8.4 Comparison of first-order, second-order, and higher-order logics
4 min read•august 7, 2024
First-order, second-order, and higher-order logics form a hierarchy of increasing expressive power. Each level builds on the previous, allowing quantification over more complex entities and capturing more sophisticated mathematical concepts.
However, this increased comes at a cost. While has nice properties like and compactness, higher-order logics sacrifice these for greater power. This trade-off shapes their use in different areas of mathematics and computer science.
Logical Properties
Expressive Power and Limitations
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- First-order logic has limited expressiveness compared to higher-order logics
- Cannot quantify over predicates or functions, only variables
- Struggles to express concepts like infinity, finiteness, or connectedness
- extends first-order logic by allowing quantification over predicates and functions
- Can express properties of relations and functions directly (continuity)
- Captures important mathematical concepts first-order logic cannot (Peano arithmetic, real analysis)
- Higher-order logics further extend quantification to predicates and functions of higher types
- Even more expressive than second-order logic (can define truth predicates)
- Useful for formalizing mathematics and computer science (type theory, category theory)
Decidability and Completeness Properties
- First-order logic is semi-decidable
- Valid formulas are recursively enumerable, but invalid ones may not be
- Has a complete proof system (all valid sentences are provable)
- Second-order logic and higher-order logics are not decidable in general
- Validity is not recursively enumerable due to higher expressiveness
- Do not have complete proof systems (Gödel's incompleteness theorems)
- Restricted fragments of higher-order logics can be decidable (propositional, monadic)
Compactness and Löwenheim-Skolem Properties
- First-order logic has the compactness property
- If every finite subset of a set of sentences has a model, the whole set has a model
- Useful for constructing models (ultraproducts, nonstandard analysis)
- First-order logic satisfies the Löwenheim-Skolem theorems
- If a sentence has an infinite model, it has models of all infinite cardinalities
- Leads to non-categoricity of first-order theories (Peano arithmetic has nonstandard models)
- Second-order logic and higher-order logics do not have compactness or Löwenheim-Skolem properties in general
- Due to the expressive power to characterize infinite structures up to isomorphism (standard model of arithmetic)
Theoretical Foundations
Model-Theoretic Semantics
- First-order logic has a well-developed based on structures and interpretations
- Structures consist of a domain of discourse and interpretations of function and symbols
- Satisfaction relation defined inductively on the structure of formulas
- Second-order logic extends first-order with second-order variables and quantifiers
- Variables range over subsets and functions on the domain
- Full semantics vs. Henkin semantics (nonstandard models)
- Higher-order logics have a more complex model theory due to the hierarchy of types
- Often interpreted in typed structures or extensional models (all functions and predicates included)
- Can also have intensional or nonstandard semantics (general models)
Proof-Theoretic Aspects
- First-order logic has a variety of complete proof systems
- Hilbert-style systems, , sequent calculus
- Proofs are finite syntactic objects, can be checked algorithmically
- Second-order logic does not have a complete proof system with standard semantics
- Categorical theories like Peano arithmetic are not recursively axiomatizable
- Henkin semantics allow for a wider class of models and complete proof systems
- Higher-order logics have proof systems based on typed lambda calculi
- Curry-Howard correspondence relates proofs and programs (dependent type theory)
- Proof assistants implement higher-order proof systems (Coq, Isabelle/HOL)
Computational Aspects
Computational Complexity and Decidability
- First-order logic has many decidable fragments used in computer science applications
- Propositional logic is decidable and NP-complete (SAT solvers)
- Monadic first-order logic and two-variable logic are decidable (description logics, modal logics)
- Second-order logic is not decidable, but has decidable fragments
- Monadic second-order logic over finite structures (WS1S) used in verification (Buchi automata)
- Weak second-order logic restricts quantification to finite sets (decidable)
- Higher-order logics are generally not decidable, but have important computable fragments
- Higher-order recursive functions (System T) and polymorphic lambda calculus (System F)
- Constructive type theories used in proof assistants (Calculus of Constructions)
Abstraction Levels and Applications
- First-order logic is used in many areas of computer science for specification and modeling
- Database query languages (SQL), knowledge representation (description logics)
- Program verification (Hoare logic), model checking (temporal logics)
- Second-order logic can express important program properties not captured by first-order logic
- Transitive closure, graph connectivity (monadic second-order logic)
- Program schemes and software synthesis (weak second-order logic)
- Higher-order logics are used in programming language theory and formal verification
- Type systems and semantics of functional programming languages (polymorphic lambda calculus)
- Hardware and software verification using proof assistants (Coq, Isabelle/HOL)
- Automated theorem proving in (TPS, LEO)
Key Terms to Review (22)
Church's Theorem: Church's Theorem states that every effectively calculable function is a lambda-definable function, which implies that the class of functions computable by a Turing machine is the same as those definable in lambda calculus. This theorem connects the fields of logic and computer science by establishing a fundamental equivalence between these two systems, illustrating the limits of computation and the foundational concepts of higher-order logics.
Completeness: Completeness refers to a property of a formal system where every statement that is true in all models of the system can be proven within that system. This concept is crucial because it connects syntax and semantics, ensuring that if something is logically valid, there's a way to derive it through formal proofs.
Decidability: Decidability refers to the property of a logical system or problem being able to be resolved or determined definitively, typically by a mechanical procedure or algorithm. This concept is crucial in understanding which questions can be answered through formal proofs and how they relate to the structure and nature of various logical systems.
Existential Quantifier: The existential quantifier is a symbol used in logic that indicates the existence of at least one element in a given domain that satisfies a particular property. It is usually denoted by the symbol $$\exists$$, and is crucial in formulating statements about existence within first-order logic, connecting it to the syntax and structure of logical expressions, the properties of quantifiers, proof systems that incorporate existential claims, and comparisons with higher-order logics.
