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The axiomatic method . . . as, . . . a proposition lid down as one from which we may begin, an assertion that we have taken as fundamental, at least for the branch of enquiry in hand. The axiomatic method is that of defining as a set of such propositions, and the proof procedures or finding of how a proof ever gets started. Suppose I have as premises (1) p and (2) p   q. Can I infer q? Only, it seems, if I am sure of, (3) (p & p   q)   q. Can I then infer q? Only, it seems, if I am sure that (4) (p & p   q)   q)   q. For each new axiom (N) I need a further axiom (N + 1) telling me that the set so far implies q, and the regress never stops. The usual solution is to treat a system as containing not only axioms, but also rules of reference, allowing movement fro the axiom. The rule modus ponens allow us to pass from the first two premises to q. Charles Dodgson Lutwidge (1832-98) better known as Lewis Carrolls puzzle shows that it is essential to distinguish two theoretical categories, although there may be choice about which to put in which category.


This type of theory (axiomatic) usually emerges as a body of (supposes) truth that are not nearly organized, making the theory difficult to survey or study a whole. The axiomatic method is an idea for organizing a theory (Hilbert 1970): one tries to select from among the supposed truth a small number from which all others can be seen to be deductively inferable. This makes the theory rather more tractable since, in a sense, all the truth are contained in those few. In a theory so organized, the few truth from which all others are deductively inferred are called axioms. In that, just as algebraic and differential equations, which were used to study mathematical and physical processes, could themselves be made mathematical objects, so axiomatic theories, like algebraic and differential equations, which are means of representing physical processes and mathematical structures, could be made objects of mathematical investigation.

In the traditional (as in Leibniz, 1704), many philosophers had the conviction that all truth, or all truth about a particular domain, followed from a few principles. These principles were taken to be either metaphysically prior or epistemologically prior or in the fist sense, they were taken to be entities of such a nature that what exists is caused by them. When the principles were taken as epistemologically prior, that is, as axioms, either they were taken to be epistemologically privileged, e.g., self-evident, not needing to be demonstrated or (again, inclusive or) to be such that all truth do follow from them (by deductive inferences). Gödel (1984) showed that treating axiomatic theories as themselves mathematical objects, that mathematics, and even a small part of mathematics, elementary number theory, could not be axiomatized, that, more precisely, any class of axioms that in such that we could effectively decide, of any proposition, whether or not it was in the class, would be too small to capture all of the truth.

Gödel proved in 1929 that first-order predicate calculus is complete: any formula that is true under every interpretation is a theorem of the calculus: The propositional calculus or logical calculus whose expressions are letter present sentences or propositions, and constants representing operations on those propositions to produce others of higher complexity. The operations include conjunction, disjunction, material implication and negation (although these need not be primitive). Propositional logic was partially anticipated by the Stoics but researched maturity only with the work of Frége, Russell, and Wittgenstein.

The concept introduced by Frége of a function taking a number of names as arguments, and delivering one proposition as the value. The idea is that '÷' loves 'y' is a propositional function, which yields the proposition John loves Mary from those two arguments (in that order). A propositional function is therefore roughly equivalent to a property or relation. In Principia Mathematica, Russell and Whitehead take propositional functions to be the fundamental function, since the theory of descriptions could be taken as showing that other expressions denoting functions are incomplete symbols.

Keeping in mind, the two classical truth-values that a statement, proposition, or sentence can take. It is supposed in classical (two-valued) logic, that each statement has one of these values, and none has both. A statement is then false if and only if it is not true. The basis of this scheme is that to each statement there corresponds a determinate truth condition, or way the world must be for it to be true, and otherwise false. Statements may be felicitous or infelicitous in other dimensions, polite, misleading, apposite, witty, etc., but truth is the central normative governing assertion. Considerations of vagueness may introduce greys into black-and-white scheme. For the issue of whether falsity is the only way of failing to be true.

