Predicate Logic: Semantics and Validity

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SUMMARY

The discussion centers on the evaluation of the predicate logic formula (∀x)[Bx ⊃ (Lxx ⊃ Lxa)] within the domain D = {a,b} and the specific interpretations ~Ba, Bb, Laa, ~Lab, Lba, and ~Lbb. The participants clarify that the formula evaluates to true under the provided conditions, particularly noting that the false case Ba ⊃ (Laa ⊃ Laa) is indeed true. The confusion arises from the misinterpretation of the implications involved in the logical structure.

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joyofbitz
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Hello,

Given the domain as:

D = {a,b}; ~Ba & Bb & Laa & ~Lab & Lba & ~Lbb

Why is the interpretation false? (∀x)[Bx ⊃ (Lxx ⊃ Lxa)]

I am having trouble understanding why that is the case because (Lxx ⊃ Lxa) evaluates to true in any case as long as Lxa is true in all cases, so the overall interpration should be true in all cases.

The false case that is given is: Ba ⊃ (Laa ⊃ Laa), but isn't this case true as well?
 
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You are right: the formula (∀x)[Bx ⊃ (Lxx ⊃ Lxa)] is true in the given interpretation.
 

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