David Hume proposed that is the passions of people that is delimited by rules governing interaction, connection, and possession (i.e., the occupation, prescription, accession and succession of property). To resolve the problem of credibility, the communication of a commitment to the rules of social convention is necessary. The institution of signs and symbols signal to the community as a representation of a promise of obligation, motive and conduct. These self-constituted rules of action (e.g., intention of subject and object) are the moral virtues of stability, necessity, liberty, benefice, and justice. They increase confidence and security in social and commercial affairs. Ludwig Wittgenstein concurred with this conviction to ground philosophy with customary rules, but instead argued language is common to these rules. Wittgenstein was a protégé of the mentor (after the guardian disguise of Athena in the Odyssey) Bertrand Russell. Philosophy promises to elucidate the essence of human condition, mind, nature, and world. Wittgenstein eventually received the ire from traditional philosophy for his insistence that there are no realms of fact that are uniquely accessible to the intuition and conception of philosophy and not to the methods of science.
The doctrine Wittgenstein proposed asserts that the traditional philosophy is scientistic, with the objective of principles of general uniformity and correct profundity. The non-empirical quality of philosophical investigation, which focuses on a complex and variable conceptual verity, is in tension with those objectives. It is impossible to adapt superficial complexity by basic simplicity. Cognitive concepts interact with individual contingencies of indeterminate nature and culture. In the paradigm of philosophy, the question of what is verity (truth) is principal. Philosophers have debated the existence of truth, doubting an absolute and favouring a relative. Others have maintained that is an intrinsic paradox with an incomprehensible essence that is impossible to demonstrate probability (manifest, discover and prove). Wittgenstein contended that philosophy is erred in its extrapolation from the fact that empirical concepts possessing properties with an identifiable and natural basis to the presumption that the concept of truth must also possess such a property. That is, it is not inferable that truth is reducible to a thing more basic. In fact, the concept of truth is an artifice of the generality and communication of language that is only an attribution of equivalence (similar to the concepts of good, person, now and necessary). The source of the irrational error in metaphysics is this unification by generalisation and simplification. In light of its illusory promise, philosophy is merely descriptive in the therapy of exposing irrationality. There is satisfaction in this clarity and purity.
Science is mathematical and natural philosophy. The foundation of the scientific method is consistency, causality and induction. Causality or causal inference is the association of causes and effects originating from perceived temporal order and conceived event structure. Induction is the presumption (a reasonable belief) that the future will be similar to the past. In the method of science, (1) propositions or hypotheses (2) are tested in empirical experimentation (3) to collect a basis of data (4) from which conclusions are formed by the process of inference (5) that possibly confirm or validate (accept and verify or reject and falsify) the hypothesis. All that a conscious mind knows is its cognisance (wit, wisdom and knowledge), which is only known by inference. Credence or belief are informed by the memory of these associations. Inference reflects a matrix of axioms or improbable speculation (presumptions and suppositions). The sensory flow of perception of present experience is real-time, empirical, and indubitable, whereas the conception and memory of past experience is dubious and fallible. The basis of expectations of the future is the result of inferential processes, which are possibly (but not probably) true or correct. Cognisance informed exterior to sensory experience is contingent, or of imaginary confidence or faith. Humanity possesses the intellectual liberty to elect their personal axioms, or values moderated with humility and respect that construct the consensual and minimal norms of society.
Errors, as the obstacles to verity and cognisance, are consequences of:
the example of false (inadequate, improper and fragile) authority,
the durability of habit and customs,
the human propensity to obfuscate ignorance with the ostentation of apparent sapience.
Ignorance is a menace, and those who pretend and feign to utilise cognitive scientific methods debilitates confidence. Scrutiny and examination promote perspective (multiple points of view) necessary to prevent corruption. The mental practice of investigation is a personal responsibility and natural capacity of the scientists of the inquisitive and curious human spirit. The empiricism of experimentation, as an opportunity to prove and a privilege to fail, counteracts the tendencies resultant of credulity and conformity of ideas. Humans acquire cognisance in the perception and conception of experience. Reason in conception concludes with uncertainty. Experience, in the perception of actual reality, proves with certainty and reinforces the intuition of verity in the mental elimination of doubt.
