All of these arguments hold true if a Pigouvian tax is not substantial enough.
A Pigouvian tax has the following effects on a market: outcome won't optimize surplus; result will be inefficient; quantity will be too large. If the Pigouvian tax is too small, private costs will be lower than societal costs. Pigouvian taxes can also increase an economy's efficiency, particularly if they pay for the external damage. The actual cost of creating the good or service is created. The marginal social cost of pollution at the socially ideal level of pollution is equal to the ideal Pigouvian tax.
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