Formally, it is nonetheless, that any suppressed premise or background framework of thought necessary to make an argument valid, or a position tenable. More formally, a presupposition has been defined as a proposition whose truth is necessary for either the truth or the falsity of another statement. Thus, if p presupposes q, q must be true for p to be either true or false. In the theory of knowledge of Robin George Collingwood (1889-1943), any propositions capable of truth or falsity stand on a bed of absolute presuppositions which are not properly capable of truth or falsity, since a system of thought will contain no way of approaching such a question. It was suggested by Peter Strawson, 1919-in opposition to Russells theory of definite descriptions, that there exists a King of France is a presupposition of the King of France is bald, the latter being neither true, nor false, if there is no King of France. It is, however, a little unclear weather the idea is that no statement at all is made in such a case, or whether a statement is made, but fails of being either true or false. The former option preserves classical logic, since we can still say that every statement is either true or false, but the latter does not, since in classical logic the law of bivalence holds, and ensures that nothing at all is presupposed for any proposition to be true or false. The introduction of presupposition therefore means that either a third truth-value is found, intermediate between truth and falsity, or that classical logic is preserved, but it is impossible to tell whether a particular sentence expresses a proposition that is a candidate for truth ad falsity, without knowing more than the formation rules of the language. Each suggestion carries costs, and there is some consensus that at least where definite descriptions are involved, examples like the one given are equally well handed by regarding the overall sentence false when the existence claim fails.

A proposition may be true or false it be said to take the truth-value true, and if the latter the truth-value false. The idea behind the term is the analogy between assigning a propositional variable one or other of these values, as a formula of the propositional calculus, and assigning an object as the value of many other variable. Logics with intermediate values are called many-valued logics. Then, a truth-function of a number of propositions or sentences is a function of them that has a definite truth-value, depend only on the truth-values of the constituents. Thus (p & q) is a combination whose truth-value is true when 'p' is true and 'q' is true, and false otherwise,'¬ p' is a truth-function of 'p', false when 'p' is true and true when 'p' is false. The way in which the value of the whole is determined by the combinations of values of constituents is presented in a truth table.

In whatever manner, truth of fact cannot be reduced to any identity and our only way of knowing them is empirically, by reference to the facts of the empirical world.

A proposition is knowable deductively if it can be known without experience of the specific course of events in the actual world. It may, however, be allowed that some experience is required to acquire the concepts involved in an deductive proposition. Some thing is knowable only empirical if it can be known deductively. The distinction given one of the fundamental problem areas of epistemology. The category of deductive propositions is highly controversial, since it is not clear how pure thought, unaided by experience, can give rise to any knowledge at all, and it has always been a concern of empiricism to deny that it can. The two great areas in which it seems to be so are logic and mathematics, so empiricists have commonly tried to show either that these are not areas of real, substantive knowledge, or that in spite of appearances their knowledge that we have in these areas is actually dependent on experience. The former ligne tries to show sense trivial or analytic, or matters of notation conventions of language. The latter approach is particularly y associated with Quine, who denies any significant slit between propositions traditionally thought of as speculatively, and other deeply entrenched beliefs that occur in our overall view of the world.

Another contested category is that of speculative concepts, supposed to be concepts that cannot be derived from experience, bu t which are presupposed in any mode of thought about the world, time, substance, causation, number, and self are candidates. The need for such concept s, and the nature of the substantive a prior I knowledge to which they give rise, is the central concern of Kant s Critique of Pure Reason.

Likewise, since their denial does not involve a contradiction, there is merely contingent: Their could have been in other ways a hold of the actual world, but not every possible one. Some examples are Caesar crossed the Rubicon and Leibniz was born in Leipzig, as well as propositions expressing correct scientific generalizations. In Leibniz's view truth of fact rest on the principle of sufficient reason, which is a reason why it is so. This reason is that the actual world (by which he means the total collection of things past, present and future) is better than any other possible world and therefore created by God. The foundation of his thought is the conviction that to each individual there corresponds a complete notion, knowable only to God, from which is deducible all the properties possessed by the individual at each moment in its history. It is contingent that God actualizes te individual that meets such a concept, but his doing so is explicable by the principle of sufficient reason, whereby God had to actualize just that possibility in order for this to be the best of all possible worlds. This thesis is subsequently lampooned by Voltaire (1694-1778), in whom of which was prepared to take refuge in ignorance, as the nature of the soul, or the way to reconcile evil with divine providence.

In defending the principle of sufficient reason sometimes described as the principle that nothing can be so without there being a reason why it is so. But the reason has to be of a particularly potent kind: eventually it has to ground contingent facts in necessities, and in particular in the reason an omnipotent and perfect being would have for actualizing one possibility than another. Among the consequences of the principle is Leibniz's relational doctrine of space, since if space were an infinite box there could be no reason for the world to be at one point in rather than another, and God placing it at any point violate the principle. In Abelards' (1079-1142), as in Leibniz, the principle eventually forces te recognition that the actual world is the best of all possibilities, since anything else would be inconsistent with the creative power that actualizes possibilities.