Positivism was revitalised by Karl Popper, a philosopher of science like Auguste Comte, after criticising the logical positivist and empiricist inheritors. To positivism, the basis of "positive" cognisance is the natural phenomena and their properties and relations. Verified data or positive facts received from the senses are empirical evidence. The interpretation (the process of filtration, attention and construction) of this information derived from sensory experience forms the cognisance of the mind. Popper reintroduced the principle of falsification to replace that of verification. That is, in this epistemology, the scientific method is not to verify or confirm a theory but to corroborate the verisimilitude (truthlikeness) of such theory with empirical evidence from testing (experiment or observation). A scientific theory of any physical concept is a mathematical model that describes the observations of experience. Based on postulates, a theory predicts a testable and definite phenomenon. Theories being claims of universal knowledge or beliefs of objective truth, in what was termed as the fact–value distinction of Hume's is–ought problem, are impossible to be accepted (confirmed as true) but can be, when falsifiable, rejected (confirmed as false).
Popper proposed that falsifiability distinguishes empirical science (the scientific) from pseudoscience (the unscientific), not the metaphysical and semantic question of the meaningful (the significant) from the meaningless (the insignificant). Science originates from conjectures (myths and guesses) and an attitude of criticism. It is not merely the invention of experiments and collection of observations. The tradition of science transmits theories but with the critical discussion and conversation of the myths, and their technical and practical magic Thomas Kuhn expanded Popper's criticism beyond scientific revolution by arguing that a paradigm (a common intellectual structure or "disciplinary matrix", or more succinctly a cosmovision or worldview) must shift in response to evidence that subverts traditional presumptions. Kuhn proposed science, as a structure of its community and reality, is based on revisable paradigms formed by unprovable presumptions (i.e., beliefs and values) about the universal nature of the Cosmos. In this "normal" science, an accumulation of anomalous phenomena in reproducible experience occurs, realising what was ignorable as a problematic crisis for the present predictive capacity of the existing paradigm. The revision of the paradigm by the scientific community in order to reconcile and resolve the incongruent mystery and the incoherent enigma is a scientific revolution. With the restoration of confidence, the field continues, and science marches on.
The egocentric approach of philosophy with the self (auto) as the as basic frame of reference is inherently solipsistic. The interior mind of the individual acquires and possesses concepts from their own case. Ideas as psychological concepts originate from personal experience, an asymmetric reflection of reality, affirmed through perception and conception. Existence is the experience of the others (persons, events, processes, and phenomena, which are physical objects and subjects that coexist as constituents in space and time) in the content of conscious perception (feeling or smacking) and conception (thinking or thanking). The experience of the other is relational and intentional. An encounter with another is an encounter with someone who comprehends me as an object. In existence, if I am (sum) therefore I think (ergo cogito), then the self (ego) reveals the cogitation of essence. The mental substance (res cogitans) is united in physical space with the corporal substance (res extensa). Any differentiation in these is an illusionary consequence of existential egocentrism. Because of this universal and conceptual connection, the implication is the correlation of the contents of the mind and the nature of the body in the animal being. From the observation of other bodies acting like my body in similar conditions and circumstances, the self can infer the mental states that are correlated to my body occur in the other bodies. The foundation of this conceptual assembly (an inferential association of the internal and external) is the resemblance (identification) of a conscious being with sensation. The cognisance and credence of comprehension are not mental states (the process of the conscious mind mechanism).
