He was educated at Harvard, where his father was a mathematics professor. His greatest philosophical influence was Kant, and he saw himself as constructing the philosophical system that Kant might have developed had he not been so ignorant of logic. But the influence of Thomas Reid and other commonsense philosophers became increasingly important: in late writings, the two influences were combined in his 'critical common-sensism'. Describing himself as a logician, Peirce made major contributions to formal logic (independently of Frege he and his students developed a logic of quantifiers and relations after 1880) and to the study of the logic of science. Indeed, he lectured on these topics at Harvard in the late 1860s and held a lectureship in logic at Johns Hopkins University from 1879 until 1884. But he also served as an experimental scientist, working at the Harvard laboratory after he had graduated in chemistry, and being employed for over twenty years by the United States Coastal Survey.

Peirce was a difficult man, widely perceived as an immoral libertine, prone to paranoia and wild mood swings, and possessing an assessment of his own intellectual powers which may have been accurate but which was sometimes accompanied by contempt for the capacities of those of lesser talents. In 1884, when confident of obtaining tenure at Johns Hopkins, information about his irregular life-style, together with suspicion of his unorthodox religious beliefs, led to his being removed from his post. From then until his death, it was understood that he could expect no orthodox academic employment: he lived precariously with his second wife in north-eastern Pennsylvania, writing extensively and giving a few important series of lectures arranged by his friend William James. He never completed the canonical statement of his philosophical position that he sought, but he published extensively and left hundreds of thousands of manuscripts; his work is gradually becoming more readily available.

In a late paper, Peirce described himself as a 'laboratory philosopher', claiming that years of laboratory experience encouraged him, like any experimentalist, to approach all issues in the distinctive manner which comprises his pragmatism. This is clearest in the approach to epistemological matters which emerges in his earliest published work, from the 1860s and 1870s - most clearly in a series of papers in the Popular Science Monthly (1877-1878).

His epistemological work begins from a rejection of Cartesian strategies in philosophy. They do not, he pointed out, accord with our ordinary practice of carrying out investigations: the latter is a cooperative venture, while Descartes suggests that a responsible investigator should carry out a solitary investigation of his or her cognitive standing. Ordinary inquiry takes for granted all the propositions we find certain as we begin the inquiry, while Descartes's sceptical arguments prompt philosophical doubt about what occasions no real doubt. And ordinary inquiry is impressed by the number and variety of the arguments supporting a conclusion, while the Cartesian requires a single indubitable train of reasoning to ground any belief. Peirce proposes to begin from our everyday and scientific experience of inquiry, and to investigate the norms which govern cognition on that basis.

The first paper of the series suggests that inquiry begins only when one of our previously settled beliefs is disturbed, and it is ended as soon as we have a new answer to the question that concerns us: the aim of inquiry is to replace doubt by settled belief. What methods should we use if we are to carry out our inquiries well? He considers four, the first three being devised to bring to light the key features of the fourth. (1) The method of tenacity requires us to choose any answer, and to take all means necessary to maintain it; (2) the method of authority requires us to defer to an authority and accept whatever the authority requires (it may be no accident that Peirce wrote soon after the bull of papal infallibility had been promulgated); and (3) the a priori method requires us to go by what seems agreeable to reason. It will be no surprise that these methods fail: the second has the advantage over the first that our beliefs will escape the constant buffeting of disputes from those who have decided differently, but we are still likely to meet those who accept a different authority, and our own authority will not be able to settle matters about everything. So fixation of belief must be independent of will or human choice. The third method secures that, but it is likely to make belief a matter of fashion: selection of belief still has a subjective basis. Hence we should adopt (4) the 'method of science', which holds that 'there are Real things, whose characters are entirely independent of our opinions about them; those realities affect our senses according to regular laws, and, though our sensations are as different as our relations to the objects, yet, by taking advantage of the laws of perception, we can ascertain by reasoning how things really are'.

Peirce probably believed that this claim was a presupposition of inquiry and that we should adopt only such methods as were in accord with it. The remainder of the series of papers offers a more detailed account of what this method involves: Peirce was one of the first philosophers to arrive at a satisfactory understanding of statistical reasoning, and this is central to his account of science. He is a 'contrite fallibilist': any of our current certainties might turn out to be mistaken, but relying upon them will not prevent our making cognitive progress; any errors will emerge with time.

