MHB Predicate Logic: Semantics and Validity

AI Thread Summary
The discussion centers on the interpretation of a predicate logic formula given a specific domain and set of conditions. The user questions why the interpretation of (∀x)[Bx ⊃ (Lxx ⊃ Lxa)] is considered false, despite believing that (Lxx ⊃ Lxa) evaluates to true if Lxa is true. The false case presented, Ba ⊃ (Laa ⊃ Laa), is also debated regarding its truth value. Ultimately, the formula is confirmed to be true in the provided interpretation, indicating a misunderstanding of the conditions leading to the false interpretation. The conversation highlights the complexities of evaluating predicate logic semantics and validity.
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.
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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