Goal A common style is to declare in each base table one of the candidate keys as the primary key. All the other candidate keys would be alternate keys that will be enforce with the help of UNIQUE + NOT NULL constraints.
Type Problem detection (Each row in the result could represent a flaw in the design)
Reliability High (Few or no false-positive results)
License MIT License
Fixing Suggestion Be consistent and declare in each base table one of the candidate keys as the primary key.
Data Source INFORMATION_SCHEMA only
SQL Query
SELECT A.table_schema, A.table_name 
FROM INFORMATION_SCHEMA.tables A
INNER JOIN INFORMATION_SCHEMA.schemata B
ON A.table_schema=B.schema_name
WHERE (A.table_schema, A.table_name) NOT IN 
(SELECT table_schema, table_name FROM INFORMATION_SCHEMA.table_constraints
WHERE constraint_type IN ('PRIMARY KEY'))
AND (A.table_schema, A.table_name) IN 
(SELECT table_schema, table_name FROM INFORMATION_SCHEMA.table_constraints
WHERE constraint_type IN ('UNIQUE'))
AND A.table_type='BASE TABLE'
AND (A.table_schema = 'public'
OR B.schema_owner<>'postgres')
ORDER BY A.table_schema, A.table_name;

Collections

This query belongs to the following collections:

NameDescription
Find problems about integrity constraintsA selection of queries that return information about the state of integrity constraints in the datadabase. Contains all the types of queries - problem detection, software measure, and general overview
Find problems automaticallyQueries, that results point to problems in the database. Each query in the collection produces an initial assessment. However, a human reviewer has the final say as to whether there is a problem or not .
Categories

This query is classified under the following categories:

NameDescription
InconsistenciesQueries of this catergory provide information about inconsistencies of solving the same problem in different places.
UniquenessQueries of this category provide information about uniqueness constraints (PRIMARY KEY, UNIQUE, EXCLUDE) as well as unique indexes.
Validity and completenessQueries of this category provide information about whether database design represents the world (domain) correctly (validity) and whether database design captures all the information about the world (domain) that is correct and relevant (completeness).

Further reading and related materials:

Reference
The corresponding code smell in case of cleaning code is "G11: Inconsistency". (Robert C. Martin, Clean Code)
Rule 4 in: Delplanque, J., Etien, A., Auverlot, O., Mens, T., Anquetil, N., Ducasse, S.: CodeCritics applied to database schema: Challenges and first results. In: 2017 IEEE 24th International Conference on Software Analysis, Evolution and Reengineering (SANER), pp. 432-436. IEEE, (2017).