The majority of work in targeted sentiment analysis has concentrated on finding better methods to improve the overall results. Within this paper we show that these models are not robust to linguistic phenomena, specifically negation and speculation. In this paper, we propose a multi-task learning method to incorporate information from syntactic and semantic auxiliary tasks, including negation and speculation scope detection, to create English-language models that are more robust to these phenomena. Further we create two challenge datasets to evaluate model performance on negated and speculative samples. We find that multi-task models and transfer learning via language modelling can improve performance on these challenge datasets, but the overall performances indicate that there is still much room for improvement. We release both the datasets and the source code at https://github.com/jerbarnes/multitask_negation_for_targeted_sentiment.