Grossing Up

If you are looking for logic in tax then you may of course be looking for a long time! Tax is made up of a set of rules created through statute and case law that need to be learned and applied so going forward I would advise don’t worry yourself about asking why, just learn the rules. However in relation to your specific question we include interest and dividends gross of the tax rate applicable to them within the basic rate band as they may be chargeable to higher or additional rate tax and therefore there may be more tax to pay in relation to their receipt. If we did not include them then our liability would be limited to just the basic rate charge!

As stated above we must always include income gross – before deduction of any tax at source, for example in addition to the 2 items you note we must include the gross salary of the taxpayer not the net amount received by the employee!
The reason again why we include interest and dividends is that they have only been subject to a basic rate or deemed basic rate deduction at source so that when combined with all other sources of taxable income an additional charge may arise if these amounts should then fall within the higher or additional rate bands.