The regress problem is a sceptical issue that arises in the context of providing justification for our beliefs. According to the problem, any given reason for a belief requires a further reason, which in turn requires its own justification, leading to an infinite regress. If this regress cannot be stopped, it seems that we have no firm foundation for our beliefs, rendering all knowledge unjustified. Sceptics use this problem to argue that if there are no basic beliefs that can halt the regress without themselves requiring justification, then it appears that we have no justified beliefs at all. This sceptical argument thus challenges the coherence of our belief systems and the possibility of having any justified knowledge, contributing significantly to sceptical thought by questioning the structure and foundation of epistemic justification.

The Gettier problem is a challenge to the classical definition of knowledge as justified true belief, introduced by Edmund Gettier in 1963. Gettier presented cases in which a person has a belief that is both true and well-supported by evidence (justified), yet intuitively does not seem to be knowledge. This problem is significant for discussions of scepticism because it demonstrates that having a justified true belief may not be sufficient for having knowledge. The Gettier problem thus feeds into sceptical discussions by revealing potential flaws in our understanding of knowledge and by suggesting that our criteria for knowledge might be incomplete or incorrect. It compels philosophers to refine the definition of knowledge and to investigate more deeply what it means for a belief to be justified, ultimately enriching sceptical and epistemological discourse.

The 'brain in a vat' hypothesis is a modern thought experiment related to scepticism, particularly to arguments concerning the external world and our knowledge of it. It posits a scenario in which a human brain is kept alive in a vat and connected to a computer simulation that provides it with the same experiences that it would have if it were in a skull, and the person whose brain it is had a normal perceptual experience. This thought experiment serves to question the reliability of our perceptions and the possibility that our entire experience of the world may be an elaborate illusion. It is related to sceptical arguments that challenge the assumption that our cognitive faculties provide us with direct knowledge of an external world, thereby highlighting the limits of our knowledge and the need for a strong foundation for epistemological claims.

Cartesian scepticism, as proposed by René Descartes, is methodological, employed as a strategic doubt at the outset of his philosophical enquiries to establish certain, indubitable truths. Descartes doubted everything he could, including the evidence of the senses and the existence of the physical world, to find an absolutely certain foundation for knowledge, famously arriving at the cogito, "I think, therefore I am." Pyrrhonian scepticism, named after Pyrrho of Elis, is more radical and practical, aiming to suspend judgement on all matters to attain a state of tranquillity (ataraxia). Pyrrhonists refrain from making claims about the reality beyond appearances, remaining uncommitted to any knowledge claims to avoid the unease that comes from trying to discern the true nature of things.

Scepticism can indeed be considered a valid and influential philosophical position. It is not merely a paradox; it represents a critical approach in philosophy that challenges the very possibility of certain knowledge. While scepticism raises questions that may seem paradoxical, such as suggesting we cannot know anything with certainty, it also encourages the thorough examination of our assumptions and beliefs about knowledge. The enduring discussions around scepticism demonstrate its value in philosophical discourse. It forces philosophers to confront the foundational issues of epistemology and develop more robust theories in response to sceptical challenges. As such, scepticism is an integral part of philosophical enquiry, as it pushes the boundaries of what is considered knowable and helps refine our understanding of truth, belief, and justification.