In the solipsism (from the Latin solus "alone" plus ipse "self") doctrine, reality is limited to its relation to self. Existence refers to the existential and physical (mental and corporal) states of solely the self. To a solipsist person, their sole axion supposes their self is the only known to exist, as their cognisance (the state of the cognitive, affective, imaginative and sensitive experiences of emotions and reasons) is only known to their mind. Without more, this axiom is difficult to reconcile with the human experience of pleasure and dolour in the Universe, with a multitude of provisionally and conditionally observed facts. More axioms (premises that are neither true nor false, they solely are) are necessary to explain experiences of existence, perception and conception. Solipsism is isomorphic to pantheism, where the personality of self is divided into all physical and eternal interactions of the Universe. Wittgenstein makes the distinction of privacy and publicity in experience by the exclusivity and inclusivity of possession and cognisance. Language is incoherent in private so it can only public. It is the realisation of thought as expression. The requirement of a solipsist for a language (a sign-system) to think thoughts generates this incoherence. Language is a vital form of system used in society and social contexts that is governed by conventional not personal rules. Its meaning is public. It is the conduit for processing the phenomena and ascribing the data of the real world of which the self lives and experiences with other beings. Cognisance implies the existence of the social context of a public, common, intersubjective world that is absent in solipsism.
Wittgenstein was sceptical and critical of the psychology. Whilst it can infer causes and effects, its experimental methods he argued are prone to conceptual confusion. The general presumption is that psychology is parallel in its study to that of physics. The realm of physics is the physical world (Nature). Psychology, excluding the science of neurobiology, views processes in physical realm as does physics. However, the phenomena of cognition, affection, sensation, and imagination that are subject to psychology are not equivalent to the phenomena of matter, energy, force, and spacetime motion that are subject to the study of physics. The physicist uses the phenomena of psychology to investigate (experience and measurement by observation and collection) the physical world. The psychologist observes the external reactions of these phenomena. Unlike in physics, the words of the language of psychology do not refer to physical phenomena and therefore they cannot be the subject of investigation. The psychologist studies the symptoms of the nature of a phenomenon, not its perceptual change of aspect in expression, manifestation or conception. The confusion originates from the conceptual context conferred to mental states and processes as internal conditions. If humans are mental and corporal machines, conceptual mechanisms aid the comprehension of its function in the study of operation not its actions. Diagnosis, which provides a name, in psychology is a recognition of symptoms. Medical diagnoses, which are deductive hypotheses based on observed, measured and identified pathological processes, provide a prognosis and course of treatment and therapy with potential or possible complications. Psychology, in its inductive but confused procedure, misrepresents the diagnosis of effects as the causal factor to the malady.
As with epistemological subjectivity and objectivity, in science a conflict exists between realists and relativists who respectively argue that the description of the natural world is a true reality or a social construct. Similarly, philosophers debate the existence of mathematical entities is absolute (eternal and abstract ideas, and universal and certain objects) or fallible (corrigible and incomplete beliefs, and revisable and uncertain truths). In these views, mathematics is either discovered or invented. Wittgenstein proposes that mathematics consists of "language games", which are practices governed by rules that provide significance to symbolism of concepts and ideas. These rules (norms) are of traditional, cultural and social origins, not logical necessity. Inspired by the scepticism of Hume, this fallibilism (common to Popper) argues that no mathematical definitions or proofs are final, instead they are only accepted on the basis of authority and not by the conclusive justification of logic or reason. The momentum of mathematics is frequently motivated by the resolution and application of problems. The impetus of internal modification of mathematics were external necessities and resources. Arithmetic and its study of quantity (i.e., natural, integer, rational, real and complex numbers and their operations) evolved to sustain commerce and taxation. Geometry, the spatial discipline of mathematics that studies the properties of space, originated to progress trigonometry, astronomy, and navigation. It would describe the space that human experience, through the cognisance of perception and conception, occupies, imagines and calculates.
