Wx). The symbol "≡" means "equivalent to". Whenever F ≡ G, you can substitute one for the other while preserving the truth of your formulas. Show that ∀x(Rx → Bx) entails ∀x(Rx → B∗x), i.e., show that any model that makes ∀x(Rx → Bx) also makes ∀x(Rx → (Bx ∨ W x)) true. (b) Explain why the evidence (Ra ∧ Ba) also confirms ∀x(Rx → B∗x). (c) Similarly, let G∗x ≡ (Gx ∨ Bx) be the predicate "x is grue" that is true of x if: · x is observed anytime before January 1st 2100 and is found to be green (Gx); or · x is observed anytime after January 1st 2100 and is found to be blue (Bx). Show that ∀x(Ex → Gx) ("All emeralds are green") entails ∀x(Ex → G∗x) ("All emeralds are grue"), i.e., show that any model that makes ∀x(Ex → Gx) also makes ∀x(Ex → (Gx ∨ Bx)) true. (d) Explain why the evidence (Eb ∧ Gb) also confirms ∀x(Ex → G∗x) Goodman sought the basis of our apparent willingness to generalize inductively with respect to such predicates as “black”, “white”, “green” and “blue,” but not with respect to “blite” and “grue.” In other words, what are the rules that make “black”, “white”, “green” and “blue” projectible predicates into the future; but “blite” and “grue” unprojectible? One may say that this problem only arises if we think that the good rules of inductive inference and confirmation are entirely formal. But if the good rules for inductive inference are not entirely formal, what are they? This is the new riddle of induction.
Question 5
In this problem we will seek to understand Goodman’s Riddle of Induction. Another way of understanding the scope of Hume’s problem of induction is that it is the problem of providing a justification for good rules of inductive inference.
Goodman’s Riddle of Induction shows that good rules of induction cannot be entirely formal in the same way that good rules of inference in deductive logic are formal.
The key assumption in the following examples is that on a purely formal account of confirmation the meaning or interpretation of the predicate symbols in the formal system does not matter. All that matters are which transformation rules involving the predicate symbols in the formulas are allowed and which rules are truth preserving.
(a) So we can come up with a colorful (pun intended!) predicate symbol B∗x ≡ (Bx∨Wx) for the predicate "x is blite" that is true of x if:
· x is observed anytime before January 1st 2100 and is found to be black (Bx); or
· x is observed anytime after January 1st 2100 and is found to be white (Wx).
The symbol "≡" means "equivalent to". Whenever F ≡ G, you can substitute one for the other while preserving the truth of your formulas.
Show that ∀x(Rx → Bx) entails ∀x(Rx → B∗x), i.e., show that any model that makes ∀x(Rx → Bx) also makes ∀x(Rx → (Bx ∨ W x)) true.
(b) Explain why the evidence (Ra ∧ Ba) also confirms ∀x(Rx → B∗x).
(c) Similarly, let G∗x ≡ (Gx ∨ Bx) be the predicate "x is grue" that is true of x if:
· x is observed anytime before January 1st 2100 and is found to be green (Gx); or
· x is observed anytime after January 1st 2100 and is found to be blue (Bx). Show that ∀x(Ex → Gx) ("All emeralds are green") entails ∀x(Ex → G∗x) ("All emeralds are grue"), i.e., show that any model that makes ∀x(Ex → Gx) also makes ∀x(Ex → (Gx ∨ Bx)) true.
(d) Explain why the evidence (Eb ∧ Gb) also confirms ∀x(Ex → G∗x)
Goodman sought the basis of our apparent willingness to generalize inductively with respect to such predicates as “black”, “white”, “green” and “blue,” but not with respect to “blite” and “grue.” In other words, what are the rules that make “black”, “white”, “green” and “blue” projectible predicates into the future; but “blite” and “grue” unprojectible? One may say that this problem only arises if we think that the good rules of inductive inference and confirmation are entirely formal. But if the good rules for inductive inference are not entirely formal, what are they? This is the new riddle of induction.
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