Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Mathematical Logic and Foundations → Predicate Logic ↓


Formal Systems in Predicate Logic


Formal systems in predicate logic are fundamental in mathematical logic and serve as the backbone for understanding logical inference and reasoning. The purpose of this explanation is to clarify the complex workings of formal systems, particularly in the area of predicate logic, by breaking down complex ideas into more digestible parts and presenting them in a simple and structured manner. We will cover terminology, examples, and visual representations to ensure a complete understanding of how formal systems work.

Introduction to predicate logic

Predicate logic, also known as first-order logic, extends the capability of propositional logic by introducing predicates, quantifiers, and variables. While propositional logic deals with simple statements that are either true or false, predicate logic allows us to express more complex statements involving objects and their relations.

Elements of predicate logic include:

  • Predicates: Functions that return true or false, often used to express properties of objects.
  • Quantifiers: symbols such as ∀ (for all) and ∃ (there exists) that express the scope of the variables involved.
  • Variables: Symbols representing objects in the dialog box.
  • Logical connectives: such as ∧ (and), ∨ (or), ¬ (not), → (sign), etc.

Components of a formal system

A formal system in predicate logic includes the following elements:

  1. Alphabet: A set of symbols used to form expressions.
  2. Syntax: rules for creating well-formed formulas (WFFs) with the symbols of the alphabet.
  3. Semantics: Interpretations that give meaning to formulas.
  4. Axiom: A set of elementary statements that are assumed to be true.
  5. Inference rules: Logical principles used to deduce new statements from existing statements.

Alphabet in predicate logic

The alphabet of predicate logic includes the following:

  • P, Q, R, ...: predicate symbols
  • x, y, z, ...: variable symbols
  • ∀, ∃: quantifier
  • ∧, ∨, ¬, →: logical connectives
  • ∈, =: relational symbols
  • Brackets for grouping: ( , )

Syntax of predicate logic

Syntax rules help determine which combinations of symbols are considered well-formed formulas (WFF). Here are some basic rules:

  • A predicate with terms, e.g., P(x), is a WFF.
  • If φ and ψ are WFFs, then (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), and ¬φ are also WFFs.
  • If φ is a WFF and x is a variable, then ∀x φ and ∃x φ are WFFs.

Semantic interpretation

Semantics in predicate logic involves assigning meanings to formulas. To do this, we need to interpret the symbols in the language:

  • Field of interaction: A set of objects that the predicate and variable refer to.
  • Explanation: It gives meaning to predicate symbols, variable symbols and constants in logic.

The interpretation of a formula I can be expressed as follows:

I: P(x) -> True if and only if the object corresponding to x has the property P.

Axioms and inference rules

Axioms are statements that are considered true within a formal system. They form the foundational truths upon which other true statements are built. For example, in arithmetic, statements such as 0 + x = x can be considered axioms.

Inference rules are used to infer new truths from axioms or previously inferred truths. Rules such as modus ponens are examples of this:

Modus ponens:
If (φ → ψ) is true, and φ is true, then ψ will also be true.

Example of a formal system

Consider a simple formal system where our objective is to explore properties of numbers and operations:

  1. Alphabet: P(x), Q(y), +, 0, =, ∀, ∃, ∧, ¬
  2. Self-evident:
    • ∀x (0 + x = x)
    • ∀x ∀y (x + y = y + x)
  3. Inference rule: modus ponens

The axioms above tell us the basic properties of addition. Using these and some inference rules, we can determine new facts, for example, that x + 0 = x, as well as properties such as commutativity.

Example 1: Checking well-formed formulas

Given the following strings, determine which of the formulas are well-formed:

  • ∀x (P(x) → ∃y Q(y))
  • x Q(x → y) ∧ P

Analysis:

  • ∀x (P(x) → ∃y Q(y)): This is a well-formula, because it applies quantifiers and logical implications correctly.
  • x Q(x → y) ∧ P: This is not a well-formed formula because of incorrect logical structure and lack of connective context.

Example 2: Simple semantic assignment

Suppose we have a formula ∀x (P(x) → Q(x)) to analyze within a specific domain:

Workspace Category

Let the domain be the set of natural numbers, and specify:

  • P(x): x is even
  • Q(x): x is divisible by 2

The interpretation indicates that all even numbers are divisible by 2, which is a universally true statement in this context.

Conclusion

Understanding formal systems in predicate logic is crucial to delving deeper into advanced mathematical logic. By mastering the elements of formal systems - alphabet, syntax, semantics, axioms, and inference rules - one can analyze complex logical structures and effectively formalize statements about abstract concepts. Through practical examples and visual aids, the nuances of predicate logic become more clear, leading to a deeper understanding of its application in various mathematical and logical scenarios.


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