Glossary

9.1 Foundations of intuitionistic logic

3 min read•august 7, 2024

challenges classical logic's assumptions, rejecting the and . It's rooted in , which demands mathematical objects and proofs be constructible, not just abstract concepts.

The gives intuitionistic logic meaning through proofs. It defines based on how propositions are proven, emphasizing the link between truth and in .

Constructivism and Intuitionistic Logic

Philosophical Foundations of Constructivism

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  • Constructivism holds that mathematical objects and proofs must be constructible
  • Rejects the existence of abstract mathematical objects independent of the mind
  • Contrasts with Platonism which asserts the existence of abstract mathematical objects
  • Constructive proofs provide explicit procedures for constructing mathematical objects (algorithms, finite processes)

Intuitionistic Logic and Classical Logic

  • Intuitionistic logic rejects certain principles of classical logic
  • Law of excluded middle (P∨¬PP \lor \neg P) not accepted as universally valid
    • Asserts that every proposition is either true or false
    • Constructivists require proof of PP or proof of ¬P\neg P to accept the
  • Double elimination (¬¬P→P\neg \neg P \rightarrow P) not accepted as valid
    • In classical logic, if a proposition is not false, it is considered true
    • Intuitionistic logic distinguishes between "not false" and "provably true"
  • requires explicit proof or construction of the object in question
    • Truth is equated with provability or constructibility
    • Contrasts with classical notion of truth as correspondence to objective reality

Semantics and Interpretation

Brouwer-Heyting-Kolmogorov (BHK) Interpretation

  • Provides an informal semantics for intuitionistic logic based on the notion of proof
  • Interprets logical connectives in terms of the proofs of the propositions involved
    • P∧QP \land Q is proven if both PP and QQ are proven
    • P∨QP \lor Q is proven if either PP or QQ is proven
    • P→QP \rightarrow Q is proven if there is a construction that transforms any proof of PP into a proof of QQ
    • ¬P\neg P is proven if there is a construction that transforms any hypothetical proof of PP into a contradiction
  • Gives intuitionistic logic a in terms of proofs and constructions

Proof-Theoretic Semantics and Knowability

  • assigns meaning to propositions in terms of the proofs that establish them
    • The meaning of a proposition is determined by the conditions under which it can be proven
    • Contrasts with model-theoretic semantics which assigns meaning in terms of truth conditions
  • states that every true proposition is provable in principle
    • P→◊□PP \rightarrow \Diamond \square P (if PP is true, then it is possible to prove PP)
    • Captures the constructivist idea that truth and provability are closely linked
    • Implies that there are no unknowable or unprovable truths in constructive mathematics
  • Intuitionistic logic can be seen as formalizing the notion of constructive provability
    • Proofs are understood as constructions or algorithms that establish the truth of propositions
    • The meaning of logical connectives is given by the BHK interpretation in terms of proofs

Key Terms to Review (19)

