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Having established both syntax and semantics, we now delve into the essential methods used to formally prove statements in predicate logic. Specifically, we’ll explore Natural Deduction and Hilbert-style systems, illustrating their application to real mathematical reasoning. Natural Deduction in Predicate Logic Natural deduction is a formal proof system that extends naturally from propositional logic, incorporating… Continue reading.
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In the previous posts, we’ve discussed the syntax of predicate logic, outlining how terms, predicates, and formulas are formed. Now, we’ll explore semantics, explaining how meaning is formally assigned to these formulas. Structures and Interpretations The meaning of formulas in predicate logic is given by structures (or interpretations). Intuitively, a structure assigns concrete meanings to… Continue reading.
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In my previous post, I introduced variables and explained how C++ stores and manages data using fundamental data types. Now, I will delve deeper into how memory works in C++ and introduce two powerful features: pointers and references. Understanding memory management is crucial for becoming proficient in C++. It will give you greater control over… Continue reading.
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In numerical computations, errors can propagate through calculations, potentially leading to significant inaccuracies in results. Understanding how errors propagate and how the conditioning of a problem affects numerical stability is crucial for designing robust numerical algorithms. In this post, I will discuss error propagation and the concept of conditioning in numerical problems. Error Propagation Errors… Continue reading.
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Building on our understanding of stochastic processes and Brownian motion, we now delve deeper into the mathematical framework essential for modeling financial systems—Itô calculus and stochastic differential equations (SDEs). These tools allow us to rigorously handle randomness in continuous-time finance models. Stochastic Differential Equations Explained A stochastic differential equation (SDE) describes how a stochastic process… Continue reading.
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In the previous post, we saw why propositional logic is not sufficient to express general mathematical statements, and why predicate logic is required.In this post, we’ll explore the formal syntax of predicate logic, detailing how statements are constructed using terms, predicates, quantifiers, and formulas. Terms In predicate logic, terms represent objects within a given domain.… Continue reading.
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Why Predicate Logic? In our journey through formal mathematics, we’ve explored propositional logic—a powerful yet limited tool. Propositional logic allows us to reason about the truth of statements built from simpler ones using logical connectives. However, it falls short when we need to express statements about generalizations, properties, or specific relationships involving objects. Predicate logic… Continue reading.
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Humans have always sought knowledge, driven by an insatiable curiosity that compels us to explore, learn, and question. From ancient civilizations charting the stars to modern scientists probing the quantum world, our relentless desire to understand our surroundings defines us uniquely among Earth’s creatures. Unlike most animals, whose curiosity fades with maturity, humans maintain a… Continue reading.
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Before we delve deeper into predicate logic, it’s important to clarify a few essential concepts from set theory. Predicate logic itself relies on some basic set-theoretic notions for its formal definitions and interpretations. This short introduction provides the minimal set theory you’ll need. Introduction to Sets A set is a collection of distinct objects, called… Continue reading.
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In my previous post of this thread, I introduced the basic structure of a simple C++ program. Before moving on to more advanced topics like memory management, pointers, and references, I want to cover a fundamental concept: variables. Variables are an essential building block of programming. They let you store, access, and manipulate data in… Continue reading.
