Enabling machines to interact with humans requires understanding what people mean, even when they do not say it explicitly. For example, machines should understand that ``selfish people oppose gun control'' implies a pro-gun control viewpoint (i.e., is taking a stance in support of gun control) despite the negative tone of the statement. Understanding these types of pragmatic inferences allows humans to grasp meaning (e.g., intentions, relevant facts) beyond what is literally expressed in an utterance. Furthermore, pragmatic inferences conveyed through generalizations (e.g., referring to generic ``selfish people'' rather than specific individuals in order to be more persuasive) support flexible and efficient reasoning. Therefore, in this thesis we focus on improving computational understanding of two inter-related types of pragmatic inferences: stancetaking and linguistic generalizations. This thesis is divided into two parts. In Part II, we focus on stance detection. One major challenge for stance detection models is the large and continually growing set of stance targets (i.e., topics to take a stance on). Therefore, to address this we define and study zero-shot stance detection (i.e., evaluation on topics for which there is no training data). Our work develops both datasets and models for this task and analyzes the ongoing challenges for future work. This work has stimulated increasing and ongoing research in zero-shot stance detection in NLP. Then in Part I we study generics --- a specific type of linguistic generalization that does not contain explicit quantifiers (e.g., ``most'', ``some''). These statements can have strong persuasive force and are also related to complex patterns of reasoning. To probe the current understanding capabilities of computational models, we focus on generating generics exemplars --- specific cases when a generic holds true or false. In particular, we propose computational frameworks grounded in linguistic theory to generate the first datasets of exemplars. We then use our datasets to highlight the challenges generics pose for natural language reasoning and the current generic-understanding capabilities of large language models.