The 'pragmatist principle' forms part of this theory of inquiry, and was elaborated in the second paper of the series, 'How to Make Our Ideas Clear'. When William James won notoriety for pragmatism, crediting it to Peirce, the latter renamed his principle 'pragmaticism'. It is a rule for clarifying the content of concepts and hypotheses, and is supposed to reveal all features of the meaning of concepts and hypotheses that are relevant to scientific investigations. Suppose I wish to test whether a sample before me is sodium. In the light of my knowledge of sodium, I can predict that if it is sodium then, if I were to drop it into hot water, it would ignite: I make predictions about the consequences of actions if the hypothesis is true. Peirce expresses his principle: 'Consider what effects, which might conceivably have practical bearings, we conceive the object of our conception to have. Then our conception of those effects is the whole of our conception of the object.' When I have listed all the predictions I would make about the consequences of my actions if the substance were sodium, I have a complete clarification of my understanding of the hypothesis: nothing which could be relevant to testing it scientifically has been omitted.

As well as showing its value in clarifying hypotheses, and arguing that it can be used to dismiss some metaphysical 'hypotheses' as empty, Peirce illustrates the value of his pragmatism by clarifying our conception of truth and reality. If a proposition is true, then anyone who investigated the matter long enough and well enough would eventually acknowledge its truth: truth is a matter of long-term convergence of opinion. 'The opinion which is fated to be ultimately agreed upon by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real.' Although the principle bears a superficial resemblance to the verification principle of the later Logical Positivists, there are important differences. First, there is no suggestion that, in clarifying our conception, we list only those conditional expectations that are analytic or true by definition: Peirce expects the content of a conception or hypothesis to develop as our scientific knowledge advances. And, second, as he developed his philosophical position, he insisted that the principle could only be taken seriously by someone who shared his realism about natural necessity: the conceptual clarifications are expressed as subjunctive conditionals ('would-bes'); and such conditionals report real facts about the world.

Peirce's logic is a theory of cognitive norms: methods of inquiry, standards of inference, rules for identifying plausible hypotheses, principles for clarifying meanings, and so on. He was unsatisfied with the kind of grounding he provided for cognitive norms in the papers just discussed, and his attempts to correct the Kantian framework were directed at remedying this. His sophisticated architectonic approach to philosophy rested upon a classification of the sciences. Logic was the least fundamental of three normative sciences, being a special application a system of norms initially developed in ethics and aesthetics. All of these investigations made use of a system of categories, a correction of Kant's system, which was defended through a kind of phenomenological investigation. And these philosophical and phenomenological inquiries used mathematical methods to study experience and reality, mathematics being the only discipline which had, and needed, no foundations. So Peirce's later work developed a highly sophisticated account of how we can have knowledge of cognitive or logical norms.

His system of categories is most easily understood from the perspective of his logic of relations. Properties and relations can be classified according to the number of relata they have: '... is blue' is a one-place predicate, '... respects ...' is a dyadic, two-place relation, and '... gives ... to ...' is a triadic, three-place relation. Peirce argued that a language adequate for scientific or descriptive purposes must contain terms of all these three kinds, but that there are no phenomena which can only be described in a language which contains expressions for four-place relations. Thus he classified phenomena and elements of reality numerically: according to whether they are forms of firstness, secondness, or (like giving) thirdness. The irreducibility of thirdness is, he thinks, a distinctive part of his philosophical outlook, something which allies him with realist philosophers in opposition to nominalism. In early work, his defence of his categories was largely found in his work on formal logic, but later he turned to phenomenology: reflection on experience of all kinds was to convince us that triadicity was ineliminable but that no more complex phenomena were involved in experience.

Thus we are aware that our experiences have raw qualitative characters which do not directly involve relations with other things: they exhibit firstness. They also stand in relations to each other, interacting against one another and so on: this involves secondness, as when fire immediately follows our dropping the sodium in hot water. But we are aware that this interaction is intelligible, it is 'mediated': we can bring it down into a continuous spread of small changes which go together to make up the big one; and we are aware that it conforms to a law. Finding it intelligible introduces thirdness: we understand the two elements of the interaction by reference to a third mediating fact. The aim of inquiry, for Peirce, is to find the thirdness (law and pattern) in the manifold of sensory experiences that we undergo. The norms employed by the scientific method are to be vindicated by showing how they provide means for finding more and more pattern and mediation (more and more thirdness) in the world of our experience.

According to Peirce, the most important forms of thirdness involve meaning and representation, and all of his work is underpinned by a sophisticated theory of meaning: his semiotics. He probably believed that everything was a sign, but the signs of most interest to him were thoughts and 'the assertions of a scientific intelligence'. This theory of meaning ('speculative grammar') was to provide foundations for his writings in logic.

