Question 4 In this problem we will seek to understand another limitation of trying to use deductive logic to do inductive inference. This is the tacking problem or the problem of irrelevant conjunctions. We say that an inference in an argument is monotonic if adding additional premises to the argument does not make the new argument invalid. For example, we know the inference from F to (F ∨ G) is valid. But if we add any formula H to the premises, then the inference from F and H to (F ∨ G) is still valid because ((F ∧ H) → (F ∨ G)) is a tautology. Using truth tables, show that the following truth-preserving rules of inference are monotonic. In fact, deductive inference is characterized by being monotonic. (a) From F and (F → G), infer G (b) From (F ∧ G), infer G. Now, according to the Hypothetico-Deductive (H-D) account of confirmation, evidence E confirms H relative to background knowledge K if and only if: 1. (H ∧ K) is satisfiable. 2. (H ∧ K) entails E. 3. K alone does not entail E. (c) Suppose E confirms H. Here we show that the H-D account of confirmation suffers from the problem of irrelevant conjunctions or the tacking problem. Let H′ = Pa, H = ∀x(Rx → Bx), K = Ra and E = Ba (i) Show that ((H ∧ H′) ∧ K) is satisfiable by finding a model that makes H, H′ and K true. H′ is the irrelevant conjunction or hypothesis that we have "tacked" to H. It can say anything but here it says that "a is a piece of paper". (ii) Explain why K alone does not entail E. Hint: Is (K → E) a tautology? (iii) Since (H ∧ K) entails E, explain why ((H ∧ H′) ∧ K) entails E. Hint: is the inference monotonic? Suppose someone accepts something like the following principle, which we shall call the special conse- quence condition: If E confirms H and H entails H′, then E confirms H′. (iv) Show that (H ∧ H′) entails H. (v) Explain why E confirms H′. Hint: Use the special consequence condition. Do you find it plausible that E confirms H′? Why or why not?
Question 4
In this problem we will seek to understand another limitation of trying to use deductive logic to do inductive inference. This is the tacking problem or the problem of irrelevant conjunctions. We say that an inference in an argument is monotonic if adding additional premises to the argument does not make the new argument invalid. For example, we know the inference from F to (F ∨ G) is valid. But if we add any formula H to the premises, then the inference from F and H to (F ∨ G) is still valid because
((F ∧ H) → (F ∨ G)) is a tautology.
Using truth tables, show that the following truth-preserving rules of inference are monotonic. In fact, deductive inference is characterized by being monotonic.
(a) From F and (F → G), infer G
(b) From (F ∧ G), infer G.
Now, according to the Hypothetico-Deductive (H-D) account of confirmation, evidence E confirms H relative to background knowledge K if and only if:
1. (H ∧ K) is satisfiable.
2. (H ∧ K) entails E.
3. K alone does not entail E.
(c) Suppose E confirms H. Here we show that the H-D account of confirmation suffers from the problem of irrelevant conjunctions or the tacking problem. Let H′ = Pa, H = ∀x(Rx → Bx), K = Ra and E = Ba
(i) Show that ((H ∧ H′) ∧ K) is satisfiable by finding a model that makes H, H′ and K true. H′ is the irrelevant conjunction or hypothesis that we have "tacked" to H. It can say anything but here it says that "a is a piece of paper".
(ii) Explain why K alone does not entail E. Hint: Is (K → E) a tautology?
(iii) Since (H ∧ K) entails E, explain why ((H ∧ H′) ∧ K) entails E. Hint: is the inference monotonic?
Suppose someone accepts something like the following principle, which we shall call the special conse- quence condition: If E confirms H and H entails H′, then E confirms H′.
(iv) Show that (H ∧ H′) entails H.
(v) Explain why E confirms H′. Hint: Use the special consequence condition. Do you find it plausible that E confirms H′? Why or why not?
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I apologize, for part 4 (c) I typed (iv) incorrectly, there is a "prime" missing. Please correct the answer to the respected problem :
(iv) Show that (H ∧ H′) entails H′
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