where the universal (all) has been replaced with an existential (some/exists) then the argument would not be valid. In Ancient Greece, Aristotle exhaustively considered all possible combinations of universals and existentials in syllogisms, allowing also for the possibility of negations, and collected those corresponding to valid inferences in a classification theorem. For many centuries, that ...

Recently, the old logical notion of forgetting propositional symbols (or reducing the logical vocabulary) has been generalized to a new notion: forgetting literals. The aim was to help the automatic computation of various formalisms which are currently used in knowledge representation, particularly for nonmonotonic reasoning. We develop here a further generalization, allowing propositional symb...

The notion of a compositional language of thought (LOT) has been central in computational accounts of cognition from earliest attempts (Boole, 1854; Fodor, 1975) to the present day (Feldman, 2000; Penn, Holyoak, & Povinelli, 2008; Fodor, 2008; Kemp, 2012; Goodman, Tenenbaum, & Gerstenberg, 2015). Recent modeling work shows how statistical inferences over compositionally structured hypothesis sp...

Most parents clearly understand that childhood vaccinations for potentially lethal infectious diseases are safe, effective, and crucial to children’s health. Decision making on whether to vaccinate or not to vaccinate is multifactorial and complex. While doctors use scientific evidence and research to promote vaccination in vaccine hesitant and vaccine resistant parents, anti-vaccine activists ...

Quantifying the risk carried by an aggregate position Sd := ∑d i=1Xi comprising many risk factors Xi is fundamental to both insurance and financial risk management. Fréchet inequalities quantify the worst-case risk carried by the aggregate position given distributional information concerning its composing factors but without assuming independence. This marginal factor modeling of the aggregate ...

T o make sense of the information agents gather from the Web, they need to reason about it. If the information is precise and correct, they can use engines such as theorem provers to reason logically and derive correct conclusions. Unfortunately, the information is often imprecise and uncertain, which means they will need a probabilistic approach. More than 150 years ago, George Boole presented...

Boolean algebras are those mathematical systems first developed by George Boole in the treatment of logic by symbolic methodsf and since extensively investigated by other students of logic, including Schröder, Whitehead, Sheffer, Bernstein, and Huntington.J Since they embody in abstract form the principal algebraic rules governing the manipulation of classes or aggregates, these systems are of ...

Boolean problems are conventionally formulated and solved using the algebraic framework originally proposed by George Boole for application to mathematical logic problems. Boole’s algebra was used by Claude Shannon for modeling networks of switching relays and later more general manipulation of information in the form of binary digits, or bits. An alternative approach is to model data as an ele...

Contemporary students of logic tend to think of the logic of the connectives as the most basic area of the subject, which can then be extended with a logic of quantifiers. Historically, however, the logic of the quantifiers, in the form of the Aristotelian theory of the syllogism, came first. Truth conditions for negation, conjunction, and disjunction were well understood in ancient times, thou...