Recent advancements in large language models (LLMs) have revitalized philosophical debates surrounding artificial intelligence. Two of the most fundamental challenges - namely, the Frame Problem and the Symbol Grounding Problem - have historically been viewed as unsolvable within traditional symbolic AI systems. This study investigates whether modern LLMs possess the cognitive capacities required to address these problems. To do so, I designed two benchmark tasks reflecting the philosophical core of each problem, administered them under zero-shot conditions to 13 prominent LLMs (both closed and open-source), and assessed the quality of the models' outputs across five trials each. Responses were scored along multiple criteria, including contextual reasoning, semantic coherence, and information filtering. The results demonstrate that while open-source models showed variability in performance due to differences in model size, quantization, and instruction tuning, several closed models consistently achieved high scores. These findings suggest that select modern LLMs may be acquiring capacities sufficient to produce meaningful and stable responses to these long-standing theoretical challenges.