A Propositional Modal Logic for the Liar Paradox Martin Dowd
... is impossible since statements of formal arithmetic must have a truth value. One is then prompted to ask whether the truth modality cannot simply be added to the language. The answer to this question is, as Kripke pointed out, yes, provided the basic semantics of formal logic is altered to allow for ...
... is impossible since statements of formal arithmetic must have a truth value. One is then prompted to ask whether the truth modality cannot simply be added to the language. The answer to this question is, as Kripke pointed out, yes, provided the basic semantics of formal logic is altered to allow for ...
notes - Department of Computer Science and Engineering, CUHK
... This is an example of a homogeneous linear recurrence. Such recurrences can be solved using the following guess verify method. Initially, we forget about the “base cases” f (0) = 1, f (1) = 1 and focus on the equations (4). We look for solutions of the type f (n) = xn for some nonzero real number x. ...
... This is an example of a homogeneous linear recurrence. Such recurrences can be solved using the following guess verify method. Initially, we forget about the “base cases” f (0) = 1, f (1) = 1 and focus on the equations (4). We look for solutions of the type f (n) = xn for some nonzero real number x. ...
an interpretation of aristotle`s syllogistic and a certain fragment of set
... formulas (propositions) on a strictly defined form (comp. textbooks and also [2]). But in the system presented by Lukasiewicz all formula are considered which can be formed by means of categorical formulas and classical connectives (see below the definition of the set of syllogistic formulas AS and ...
... formulas (propositions) on a strictly defined form (comp. textbooks and also [2]). But in the system presented by Lukasiewicz all formula are considered which can be formed by means of categorical formulas and classical connectives (see below the definition of the set of syllogistic formulas AS and ...
ON DICKSON`S THEOREM CONCERNING ODD PERFECT
... with equality on the right exactly when A = 1. Suppose A 6= 1. Then the preceding inequalities show that σ(B) - 2B, and so there is a prime p1 dividing σ(B) to a higher power than that to which it divides 2B; for definiteness, fix p1 as the least such prime. It now follows from (2) that p1 | A. Let ...
... with equality on the right exactly when A = 1. Suppose A 6= 1. Then the preceding inequalities show that σ(B) - 2B, and so there is a prime p1 dividing σ(B) to a higher power than that to which it divides 2B; for definiteness, fix p1 as the least such prime. It now follows from (2) that p1 | A. Let ...
2 y x = − 6 y x = y x = − 2 y x = 5 y = − y x = 2 3 y x = 9 y x
... (If the divisor is not linear then the remainder might not be just one constant number but a polynomial – and this theorem will not work) Rational Root Test A polynomial ...
... (If the divisor is not linear then the remainder might not be just one constant number but a polynomial – and this theorem will not work) Rational Root Test A polynomial ...
Math Lesson-2.notebook
... The 6 members of a Model UN club must choose 2 representatives to attend a state convention. Use Pascal's Triangle to find the number of combinations of 2 members that can be chosen as representatives. ...
... The 6 members of a Model UN club must choose 2 representatives to attend a state convention. Use Pascal's Triangle to find the number of combinations of 2 members that can be chosen as representatives. ...
BASIC SET THEORY
... an element has to satisfy: {x: x is at least 5 feet tall} {x: x is divisible by 2} these are the {y: y is divisible by 2} same set! The variables x and y above are known as “dummy variables”; they have no meaning outside the expression. That’s why we can change the name. ...
... an element has to satisfy: {x: x is at least 5 feet tall} {x: x is divisible by 2} these are the {y: y is divisible by 2} same set! The variables x and y above are known as “dummy variables”; they have no meaning outside the expression. That’s why we can change the name. ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.