9.2 Syntax and semantics of intuitionistic logic

Intuitionistic logic rejects certain classical principles, like the law of excluded middle. It demands constructive proofs, requiring concrete evidence for each logical step. This approach aligns with the chapter's focus on constructive reasoning in mathematics.

The syntax and semantics of intuitionistic logic differ from classical logic. Kripke semantics and Heyting algebras provide frameworks for understanding intuitionistic logic, while properties like disjunction and existence reflect its constructive nature.

Intuitionistic Propositional and Predicate Logic

Intuitionistic Propositional Calculus

• Formal system of logic based on constructive principles
• Differs from classical propositional calculus by rejecting the law of excluded middle ($A \lor \lnot A$) and double negation elimination ($\lnot\lnot A \rightarrow A$)
• Proofs in intuitionistic propositional calculus require constructive evidence for each step
• Connectives in intuitionistic propositional calculus include conjunction ($\land$), disjunction ($\lor$), implication ($\rightarrow$), and negation ($\lnot$)

Intuitionistic Predicate Logic

• Extension of intuitionistic propositional calculus to include quantifiers ($\forall$ and $\exists$)
• Allows reasoning about properties of objects and relationships between them
• Intuitionistic predicate logic is more expressive than intuitionistic propositional calculus
• Proofs in intuitionistic predicate logic require constructive evidence for quantified statements (e.g., to prove $\forall x P(x)$, one must provide a method for constructing a proof of $P(x)$ for any given $x$)

Intuitionistic Implication and Negation

• Intuitionistic implication ($A \rightarrow B$) asserts the existence of a constructive proof that transforms a proof of $A$ into a proof of $B$
• Intuitionistic negation ($\lnot A$) asserts the existence of a constructive proof that derives a contradiction from $A$
• Intuitionistic implication and negation are weaker than their classical counterparts
• Example: In classical logic, $\lnot\lnot A \rightarrow A$ is a tautology, but in intuitionistic logic, it is not valid without additional assumptions

Semantics and Algebraic Structures

Kripke Semantics

• Provides a semantics for intuitionistic logic using partially ordered sets (posets) of possible worlds or states of knowledge
• Each world in a Kripke model is associated with a set of propositions considered true at that world
• Accessibility relation between worlds represents the growth of knowledge or the availability of new information
• A proposition is true in a Kripke model if it is true in all accessible worlds from the current world
• Example: In a Kripke model with worlds $w_1$ and $w_2$, where $w_1 \leq w_2$, if a proposition $P$ is true at $w_1$, it must also be true at $w_2$

Heyting Algebra

• Algebraic structure that provides a semantics for intuitionistic logic
• Generalizes the notion of a Boolean algebra by replacing the complement operation with a pseudo-complement (intuitionistic negation)
• Elements of a Heyting algebra represent propositions, and the operations correspond to intuitionistic connectives
• The ordering relation in a Heyting algebra represents implication: $a \leq b$ means $a \rightarrow b$ is true
• Example: The set of open subsets of a topological space forms a Heyting algebra, with intersection as conjunction, union as disjunction, and interior of the complement as pseudo-complement

Logical Properties

Disjunction Property

• In intuitionistic logic, if a disjunction ($A \lor B$) is provable, then either $A$ is provable or $B$ is provable
• Reflects the constructive nature of intuitionistic logic: a proof of a disjunction must provide evidence for at least one of the disjuncts
• Contrasts with classical logic, where a disjunction can be true without either disjunct being true (e.g., $P \lor \lnot P$ is a tautology in classical logic)
• Example: If a theorem in intuitionistic logic states $\forall x (P(x) \lor Q(x))$, then for any given $x$, we can constructively find a proof of either $P(x)$ or $Q(x)$

Existence Property

• In intuitionistic predicate logic, if an existential statement ($\exists x P(x)$) is provable, then there exists a specific term $t$ such that $P(t)$ is provable
• Reflects the constructive nature of intuitionistic logic: a proof of an existential statement must provide a witness for the existential quantifier
• Contrasts with classical logic, where an existential statement can be true without a specific witness being known
• Example: If a theorem in intuitionistic predicate logic states $\exists x \forall y R(x, y)$, then there exists a specific term $t$ such that $\forall y R(t, y)$ is provable