If truth consists in concept containment, then it seems that all truth are analytic and hence necessary.  If they are all necessary, surely they are all truth of reason. In that not every truth can be reduced to an identity in a finite number of steps; in some instances revealing the connexion between subject and predicate concepts would require an infinite analysis, while this may entail that we cannot prove such proposition as a prior, it does not appear to show that proposition could have ben false. Intuitively, it seems a better ground for supposing that it is a necessary truth of a special sort. A related question arises from the idea that truth of fact depend on Gods decision to create the best world: If it is part of the concept of this world that it is best, how could its existence be other than necessary? An accountable and responsively answered explanation would be so, that any relational question that brakes the norm lay eyes on its existence in the manner other than hypothetical necessities, i.e., it follows from Gods decision to create the world, but God had the power to create this world, but God is necessary, so how could he have decided to do anything else? Leibniz says much more about these matters, but it is not clear whether he offers any satisfactory solutions.

The view that the terms in which we think of some area is sufficiently infected with error for it to be better to abandon them than to continue to try to give coherent theories of their use. Eliminativism should be distinguished from scepticism that claims that we cannot know the truth about some area; eliminativism claims rather that there is no truth there to be known, in the terms that we currently think. An eliminativist about theology simply counsels abandoning the terms or discourse of theology, and that will include abandoning worries about the extent of theological knowledge.

Eliminativists in the philosophy of mind counsel abandoning the whole network of terms mind, consciousness, self, qualia that usher in the problems of mind and body. Sometimes the argument for doing this is that we should wait for a supposed future understanding of ourselves, based on cognitive science and better than any our current mental descriptions provide, sometimes it is supposed that physicalism shows that no mental description of ourselves could possibly be true.

Sceptical tendencies emerged in the 14th-century writings of Nicholas of Autrecourt. His criticisms of any certainty beyond the immediate deliverance of the senses and basic logic, and in particular of any knowledge of either intellectual or material substances, anticipate the later scepticism of Balye and Hume. The; latter distinguishes between Pyrrhonistic and excessive scepticism, which he regarded as unlivable, and the more mitigated scepticism that accepts every day or commonsense beliefs (not as the delivery of reason, but as due more to custom and habit), but is duly wary of the power of reason to give us much more. Mitigated scepticism is thus closer to the attitude fostered by ancient scepticism from Pyrrho through to Sexus Empiricus. Although the phrase Cartesian scepticism is sometimes used, Descartes himself was not a sceptic, but in the method of doubt, uses a sceptical scenario in order to begin the process of finding a secure mark of knowledge. Descartes himself trusts a category of clear and distinct ideas, not far removed from the phantasia kataleptiké of the Stoics.

Scepticism should not be confused with relativism, which is a doctrine about the nature of truth, and may be motivated by trying to avoid scepticism. Nor is it identical with eliminativism, which counsels abandoning an area of thought altogether, not because we cannot know the truth, but because there are no truth capable of being framed in the terms we use.

Descartes theory of knowledge starts with the quest for certainty, for an indubitable starting-point or foundation on the basis alone of which progress is possible. This is eventually found in the celebrated Cogito ergo sum: I think therefore I am. By locating the point of certainty in my own awareness of my own self, Descartes gives a first-person twist to the theory of knowledge that dominated them following centuries in spite of various counter-attacks on behalf of social and public starting-point. The metaphysics associated with this priority is the famous Cartesian dualism, or separation of mind and matter into two different but interacting substances, Descartes rigorously and rightly sees that it takes divine dispensation to certify any relationship between the two realms thus divided, and to prove the reliability of the senses invokes a clear and distinct perception of highly dubious proofs of the existence of a benevolent deity. This has not met general acceptance: as Hume drily puts it, to have recourse to the veracity of the supreme Being, in order to prove the veracity of our senses, is surely making a very unexpected circuit.

In his own time Descartes conception of the entirely separate substance of the mind was recognized to give rise to insoluble problems of the nature of the causal connexion between the two. It also gives rise to the problem, insoluble in its own terms, of other minds. Descartes notorious denial that non-human animals are conscious is a stark illustration of the problem. In his conception of matter Descartes also gives preference to rational cogitation over anything derived from the senses. Since we can conceive of the matter of a ball of wax surviving changes to its sensible qualities, matter is not an empirical concept, but eventually an entirely geometrical one, with extension and motion as its only physical nature. Descartes thought, as reflected in Leibniz, that the qualities of sense experience have no resemblance to qualities of things, so that knowledge of the external world is essentially knowledge of structure rather than of filling. On this basis Descartes erects a remarkable physics. Since matter is in effect the same as extension there can be no empty space or void, since there is no empty space motion is not a question of occupying previously empty space, but is to be thought of in terms of vortices (like the motion of a liquid).