The human mind processes the displacement (motion), affine transformations (translation, reflection, dilatation, contraction, rotation, and transvection), and perspective (projection) observed in the visual field (sensory vision and object recognition by the collection and transduction of a signal). In descriptive graphical representation, the rectilinear rays of projection of an object in three-dimensional space are parallel, intersect orthogonal or oblique with the two-dimensional picture or plane of image. In perspective, parallel lines appear to converge at a point of fugue or flight (if the parallel lines are orthogonal to the plane of image, the point corresponds to the oculus, the location or station of the ocular observer). Algebra, with its algorithmic foundations and regulations, extended arithmetic (and its binary operations, varying in the properties of association, commutation, and distribution) with the implementation of abstract structures (e.g., variables, functions, matrices, and vectors). These vectors, or geometric quantities with magnitude (module or absolute value norm as a scalar with a unit) and direction (orientation and sense in reference to the referential basis and order), have a course in space and momentum in motion. They can be normalised to unit vectors whose linear combination (with the coordinates as coefficients) can be written as each vector in space, if its basis is formed by a linearly independent system of these unit vectors as elements that generate the vector space (whose dimension is the cardinality of the basis). In the canonical basis, the unit vectors are mutually orthogonal (perpendicular, or normal to the tangent plane of a surface). A vector is an eigenvector ("own, proper, self") if a linear transformation (operator or application) is a scalar (called an eigenvalue) multiple of that vector. In finite-dimensional vector space, the linear transformation, which does not mutate the orientation of the vector, can be expressed as a matrix. With differentiation (continuous and instantaneous variation) and integration (summation of definite and infinite series of quantities), infinitesimals (functional limits) would progress this further (e.g., convolution and correlation).
The concept of verity is related to the certainty of a physical (spatial and temporal) occurrence of an event. The subjectivity of memory, in its formation of myth and history, results in the virtual aspect of reality. The virtual bipolarity (dichotomy) of the construction of verity is simulation and imitation versus action and creation. Gilles Deleuze proposed the concept of virtual ideas as the conditions of experience. They are identical to the object in the spacetime imposed by the subject. The "virtual" is an ideal with potential; it is opposed to "actual" but not to "real", which is opposed to "possible". The "ideal-real" is denoted the "virtual" like how "probable" is the "actual-possible", "abstract" is the "ideal-possible", and "material" is the "actual-real". The virtual is analogous to a reflection in a mirror. Deleuze constructed this concept of the virtual from the work of Marcel Proust, who describes the states of resonant states of memory as "simultaneously in the present and in the past, real without being actual, ideal without being abstract". The virtual has no resemblance to the actual. The reality of the virtual is structure. The structure, or composition, of a substance consists of its relations of individual entities. A substance or individual being is independent of its relations to other substances. Integral to his doctrine is multiplicity, an idea of complex structure without a referential unity that replaces the substance of nature. The distinction of multiplicity (from the mathematical concept) is made between the continuous (virtual, qualitative, subjective, typical, temporal, and organic) and the discrete (actual, quantitative, objective, gradual, spatial, and ordinal) differentiation of relations and elements. Nietzsche criticises philosophers that ignore the fidelity of their senses to construct an idol (deformed copy and distorted replication) of reality with language (logic and reason). Deleuze defines this simulacrum (phantasm) as a fantasy—i.e. imitation or image of internal dissimilarity or difference, without resemblance or relation to a model or reality. Representation of reality can be categorised by reflection, perversion, absence, and simulation. With this hyperreality being a sign, its simulation is an imaginary immersion that mimics without referential origin. Any manifestation of external identity is an illusion.
