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By J.R. Büchi, D. Siefkes, G.H. Müller

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Extra resources for Decidable Theories: Vol. 2: The Monadic Second Order Theory of All Countable Ordinals

Example text

So Ralph is a dog. Arguing that ‘either . . or . ’ should be formalized as classical disjunction and ‘no way’ as classical negation, this is formalized as: Ï {Ralph is a dog ∨ Howie is a duck, (Howie is a duck)} Å Ï Å Ralph is a dog And that is a valid inference, as it has the form: {p 7 ∨ p10 , p10} p 7 . We can also talk about two propositions being semantically equivalent. For example, ‘ (Ralph is a dog ∧ (George is a duck))’ is semantically equivalent to ‘Ralph is a dog → George is a duck’.

I. For each i = 0, 1, 2, . . , ( pi ) is an atomic wff, to which we assign the number 1. Ï ii. If A and B are wffs and the maximum of the numbers assigned to A and to B is n, then each of ( A), (A→ B), (A∧ B), (A∨ B) is a compound wff to which we assign the number n + 1. These are compound wffs. iii. A concatenation of symbols is a wff if and only if it is assigned some number n ≥ 1 according to (i) or (ii) above. ___________________________________________________________________ Ï We usually refer to this language as L( , → , ∧ , ∨ ) since we usually use the same propositional variables.

D. e. f. g. 25 7 is not even. If 7 is even, then 7 is not odd. Unless Milt has a dog, Anubis is the best-fed dog in Cedar City Horses eat grass because grass is green. It’s impossible that 2 + 2 ≠ 4. Anubis, you know, eats, sleeps, and barks at night. You can’t make an omelette without breaking eggs. 6. Formalize and discuss an example from a mathematics text. 7. Formalize each of the following inferences. Then evaluate it for validity in classical logic. To do that, attempt to determine if it is possible for the premises to be true and the conclusion false.

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