Arend Heyting: Arend Heyting was a significant figure in the development of intuitionistic logic, particularly known for formalizing the syntax and semantics of this logical system. His work laid the groundwork for understanding how intuitionistic logic differs from classical logic, especially regarding proof systems and the underlying philosophical foundations. Heyting’s contributions highlight the importance of constructivism in mathematics, where the existence of a mathematical object is tied to the ability to construct it explicitly.
Brouwer-Heyting-Kolmogorov Interpretation: The Brouwer-Heyting-Kolmogorov interpretation provides a philosophical foundation for intuitionistic logic, emphasizing the constructive nature of mathematical truth. This interpretation connects mathematical statements to the existence of proofs, where a statement is only considered true if there is a method to constructively verify it. It highlights the idea that mathematical objects exist only when they can be explicitly constructed, contrasting sharply with classical logic's more abstract notions of truth.
Conjunction: In logic, a conjunction is a compound statement formed by connecting two or more propositions using the logical connective 'and', symbolized as $$\land$$. This operation is fundamental as it defines how multiple statements can be combined to yield a true or false value based on their individual truth values.
Constructive mathematics: Constructive mathematics is a branch of mathematical logic that emphasizes the constructive aspects of mathematical objects and proofs, requiring that existence claims be supported by explicit examples or algorithms. This approach contrasts with classical mathematics, where existence can often be asserted without providing a constructive method. The philosophy behind constructive mathematics fosters a more intuitive understanding of mathematics and aligns closely with computational practices.
Constructive meaning: Constructive meaning refers to the interpretation of statements within intuitionistic logic that emphasizes the necessity of providing a constructive proof to demonstrate the truth of those statements. In this framework, asserting that a mathematical object exists must be accompanied by a method for constructing that object, making the act of proof integral to understanding and validating mathematical claims.
Constructive Proof: A constructive proof is a type of mathematical proof that not only demonstrates the existence of a mathematical object but also provides a method for actually constructing such an object. This approach emphasizes the idea that existence claims must be supported by explicit examples or algorithms, distinguishing it from non-constructive proofs which may rely on indirect arguments. Constructive proofs are foundational in various areas of logic and mathematics, particularly in contexts that prioritize intuitionistic logic and constructive mathematics.
Constructivism: Constructivism is a philosophical approach to mathematics that emphasizes the necessity of providing explicit constructions or algorithms for mathematical objects and the rejection of non-constructive proofs. It asserts that to claim the existence of a mathematical entity, one must be able to construct it or demonstrate its computability. This approach significantly influences various branches of logic and mathematics, shaping perspectives on proof and validity.
Disjunction: Disjunction is a logical operation that combines two propositions with the word 'or', resulting in a statement that is true if at least one of the propositions is true. This operation plays a key role in formal systems, as it helps construct complex statements, define truth conditions, and facilitate proof strategies.
Double Negation Elimination: Double negation elimination is a principle in logic stating that if a statement is not not true, then it must be true. This concept plays a crucial role in classical logic, where one can derive a proposition directly from its double negation. The principle highlights a key difference between classical and intuitionistic logic, as the latter does not accept double negation elimination as valid without a constructive proof, leading to different interpretations in various mathematical frameworks.
Intuitionistic implication: Intuitionistic implication is a logical connective in intuitionistic logic that expresses the relationship between two propositions, indicating that if the first proposition holds, then the second proposition can be derived. This form of implication differs from classical logic, as it requires a constructive proof of the consequent based on the antecedent, reflecting the intuitionistic philosophy that truth is not merely about tautologies but about what can be constructively demonstrated.
Intuitionistic logic: Intuitionistic logic is a form of logic that emphasizes the constructive nature of mathematical proofs, where a statement is only considered true if there is a method to construct an example demonstrating its truth. This approach leads to different interpretations of logical connectives and quantifiers compared to classical logic, making it essential for understanding various proof systems, the foundations of logic, and connections between different logical frameworks.
Intuitionistic truth: Intuitionistic truth is a concept in logic that asserts that the truth of a proposition is determined by our ability to construct a proof for it, rather than relying solely on classical definitions of truth, such as the law of excluded middle. This perspective emphasizes constructive proofs and the idea that mathematical statements must be verifiable through evidence or construction, leading to a more restrictive understanding of what it means for something to be true.
Knowability Principle: The knowability principle states that if a proposition is true, then it must be knowable or can be known by some agent. This idea emphasizes that knowledge and truth are deeply interconnected, suggesting that truths about the world should be accessible to those who seek to know them. It plays a significant role in discussions around intuitionistic logic, where the focus is on constructivist approaches to knowledge and truth rather than classical logic's more absolute perspectives.
L.E.J. Brouwer: L.E.J. Brouwer was a Dutch mathematician and philosopher known for founding intuitionism, a philosophy of mathematics emphasizing constructivism and the rejection of the law of excluded middle. His work significantly influenced the development of proof theory, particularly in the context of intuitionistic logic, which contrasts with classical logic by requiring that mathematical objects be constructively defined.
Law of Excluded Middle: The law of excluded middle is a fundamental principle in classical logic that asserts for any proposition, either that proposition is true or its negation is true. This concept is pivotal in understanding formal systems and proofs, as it provides a basis for reasoning about the truth values of statements within these systems.
Logical Connectives: Logical connectives are symbols or words used to connect propositions in logic, allowing the formation of more complex statements and expressions. They play a crucial role in determining the truth values of compound statements, influencing how we analyze and construct arguments. By combining simpler propositions using logical connectives, we can express relationships such as conjunction, disjunction, negation, implication, and equivalence, which are essential in both formal proofs and reasoning.
Negation: Negation is a fundamental logical operation that takes a proposition and reverses its truth value. In other words, if a statement is true, its negation is false, and vice versa. This operation is crucial for understanding the structure of logical arguments, the formulation of sentences in first-order logic, and the development of various proof systems.
Proof-theoretic semantics: Proof-theoretic semantics is an approach in the philosophy of logic that focuses on the meanings of logical expressions based on their proof conditions rather than their truth conditions. This perspective emphasizes the role of formal proofs and how they establish the meaning of statements through the rules and processes used to derive them. It connects closely to various branches of logic, particularly intuitionistic logic, which relies heavily on constructive proofs, and it provides a framework for understanding the nature of logical consequence.
Provability: Provability refers to the property of a statement being demonstrable within a formal system based on its axioms and inference rules. It connects deeply with various foundational aspects of mathematical logic, emphasizing the importance of formal proofs in establishing the truth of mathematical propositions.
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