Simultaneously with this metaphysical doctrine, in ontology Deleuze asserted that being is univocal, i.e. all of its senses are affirmed in one voice, in difference. The self and its individuality of person that are equivocal in and for a univocal being. He argues in this echo and inversion that this univocity is implicit in the cosmic organisation of Spinoza where all existence is a modification of one substance, God or Nature. Deleuze, though, maintains that there is not a single cosmic substance but an impermanent process of multiplicity. In this paradoxical reality of difference being is becoming, and monism is pluralism. The immanence proposed by Spinoza organised the unitary and infinite substance God or Nature such that there is no transcendent principle to the Cosmos, and that the panentheistic process of natural creation is contained within immanent Nature. Deleuze advanced this theory with the plane of immanence. His conception, in metaphysical consistency with Spinoza, includes life and death, existence and annihilation, and creation and destruction. Real distinctions (mental and corporal, spatial and temporal, spiritual and material, and interior and exterior) are collapsed to a pure plane of immanence without opposition. This geometric plane of differentiation through Chaos, is without preeminent forms or transcendental subjects, but with physical forces, particles, connections, relations, affects, and objects (things and beings). The plane of immanence is the absolute ground of philosophy, a horizon of concepts and events. Concepts combine, in coincidence, condensation, and accumulation, the components of the plane as points to express ideas of thought.
Conjoined, this ontological pluralistic monism is compatible to the holism of Wittgenstein as opposed to atomism. Spinoza, in his Ethics (Ethica, ordine geometrico demonstrata, where he concludes his definitions, axioms, and propositions with "what was to be demonstrated", quod erat demonstrandum, or QED), proposes that it is conation or effort (conatus) for self-preservation that is universal to organic (human and animal) things. He writes, "Each thing, as far as it can by its own power, strives to persevere in its being" (Unaquaeque res, quantum in se est, in suo esse perseverare conatur). This striving doctrine of egoism is a resistance to annihilation, a desire to be and become, that characterises individuality. Deleuze continues the naturalist ethics of this motivational force and the will to might (desire to power) of Nietzsche where morality in society originates from individuals who possess abstract natural rights (obligations of life and liberty that in the egoist model are might and contract). This vital force is natural to the physical reality, conduct and mind of individuals not solely self-preservation, but a violation or defence of the self from the violation of the other in material manifestation, potent augmentation and the extension of the ego. Its initiation of the eternal return of memory and experience is related to the sense of a cyclic universe of perpetual entropy. The premise of the eternal returns is that existence and its finite events (a conservation of energy in an affirmation of multiplicity and difference) recur and repeat in a self-(dis)similar form over infinite spacetime. It represents the nature of absurdity of existence. With his rejection of metaphysical identity, Deleuze criticised the idea that the individual arrests differentiation in its nominal significance of "indivisible" and "atomic". Instead he attempted to comprehend individual and personal morality as a social construction and product of the pre-individual and impersonal desires. Ethics are products of immanent evaluation and not of transcendent judgment. The invalidation of one's established identity to become (of what is impossible to anticipate) is an affirmation of reality, which is a flux of impermanence and difference. This means to live well is to express one's potential as authority and experience as creativity.
The following is a gloss (i.e., a paraphrase in abbreviation and modification) of the is the seven propositions of a tractate (Tractatus Logico-Philosophicus) by Wittgenstein that defined the mutual relations of language and reality as the limit to the expression of thought. Beyond the limit is nonsense. Its title is an allusion to Tractatus Theologico-Politicus by Spinoza. In the work, Wittgenstein proposes a correspondence theory of truth where: a sentence is true if and only if (1) it represents a certain possible fact, and (2) that possible fact is actual. This is equivalent to: S is a true sentence ≡ S represents a fact (= an actualised possible fact). A sentence, as a picture, is a cord of significant signs. Pictures describe complex ideas, forms or concepts. It is a pictorial representation that consists of:
elements arranged with respect to one another in a certain way,
each such element has a referent; and
the actual fact that the pictorial elements are arranged as they are represents the possible fact that the referents of those elements
are also arranged just in that way.
