In PostgreSQL, use the boolean data type. Done.

When using pgloader to migrate your MySQL database to PostgreSQL, the following cast rule is the default:

CAST type tinyint to boolean using tinyint-to-boolean

It’s of course then possible to migrate a tinyint column from MySQL to a PostgreSQL smallint with the following specific rule:

CAST column table_a.count to smallint

Boolean Columns

When you have a boolean column in PostgreSQL then things work as you would expect. You can query rows being true:

   select *
     from table
    where col is true;

Three-Valued Logic

Be careful with boolean logic in the standard SQL though, as the standard implements Three-Valued Logic, as seen here:

select a::text, b::text,
       (a=b)::text as "a=b",
       format('%s = %s',
              coalesce(a::text, 'null'),
              coalesce(b::text, 'null')) as op,
       format('is %s',
              coalesce((a=b)::text, 'null')) as result
  from (values(true), (false), (null)) v1(a)
       cross join
       (values(true), (false), (null)) v2(b);

Here’s our SQL boolean truth table:

   a   │   b   │  a=b  │      op       │  result  
═══════╪═══════╪═══════╪═══════════════╪══════════
 true  │ true  │ true  │ true = true   │ is true
 true  │ false │ false │ true = false  │ is false
 true  │ ¤     │ ¤     │ true = null   │ is null
 false │ true  │ false │ false = true  │ is false
 false │ false │ true  │ false = false │ is true
 false │ ¤     │ ¤     │ false = null  │ is null
 ¤     │ true  │ ¤     │ null = true   │ is null
 ¤     │ false │ ¤     │ null = false  │ is null
 ¤     │ ¤     │ ¤     │ null = null   │ is null
(9 rows)

If you want to avoid the trap of null = null returning NULL, then you can use the standard operators IS DISTINCT FROM and IS NOT DISTINCT FROM.

select a::text as left, b::text as right,
       (a = b)::text as "=",
       (a <> b)::text as "<>",
       (a is distinct from b)::text as "is distinct",
       (a is not distinct from b)::text as "is not distinct from"
  from            (values(true),(false),(null)) t1(a)
       cross join (values(true),(false),(null)) t2(b);

And here’s the table this time:

 left  │ right │   =   │  <>   │ is distinct │ is not distinct from 
═══════╪═══════╪═══════╪═══════╪═════════════╪══════════════════════
 true  │ true  │ true  │ false │ false       │ true
 true  │ false │ false │ true  │ true        │ false
 true  │ ¤     │ ¤     │ ¤     │ true        │ false
 false │ true  │ false │ true  │ true        │ false
 false │ false │ true  │ false │ false       │ true
 false │ ¤     │ ¤     │ ¤     │ true        │ false
 ¤     │ true  │ ¤     │ ¤     │ true        │ false
 ¤     │ false │ ¤     │ ¤     │ true        │ false
 ¤     │ ¤     │ ¤     │ ¤     │ false       │ true
(9 rows)