Systems of formal logic

Systems of formal logic are structured frameworks used to evaluate the validity of arguments and reason about propositions through a series of formal rules and symbols. These systems aim to provide a precise method for deducing truths and identifying logical relationships. Here are some key components and concepts involved in formal logic: 1. **Syntax**: This refers to the formal rules that govern the structure of sentences in a logic system.

Substructural logic is a category of non-classical logics that arise from modifying or rejecting some of the structural rules of traditional logics, such as classical propositional logic. The term "substructural" reflects the idea that these logics investigate the structural properties of logical inference. In classical logic, some key structural rules include: 1. **Weakening**: If a conclusion follows from a set of premises, it also follows from a larger set of premises.

Alternative semantics is a theoretical framework in the field of linguistics and philosophy of language that seeks to explain how the meaning of sentences can be understood in relation to possible alternatives. This approach often contrasts with traditional truth-conditional semantics, which primarily focuses on the conditions under which a statement is true or false. The core idea of alternative semantics is that speakers often convey meanings that extend beyond mere truth conditions by considering different perspectives, contexts, or alternatives.

Attributional calculus, often referred to in the context of reasoning and inference systems, is a formal framework used to model and manipulate complex relationships between events, entities, or concepts. Although not a widely recognized term in standard mathematical literature, the concept can generally relate to reasoning about causation and the attribution of causes and effects within a logical framework.

The Aṣṭādhyāyī is a foundational text of Sanskrit grammar composed by the ancient scholar Pāṇini around the 4th century BCE. The title "Aṣṭādhyāyī" translates to "eight chapters," which reflects the structure of the work. It consists of around 4,000 sutras (aphorisms or rules) that systematically describe the phonetics, morphology, and syntax of the Sanskrit language.

Dependence logic is a type of logic that extends classical first-order logic by incorporating the concept of dependence between variables. It was introduced by the logician Johan van Benthem in the early 2000s. The key idea is to formalize the notion of dependency between variables, allowing for the expression of statements about how the value of one variable affects or is determined by the values of others.

Discourse Representation Theory (DRT) is a framework in semantics and computational linguistics that seeks to represent the meaning of sentences in a way that accounts for context and the relationships between entities mentioned in discourse. Developed primarily by Hans Kamp in the 1980s, DRT focuses on how information is structured in language, particularly in relation to an unfolding narrative or conversation.

Dynamic semantics is a theoretical approach to understanding the meaning of linguistic expressions that focuses on how context and discourse evolve over time during communication. Unlike static semantics, which views meaning as fixed and derived from the lexical and grammatical properties of expressions alone, dynamic semantics considers how the meaning of sentences can change based on the discourse context and how they interact with previous statements.

Epsilon calculus, also known as epsilon substitution or epsilon calculus of constructions, is a formal system and a framework within mathematical logic and particularly in the foundation of mathematics. It extends first-order logic by incorporating a special operator, usually denoted by the Greek letter epsilon (ε), which is used to express the idea of "the witness" or "the choice" in logical statements. The central idea in epsilon calculus is to allow assertions involving existence to be represented in a more constructive way.

Formal ethics, often referred to as deontological ethics, is a branch of ethical theory that emphasizes the importance of rules, duties, and obligations in determining what is moral. It is characterized by the idea that certain actions are inherently right or wrong, regardless of their consequences. This approach to ethics is concerned with the principles that govern moral behavior and often involves the formulation of universal laws or rules that apply to all individuals.

Frege's propositional calculus, developed by Gottlob Frege in the late 19th century, is one of the earliest formal systems in logic. It represents a significant milestone in the development of mathematical logic and formal reasoning. ### Key Features of Frege's Propositional Calculus: 1. **Propositions and Truth Values**: Frege's calculus deals with declarative sentences (propositions) that can be classified as either true or false.

Higher-order logic (HOL) is an extension of first-order logic that allows quantification not only over individual variables (as in first-order logic) but also over predicates, functions, and sets. This increased expressive power makes higher-order logic more flexible and capable of representing more complex statements and concepts, particularly in areas like mathematics, computer science, and formal semantics.

Implicational propositional calculus is a subset of propositional logic focused specifically on implications, a fundamental logical connective. In propositional logic, the primary logical connectives include conjunction (AND), disjunction (OR), negation (NOT), implication (IF...THEN), and biconditional (IF AND ONLY IF). ### Key Features 1.

Independence-friendly logic (IF logic) is a type of logical framework that extends classical propositional logic and first-order logic by allowing for the expression of certain forms of independence among variables or propositions. It was introduced by the philosopher and logician Johan van Benthem in the context of epistemic and modal logic.

Infinitary logic is an extension of classical logic that allows for formulas to have infinite lengths, enabling the expression of more complex properties of mathematical structures. Unlike standard first-order or second-order logics, where formulas are made up of a finite number of symbols, infinitary logic permits formulas with infinitely many variables or connectives.

Intermediate logic refers to a class of logical systems that occupy a middle ground between classical logic and intuitionistic logic. In classical logic, the Law of Excluded Middle (LEM) holds, which states that for any proposition, either that proposition or its negation must be true. Intuitionistic logic, on the other hand, does not accept the Law of Excluded Middle as a general principle, emphasizing constructive proofs where the existence of a mathematical object must be demonstrated explicitly.

"Logics for computability" generally refers to various formal systems and logical frameworks used to study computability, decidability, and related concepts in theoretical computer science and mathematical logic. This field intersects with areas such as recursion theory, model theory, and proof theory, focusing on the relationship between logic and computational processes.

Many-sorted logic is a type of logic that extends classical first-order logic by allowing variables to take values from multiple distinct types or sorts. In a many-sorted logic system, the domain of discourse is divided into different sorts, each representing a different type of object. This contrasts with standard first-order logic, where there is typically a single domain of discourse.

Paraconsistent logic is a type of non-classical logic that allows for the coexistence of contradictory statements without descending into triviality (where every statement would be considered true). In classical logic, if a contradiction is present, any statement can be proven true, a principle known as the principle of explosion (ex contradictione quodlibet). Paraconsistent logic, on the other hand, seeks to handle contradictions in a controlled manner.

Second-order logic (SOL) is an extension of first-order logic (FOL) that allows quantification not only over individual variables (such as objects or elements of a domain) but also over predicates or sets of individuals. This additional expressive power makes second-order logic more powerful than first-order logic in certain ways, allowing for the formulation of more complex statements about mathematical structures and relationships.

Zeroth-order logic is a concept in the realm of formal logic and mathematical logic that serves as a foundational or minimalistic framework for reasoning. It is often described as a system that lacks quantifiers, meaning it does not include the ability to express statements involving variables that can range over a domain of objects (as seen in first-order logic and higher).

Ω-logic (Omega-logic) is a term that can refer to various concepts depending on the context, usually relating to formal systems in logic, mathematics, or computer science. However, it is not a widely recognized or standard term in mainstream logic or mathematics.