**Intuitionistic Logic**

Kripke semantics for the intuitionistic logic follows the same principles as the semantics of modal logic, but uses a different definition of satisfaction.

An **intuitionistic Kripke model** is a triple, where is a partially ordered Kripke frame, and satisfies the following conditions:

- if
*p*is a propositional variable, and, then (*persistency*condition), - if and only if and ,
- if and only if or ,
- if and only if for all, implies ,
- not .

Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has the Finite Model Property.

**Intuitionistic first-order logic**

Let *L* be a first-order language. A Kripke model of *L* is a triple, where is an intuitionistic Kripke frame, *M _{w}* is a (classical)

*L*-structure for each node

*w*∈

*W*, and the following compatibility conditions hold whenever

*u*≤

*v*:

- the domain of
*M*is included in the domain of_{u}*M*,_{v} - realizations of function symbols in
*M*and_{u}*M*agree on elements of_{v}*M*,_{u} - for each
*n*-ary predicate*P*and elements*a*_{1},…,*a*∈_{n}*M*: if_{u}*P*(*a*_{1},…,*a*) holds in_{n}*M*, then it holds in_{u}*M*._{v}

Given an evaluation *e* of variables by elements of *M _{w}*, we define the satisfaction relation :

- if and only if holds in
*M*,_{w} - if and only if and ,
- if and only if or ,
- if and only if for all, implies ,
- not ,
- if and only if there exists an such that ,
- if and only if for every and every, .

Here *e*(*x*→*a*) is the evaluation which gives *x* the value *a*, and otherwise agrees with *e*.

Read more about this topic: Saul Kripke

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### Famous quotes containing the word logic:

“You can no more bridle passions with *logic* than you can justify them in the law courts. Passions are facts and not dogmas.”

—Alexander Herzen (1812–1870)