Although the structure of Descartes epistemology, theory of mind, and theory of matter have ben rejected many times, their relentless exposure of the hardest issues, their exemplary clarity, and even their initial plausibility, all contrive to make him the central point of reference for modern philosophy.

The self conceived as Descartes presents it in the first two Meditations: aware only of its own thoughts, and capable of disembodied existence, neither situated in a space nor surrounded by others. This is the pure self of I-ness that we are tempted to imagine as a simple unique thing that make up our essential identity. Descartes view that he could keep hold of this nugget while doubting everything else is criticized by Lichtenberg and Kant, and most subsequent philosophers of mind.

Descartes holds that we do not have any knowledge of any empirical proposition about anything beyond the contents of our own minds. The reason, roughly put, is that there is a legitimate doubt about all such propositions because there is no way to deny justifiably that our senses are being stimulated by some cause (an evil spirit, for example) which is radically different from the objects that we normally think affect our senses.

He also points out, that the senses (sight, hearing, touch, etc., are often unreliable, and it is prudent never to trust entirely those who have deceived us even once, he cited such instances as the straight stick that looks ben t in water, and the square tower that looks round from a distance. This argument of illusion, has not, on the whole, impressed commentators, and some of Descartes contemporaries pointing out that since such errors become known as a result of further sensory information, it cannot be right to cast wholesale doubt on the evidence of the senses. But Descartes regarded the argument from illusion as only the first stage in a softening up process which would lead the mind away from the senses. He admits that there are some cases of sense-base  belief about which doubt would be insane, e.g., the belief that I am sitting here by the fire, wearing a winter dressing gown.

Descartes was to realize that there was nothing in this view of nature that could explain or provide a foundation for the mental, or from direct experience as distinctly human. In a mechanistic universe, he said, there is no privileged place or function for mind, and the separation between mind and matter is absolute. Descartes was also convinced, that the immaterial essences that gave form and structure to this universe were coded in geometrical and mathematical ideas, and this insight led him to invent algebraic geometry.

A scientific understanding of these ideas could be derived, said Descartes, with the aid of precise deduction, and he also claimed that the contours of physical reality could be laid out in three-dimensional coordinates. Following the publication of Newtons Principia Mathematica in 1687, reductionism and mathematical modelling became the most powerful tools of modern science. And the dream that the entire physical world could be known and mastered through the extension and refinement of mathematical theory became the central feature and guiding principle of scientific knowledge.

Having to its  recourse of knowledge, its cental questions include the origin of knowledge, the place of experience in generating knowledge, and the place of reason in doing so, the relationship between knowledge and certainty, and between knowledge and the impossibility of error, the possibility of universal scepticism, and the changing forms of knowledge that arise from new conceptualizations of the world. All of these issues link with other central concerns of philosophy, such as the nature of truth and the natures of experience and meaning.

Foundationalism was associated with the ancient Stoics, and in the modern era with Descartes (1596-1650). Who discovered his foundations in the clear and distinct ideas of reason? Its main opponent is Coherentism, or the view that a body of propositions mas be known without a foundation in certainty, but by their interlocking strength, than as a crossword puzzle may be known to have been solved correctly even if each answer, taken individually, admits of uncertainty. Difficulties at this point led the logical passivists to abandon the notion of an epistemological foundation altogether, and to flirt with the coherence theory of truth. It is widely accepted that trying to make the connexion between thought and experience through basic sentences depends on an untenable myth of the given.

Meanwhile, the truth conditions of a statement is the condition the world must meet if the statement is to be true. To know this condition is equivalent go knowing the meaning of the statement. Although his sounds as if it gives a solid anchorage when in turns out that the truth condition can only be defined by repeating the very same statement. The truth condition of 'snow is white' is that snow is white, the truth condition of 'Britain would have capitulated had Hitler invaded' is that Britain would have capitulated had Hitler invaded. It is disputed whether this element of running-on-the-spot disqualifies truth conditions from playing the central role in a substantive theory of meaning.  Truth-conditional theories of meaning are sometimes opposed by the view that to know the meaning of a statement is to be able to use it in as a network of inferences.