By Herrmann R.A.
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4 Below are diagrams for two logic circuits. For the first circuit, note that if no circuit flows into lines A and B, then there is a current flowing out the C line. In current flows in the A and B line from left to right, then again current flows out the C line. But if current flows in the A and not in the B, or in the B and not in the A, then no current flows out the C line. I’ll let you do the “flow” analyze for the second diagram. In the first diagram, the symbol ցx means that the arrow has been removed from the pathway indicator.
F) Assume that Γ ⊢ A. Now let D be the set of all premises that have been utilized as a specific step marked “premise” that appears in the demonstration Γ ⊢ A. Obviously, we may replace Γ with D and have not altered the demonstration. 1 To show that (¬(¬A)) ⊢ A. (1) (2) (3) (4) (5) (6) (7) (8) (¬(¬A)) (¬(¬A)) → ((¬(¬(¬(¬A)))) → (¬(¬A))) (¬(¬(¬(¬A)))) → (¬(¬A)) ((¬(¬(¬(¬A)))) → (¬(¬A))) → ((¬A) → (¬(¬(¬A)))) (¬A) → (¬(¬(¬A))) ((¬A) → (¬(¬(¬A)))) → ((¬(¬A)) → A) (¬(¬A)) → A A Premise ..... MP( , ) .....
The formula variables A1 , . . , An and only these formula variables are used to construct A with the L′ propositional connectives ¬, → . Thus A is written in formula variables. Since L′ ⊂ L, the truth-table concept can be applied to L′ . Now let a be an assignment to the atoms that would appear in each Ai , i = 1, . . , n when specific formula are substituted for the formula variables. 1 For each i, we define a formula A′i as follows: (i) if v(Ai ) = T, then A′i = Ai . (ii) If v(Ai ) = F, then A′i = (¬Ai ).
Logic For Everyone by Herrmann R.A.