Expressiveness: Expressiveness refers to the ability of a logical system to represent various concepts, relationships, and propositions in a precise and versatile manner. This quality is crucial in understanding how different logical systems can articulate complex ideas and support richer reasoning. In the realm of logic, expressiveness helps distinguish between first-order, second-order, and higher-order logics, as each has a different capacity to describe mathematical structures and relationships.
First-order logic: First-order logic is a formal system used in mathematical logic and philosophy that allows the expression of statements about objects and their properties using quantifiers, predicates, and variables. It serves as a foundational framework for reasoning about the structure of arguments and proofs, making it crucial for understanding various formal systems and their applications in cut elimination, completeness theorems, compactness, and comparisons with other logical systems.
Function symbol: A function symbol is a notation used in logic and mathematics to represent a specific function that maps elements from a domain to a range. It is typically accompanied by a fixed number of arguments and allows for the formulation of expressions that can be evaluated within a given logical system. In the context of first-order, second-order, and higher-order logics, function symbols play a crucial role in defining relationships and operations among objects, enabling more complex reasoning.
Gödel's Completeness Theorem: Gödel's Completeness Theorem states that in first-order logic, every logically valid formula can be proven using a formal proof system. This theorem establishes a connection between semantic truth (truth in every model) and syntactic provability (provable from axioms), showing that if a statement is true in every model, then there is a formal proof of that statement within the system. It plays a significant role in understanding the foundational aspects of mathematical logic and formal systems.
Higher-order logic: Higher-order logic is a type of formal logic that extends first-order logic by allowing quantification not only over individual variables but also over predicates and functions. This means it can express more complex statements and properties, making it powerful for various applications in mathematics and computer science.
Hilbert-style system: A Hilbert-style system is a formal deductive system used in mathematical logic that consists of a set of axioms and inference rules, which allows for the derivation of theorems. It emphasizes a minimalistic approach by using a small number of axioms, making it suitable for exploring foundational aspects of logic, particularly in the context of comparing first-order, second-order, and higher-order logics.
Kripke semantics: Kripke semantics is a framework used for interpreting modal and intuitionistic logics through the use of possible worlds and accessibility relations. It connects the truth of propositions to various contexts, represented by these possible worlds, which can vary in their relationships, allowing for a nuanced understanding of necessity and possibility in logic.
Modal logic: Modal logic is a type of formal logic that extends classical logic to include operators expressing modality, such as necessity and possibility. This allows for reasoning about statements that are not strictly true or false, enabling discussions about what could be, must be, or might have been. Modal logic connects to various areas, enhancing our understanding of semantics, proof structures, and computational applications.
Model Theory: Model theory is a branch of mathematical logic that deals with the relationship between formal languages and their interpretations, or models. It examines how structures can satisfy various logical formulas, helping us understand the meanings and implications of different logical systems, including their syntax, proof theory, soundness, completeness, and expressive power across varying levels of logic.
Natural Deduction: Natural deduction is a proof system used in logic that allows one to derive conclusions from premises using a set of inference rules in a structured and intuitive way. It emphasizes the natural reasoning process, enabling proofs to be constructed through the application of these rules without the need for additional axioms or complex structures, making it particularly useful in various fields of mathematics and philosophy.
Predicate: A predicate is a statement or expression that asserts something about one or more subjects, typically involving a property or relation that the subjects satisfy. In first-order logic, predicates are crucial for forming sentences and defining relationships between objects, serving as the backbone for expressing facts and drawing conclusions within a logical system.
Proof Calculus: Proof calculus is a formal system that provides a set of rules and symbols for constructing proofs in logical systems. It lays the foundation for how statements can be derived and manipulated within different levels of logic, including first-order, second-order, and higher-order logics, highlighting the strengths and limitations of each level when it comes to expressiveness and proof techniques.
Quantifier: A quantifier is a logical construct used in first-order logic to express the quantity of instances that a predicate applies to. There are primarily two types of quantifiers: the universal quantifier, which indicates that a statement is true for all instances, and the existential quantifier, which indicates that a statement is true for at least one instance. Quantifiers are essential in defining the scope and validity of statements, helping to form complex expressions in logical reasoning.
Second-order logic: Second-order logic extends first-order logic by allowing quantification not just over individual variables, but also over predicates and relations. This added expressiveness enables more complex statements about sets and properties, bridging the gap towards higher-order logics that further generalize this concept.
Semantics: Semantics is the branch of linguistics and logic that deals with meaning, interpretation, and the relationships between symbols and what they represent. It plays a crucial role in understanding how different logical systems convey information, and it helps to analyze the implications of statements within various logical frameworks. The study of semantics is essential for grasping concepts like completeness and the distinctions between different orders of logic.
Set Theory: Set theory is a fundamental branch of mathematical logic that deals with the study of sets, which are collections of distinct objects considered as a whole. This area of logic serves as a foundation for various mathematical disciplines and is closely linked to proof theory, particularly in understanding how different logical systems can represent and manipulate sets.
Syntax: Syntax refers to the formal structure and rules that govern the combination of symbols in a logical language. It includes the rules for forming valid expressions and statements, which are essential for understanding how logical systems operate. Syntax is crucial for distinguishing between well-formed formulas and those that are not, enabling clear communication within proof systems and logical frameworks.
Universal Quantifier: The universal quantifier is a symbol used in logic to denote that a property or condition holds for all elements within a specific domain. It is typically represented by the symbol '$$\forall$$' and plays a crucial role in expressing statements that are true for every member of a particular set. Understanding this concept is essential for grasping the syntax and formation rules of first-order logic, as well as exploring quantifiers, proof systems, and comparisons with higher-order logics.