The key to the thirdness involved in signs was Peirce's notion of interpretation. A sign denotes an object only by being understood or interpreted as standing for an object: and this interpretation will always be another sign with the same object. Semiotics is thus primarily a theory of understanding, an account of how we are guided and constrained in arriving at interpretations of signs. Interpretation often involves inference, developing our understanding of the object in question. Thus my understanding of your assertion that you are tired may be manifested in my thinking that you want me to believe you are tired, in my believing you are tired, in my expecting you to fall asleep, in my offering you a cup of coffee, and so on. The interpreting thought mediates between the sign and its object.

Peirce was famous for his classifications of signs, and some of his terminology has acquired wide currency. For example, signs can be distinguished according to the features of them exploited in arriving at an interpretation. A symbol denotes a particular object because there exists a practice of interpreting it as denoting that object: an index denotes an object to which it stands in a direct existential relation: the conventions governing the use of ordinary indexical expressions such as 'this' do not fix the reference unaided but rather guide us in interpreting it as an index. And iconic signs share some feature with their object which each could possess if the other did not exist: maps are straight-forward examples, the conventions governing their use fixing how we are to interpret them as icons. Mathematical and logical symbolisms are iconic representations, and it was important for Peirce that sentences of natural languages have iconic elements too: formal inference exploits the fact that sentences exhibit a form which is shared with their subject-matter. Much of Peirce's later work attempted to use this systematic theory of meaning to provide a proof of the pragmatist principle.

Science itself is a process of sign interpretation. And Peirce's account of scientific reasoning has some important elements. As mentioned above, Peirce models all inductive reasoning on statistical sampling: quantitative induction involves attempting to estimate the chance of a member of a population having a particular property; and qualitative induction tests hypotheses by sampling their consequences. He denies that induction ever establishes that a conclusion is true or even probable. Rather, the practice of inductive testing is justified because continued use of it will eventually lead us to converge on the correct value for the chance of a member of the population having the property in question. The pragmatist principle teaches that probability is a propensity: if the chance of a coin coming up heads is 0.43, then, if we were to continue to toss it fairly, the proportion of times on which it comes up heads would converge on 0.43.

The logic of abduction is a logic of discovery: it studies how we are guided in constructing new hypotheses from the ruins of defeated ones; and it examines the norms guiding us in deciding which hypotheses are worth testing. All scientific activity is grounded in the hope that the universe is intelligible, and intelligible to us. And we are to take seriously no hypothesis that 'blocks the road of inquiry', forcing us to accept regularities as brute or inexplicable. It is connected to this that Peirce espouses 'synechism', the doctrine that we are to expect the universe to display continuities rather than discontinuities. Peirce contributed to the mathematical analysis of continuity, exploiting his ideas about the logic of relations and trying to use it as the basis of his realism about natural necessity: continuity is 'ultimate mediation'. The logic of abduction advises us to favour theories that posit continuities over those that allow for brute unmediated discontinuities.

Although Peirce envisaged that pragmatism would eliminate 'ontological metaphysics', he claimed that scientific progress demanded that we construct a 'scientific metaphysics'. Supposedly this was an empirical discipline, differing from the special sciences in using no sophisticated techniques of experiment and observation: it was 'coenoscopic', relying only on familiar everyday observations which are surprising only because their familiarity prevents our noticing them. In part, it was an attempt to describe how the world must be if science was to be possible - if there were to be no inexplicable phenomena, if 'realism' was to be true, if the three categories were to be as Peirce suggested. And in part it was an exercise in 'descriptive metaphysics': drawing out features of our everyday conception of mind or matter (for example) can be a valuable corrective to unthinking theoretical prejudices, especially in psychology.

Two elements of this metaphysics are especially interesting. Peirce defended an evolutionary cosmology, explaining how the world of existing things and law-governed behaviour evolved from pure possibility. Offering an evolutionary explanation of law, he argued, was the only alternative to asserting that fundamental laws are simply true, with no explanation of why they obtain being available. If every regularity must have an explanation, we avoid a regress of ever more general and abstract laws by invoking a historical explanation. And Peirce's account of how this evolutionary process works leads to a form of objective idealism according to which matter is 'effete mind', and physical phenomena are modelled on thought and sign interpretation rather than the mental being reduced to the physical. This is because a 'realist' account of law involves finding 'mediation' in the natural world, and sign interpretation is our best model of mediation.

Secondly, it may accord with the importance he attached to statistical reasoning in science that he accepted tychism, the thesis that there is absolute chance, that the universe is not wholly governed by determinist laws. This partly reflects his understanding of the importance of statistical laws in science, and his understanding that observation could never establish that laws were so exact as never to permit slight deviations. He also supposed it was required to explain the evolutionary process discussed in his cosmology: without appeal to such 'chance spontaneity', he doubted that we could make sense of growth and increasing complexity.