We investigated semantic cognition in the logopenic variant of primary progressive aphasia (lvPPA), including (i) the status of verbal and non-verbal semantic performance; and (ii) whether the semantic deficit reflects impaired semantic control. Our a priori hypothesis that individuals with lvPPA would exhibit semantic control impairments was motivated by the anatomical overlap between the temporoparietal atrophy typically associated with lvPPA and lesions associated with post-stroke semantic aphasia (SA) and Wernicke's aphasia (WA), which cause heteromodal semantic control impairments. We addressed the presence, type (semantic representation and semantic control; verbal and non-verbal), and progression of semantic deficits in lvPPA. Since most people with lvPPA have Alzheimer's disease (AD) pathology and are part of a broader multidimensional phenotype space encompassing AD subtypes, we compared semantic performance in lvPPA and typical amnestic AD (tAD). Given the differences in lesion and atrophy patterns in SA and WA versus semantic dementia/semantic-variant PPA patients, our second aim was to examine atrophy patterns in people with lvPPA and tAD compared to age-matched controls. Twenty-seven patients participated in the study. People were grouped into those meeting consensus criteria for lvPPA (n = 10) and others who may have previously satisfied definitions of lvPPA but had progressed with multi-domain cognitive impairments (herein referred to as "lvPPA+"; n = 8). People with tAD (n = 9) were relatively preserved across verbal and non-verbal semantic assessments. LvPPA patients were impaired on both verbal and non-verbal semantic tasks and their impairments showed the hallmark characteristics of a semantic control deficit. LvPPA and lvPPA+ patients showed effects of varying semantic control demands, positive cueing effects, and correlated performance between semantic and executive tasks. Whole-brain voxel-based morphometry, comparing each of the patient groups to age-matched controls, revealed significantly reduced grey and white matter in the bilateral hippocampi and lateral temporal regions in tAD patients. The lvPPA group exhibited an asymmetric pattern of reduced grey and white matter intensity in the language-dominant left hemisphere, including a significant portion of the lateral and medial temporal lobe. LvPPA+ patients demonstrated reduced grey and white matter in the left temporal lobe extending subcortically, anteriorly and posteriorly, as well as right temporal involvement. Our findings could aid diagnostic subtyping of PPA by adopting semantic control features and offer improved clinical characterisation of lvPPA in the trajectory of semantic decline.