The propositional logic
of the map develops the theoretical underpinnings of maps and map use.
Using propositional logic
we can prove only the theorems of a very modest part of mathematics: Boolean algebra, "naive" set theory, etc.
In propositional logic
, the operators [for all] and [there exists] are used respectively to indicate necessity and possibility.
In the first place, the density was computed for various fragments of classical propositional logic
(see (9), (13), (1) and (5)).
2001) allowing students to build formal proofs in propositional logic
while receiving step-by-step, contextualised feedback.
He works through informal logic, including meanings and definition of language and fallacies, formal logic, including categorical propositions and syllogisms, propositional logic
, natural deduction in propositional logic
and predicate logic, inductive logic, including analogies, legal and moral reasoning, causality and Mill's methods, probability, statistical reasoning, and hypothetical and scientific reasoning.
Marenbon concentrates especially on work that shows Abelard as the rediscoverer of propositional logic
(Chris Martin); as a subtle explorer of problems about modality (Simo Knuuttila, Herbert Weidemann) and semantics (Klaus Jacobi); as a metaphysician before the reception of Aristotle's Metaphysics (Peter King); and as an ethical thinker who echoes the Stoics (Calvin Normore) and anticipates Kant (Peter King).
Bergmann (computer science emerita, Smith College) then reviews classical propositional logic
, including its language and semantics, and the language and semantics of first-order logic.
She argues that such arguments were neither part of Aristotle's dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle's logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic
, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some twentieth-century theories.
He covers Konig's Lemma (including two ways of looking at mathematics), posets and maximal elements (including order), formal systems (including post systems and compatibility as bonuses), deduction in posets (including proving statements about a poset), Boolean algebras, propositional logic
(including a system for proof about propositions), valuations (including semantics for propositional logic
), filters and ideals (including the algebraic theory of Boolean algebras), first-order logic, completeness and compactness, model theory (including countable models) and nonstandard analysis (including infinitesimal numbers).
Nevertheless, the restriction to propositional logic
has its downside.
limit's and Taylor's theorem, including series representations and Taylor polynomials, infinite series, including both the positive and the general, beginning logic, including propositional logic
, predicates and quantifiers, and proofs, real numbers, functions such as derivatives and a substantial pair of chapters on integrals.