Reasoning about Programs by exploiting the environment
... would then be incomplete for this new environment. Weakening the assumptions could add feasible behaviors; the logic for the original environment would then become unsound. For example, any of the programming logics for shared-memory concurrency (e.g. [0G76]) could be used to prove that program of F ...
... would then be incomplete for this new environment. Weakening the assumptions could add feasible behaviors; the logic for the original environment would then become unsound. For example, any of the programming logics for shared-memory concurrency (e.g. [0G76]) could be used to prove that program of F ...
Binary aggregation with integrity constraints Grandi, U. - UvA-DARE
... The second step consists in checking whether the axiomatic properties required for the aggregation procedure clash with the integrity constraint that defines the domain, making use of our characterisation results from Chapter 4. In our case we can observe that ICpair is expressed in what we called t ...
... The second step consists in checking whether the axiomatic properties required for the aggregation procedure clash with the integrity constraint that defines the domain, making use of our characterisation results from Chapter 4. In our case we can observe that ICpair is expressed in what we called t ...
Probability Captures the Logic of Scientific
... If the provisos do not hold then—as can be proved—C(H, E, D) might be false even though E is a logical consequence of H.D. The provisos make sense intuitively; if H is already certainly true or false, or if the evidence is certain to obtain even if H is false, then we do not expect E to confirm H. N ...
... If the provisos do not hold then—as can be proved—C(H, E, D) might be false even though E is a logical consequence of H.D. The provisos make sense intuitively; if H is already certainly true or false, or if the evidence is certain to obtain even if H is false, then we do not expect E to confirm H. N ...
Countable and Uncountable Sets
... = f(m). Then 2n = 2m, and so n = m. To see that it is surjective, suppose that t is some even positive integer. Then t = 2k for some positive integer k and f(k) = t. ...
... = f(m). Then 2n = 2m, and so n = m. To see that it is surjective, suppose that t is some even positive integer. Then t = 2k for some positive integer k and f(k) = t. ...
Bounded Functional Interpretation
... with the extraction of bounds is that the non-computable mathematical objects whose existence is claimed by various ineffective principles can sometimes be bounded by computable ones, and this opens the way to obtaining effective bounds for ∀∃ statements as long as these claims have the right logica ...
... with the extraction of bounds is that the non-computable mathematical objects whose existence is claimed by various ineffective principles can sometimes be bounded by computable ones, and this opens the way to obtaining effective bounds for ∀∃ statements as long as these claims have the right logica ...
exam review
... 10. Let T be the statement “ real numbers x, if -1 < x ≤ 0 then x + 1 > 0”. (a) Write the converse of T: (b) Write the contrapositive of T: ...
... 10. Let T be the statement “ real numbers x, if -1 < x ≤ 0 then x + 1 > 0”. (a) Write the converse of T: (b) Write the contrapositive of T: ...
Carmichael numbers with three prime factors
... The first two Carmichael numbers generated by (5, 1) are 7 × 73 × 103 and 17 × 53 × 1201. There are no others with p < 100. We now return to the question of bounds for q. This is rather more tricky, because q(h, ∆) increases with h. Recall that (4) gives at once q ≤ 2p(p − 1). The optimal bound, due ...
... The first two Carmichael numbers generated by (5, 1) are 7 × 73 × 103 and 17 × 53 × 1201. There are no others with p < 100. We now return to the question of bounds for q. This is rather more tricky, because q(h, ∆) increases with h. Recall that (4) gives at once q ≤ 2p(p − 1). The optimal bound, due ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.