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Unformatted text preview: Notes for Week 7 of Confirmation 10/17/07 Branden Fitelson 1 Some Properties of Confirmation Relations Hempel (in his second installment) discusses various properties that confirmation relations might have. I will discuss a longer list of properties. Here are a bunch of properties that we’ll discuss today. We will assume throughout most of our discussion that all of our E ’s, H ’s, and K ’s are logically contingent. (M E ) If E confirms H relative to K , then E & E confirms H relative to K (provided that E does not contain any constant symbols not already contained in { H,E,K } ). (M K ) If E confirms H relative to K , then E confirms H relative to K & K (provided that K does not contain any constant symbols not already contained in { H,E,K } ). (NC) [ φ x & ψ x \ confirms [ ( ∀ y)(φy ⊃ ψy) \ relative to (some/all/specific) K . (SCC) If E confirms H relative to K and H K H , then E confirms H relative to K . (CCC) If E confirms H relative to K and H K H , then E confirms H relative to K . (CC) If E confirms H relative to K and E confirms H relative to K , then K ø ∼ (H & H ) . (CC ) If E confirms H relative to K and E confirms H relative to K , then K ø ∼ (H ≡ H ) . (EC) If E K H , then E confirms H relative to K . (CEC) If H K E , then E confirms H relative to K . (EQC E ) If E confirms H relative to K and K E ≡ E , then E confirms H relative to K . (EQC H ) If E confirms H relative to K and K H ≡ H , then E confirms H relative to K . (EQC K ) If E confirms H relative to K and K K , then E confirms H relative to K . (NT) For some E , H , and K , E confirms H relative to K . And, for every E / K , there exists an H such that E does not confirm H relative to K . (ST) If E confirms H relative to K and E confirms H relative to ∼ K , then E confirms H relative to > . As exercises, let’s think about some subsets of this large set of conditions. Consider the following triples: • (NT), (CEC), (SCC) – Inconsistent. Pick an E . Then, by (NT), E does not confirm (some) H relative to (some) K . But, by (CEC), E confirms H & E , relative to K . Then, by (SCC), E confirms H , relative to K . Contradiction. • (NT), (EC), (CCC) – Inconsistent. Pick an E . Then, by (NT), E does not confirm H relative to K . By (CCC), E does not confirm H ∨ E relative to K (if it did, then, by (CCC), it would also confirm the logically stronger H , contrary to our initial assumption). By (EC), E confirms H ∨ E relative to K . Contradiction. • (NT), (CCC), (SCC) – Inconsistent. By (NT), (some) E confirms (some) H relative to (some) K . By (CCC), E confirms H & H relative to K . By (SCC), E confirms H relative to K . But, H was arbitrary here. So, we have found an E / K such that, for all H , E confirms H relative to K , which contradicts (NT)....
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 Fall '06
 FITELSON
 Logic, Raa Hempel

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