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Not a definition, but an example of use in logic:

In model theory, "canonical" is often used in the phrase "the canonical model" to mean "intended structure." For instance, in first-order logic, one may speak of "the canonical model of Peano Arithmetic" to mean the structure of the natural numbers, or "the canonical model of the theory of real-closed fields" to mean the field of real numbers. Intuitively, "the canonical model" of a theory is the structure one was trying to pin down when the axiomatisation of the theory was written. It's just that in first-order logic, it is hard to pin down (infinite) structures! No first-order theories admitting infinite models are categorical (they admit non-isomorphic models; indeed, they admit models of every infinite cardinality), and compactness/ultraproduct compactness/ultraproduct/(many other) constructions can often be used to build "non-standard" models of theories. "Non-standard" models of Peano Arithmetic or the theory of real-closed fields would in this context be called "non-canonical" (even though there are many canonically studied "non-standard" models of those theories!).

But, many commonly studied theories do not have a notion of "canonical model." For instance, one would not say "the canonical model of group theory."
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Not a definition, but an example of use in logic:

In model theory, "canonical" is often used in the phrase "the canonical model" to mean "intended structure." For instance, in first-order logic, one may speak of "the canonical model of Peano Arithmetic" to mean the structure of the natural numbers, or "the canonical model of the theory of real-closed fields" to mean the field of real numbers. Intuitively, "the canonical model" of a theory is the structure one was trying to pin down when the axiomatisation of the theory was constructed, it's written. It's just that in first-order logic, no it is hard to pin down (infinite) structures! No first-order theories admitting infinite models are categorical (e.g., a first-order theory admitting an infinite model must they admit non-isomorphic models; indeed, they admit models of every infinite cardinality), and compactness/ultraproduct constructions can often be used to build "non-standard" models of theories. "Non-standard" models of Peano Arithmetic or the theory of real-closed fields would in this context be called "non-canonical" (even though there are many canonically studied "non-standard" models of those theories!).

But, many commonly studied theories do not have a notion of "canonical model." For instance, one would not say "the canonical model of group theory."
show/hide this revision's text 1 [made Community Wiki]

Not a definition, but an example of use in logic:

In model theory, "canonical" is often used in the phrase "the canonical model" to mean "intended structure." For instance, in first-order logic, one may speak of "the canonical model of Peano Arithmetic" to mean the structure of the natural numbers, or "the canonical model of the theory of real-closed fields" to mean the field of real numbers. Intuitively, "the canonical model" of a theory is the structure one was trying to pin down when the axiomatisation of the theory was constructed, it's just that in first-order logic, no theories admitting infinite models are categorical (e.g., a first-order theory admitting an infinite model must admit models of every infinite cardinality). "Non-standard" models of Peano Arithmetic or the theory of real-closed fields would in this context be called "non-canonical" (even though there are many canonically studied "non-standard" models of those theories!).

But, many commonly studied theories do not have a notion of "canonical model." For instance, one would not say "the canonical model of group theory."