Sentential truth is the idea that only when the things signified (expressed or represented) have a certain quality of being actual (factual, or existing or corresponding with the facts of reality) can the sentences with those significances (meanings) be true. A depiction of a possible fact (a sense of a proposition), in its relation of reference, is a form of correspondence. In this model, reality or the real world is factual actuality. Wittgenstein would later propose a deflationary theory of truth, which is independent of correspondence, coherence, probability, verifiability, utility, or consensus. In its minimal thesis, where a possible fact is what is now called a proposition, the concept truth consists of the schema: the proposition that p is true ↔ p. He would eventually declare, in his transition from formal (logical) realism to idealism, "meaning is [as or qua] use". It becomes impossible to speak when terms have no coherent meaning. Significance is one aspect of the expression of a sentence in the "visual" transformation by mental experience (the impression of sensation and imagination, the cognitive attention, interpretation and representation of phenomenal information by perception and conception). Logic, in logocentrism, is an ideal representation (sign) of phonetic and graphic language that is an expression of reality and a reflection of facts. Spinoza, Hume, Russell, and Popper have been interpreted to advocate and defend the correspondence theory of truth. The condition of correspondence (an objective or factual relation of congruence, reference, accordance, signification, representation, satisfaction, and conformity) of a characteristic (trait) contrasts with coherence a coherence, where a proposition (sentence, conviction, or opinion) is true if coherent with a consistent (specific and holistic) ensemble (totality and entirety) of propositions. Correspondence reduces to coherence when ideas (objects of the conception of a subject) of the world become veritable ideas.
The biconditional equivalence or identity of "if and only if" that is connective of an antecedent p and consequent q can be symbolised with ↔ and ≡, as opposed to = for an equality, ≠ for an inequality, ≅ for a congruence, → for a conditional or functional implication, ¬ for a negation, ∧ for a conjunction, ∨ for disjunction, ⊥ for contradiction, ⊤ for tautology, ∃ for existential quantification, and ∀ for universal quantification. For example, p → q is equivalent to ¬(p ∧ ¬q) and ¬p ∨ q. Equivalence, or materially necessary and sufficient, is a binary relation that is reflexive, symmetric and transitive. The binary (dyadic) relations or operations use double operands (e.g., convolution, correlation, intersections, unions, complements, modules, products, differences, logarithms, exponentiation, multiplication, division, addition, subtraction, in contrast to their unary homologues that use single operands (e.g., negations, positives, negatives, magnitudes, exponential radices and potencies, factorials, integer and fractional parts, natural logarithms, multiplicative inverses or reciprocals, transposes, and inverse, sign and trigonometric functions). In inclusions (collections of elements in an ensemble), ∈ represents the elemental relation, ⊇ (⊃) for a (proper) super ensemble, ⊆ (⊂) for a (proper) sub ensemble, ∅ for a void ensemble, ′ for a complement, ∪ for a union, and ∩ for an intersection. A finite collection of ensembles and their possible logical relations can be visualised in a diagram where elements are depicted as points in a plane, and ensembles as internal regions of circuitous curves. Points interior to the frontier are included as elements in the ensemble whereas exterior points are excluded. Deductive inference includes the probable rules of modus ponendo ponens ("mode that by affirmation affirms") and modus tollendo tollens ("mode that by negation negates"). The propositional logic of a conditional implication is that if an antecedent implies a consequent, and the antecedent is true, then the consequent is also true. By transposition, if the consequent is false (negative not positive) then the antecedent is false. The contrapositive of a conditional implication is its inversion (the inverse of p → q is ¬p → ¬q) and conversion (the converse of p → q is q → p), where the negation of the antecedent is the consequence (consequential implication) of the negation of the consequent (¬q → ¬p).
- 1. The world is all that is the case.
- 1.0 The world is reality.
1.1 The world is the totality of facts, not of things.
- 1.11 The world is determined by the facts, and by their being all the facts.
1.12 For the totality of facts determines what is the case, and also whatever is not the case.
1.13 The facts in logical space are the world.
1.2 The world divides into facts, which can be the case or not the case while everything else remains the same.
- 2. What is the case—a fact—is the existence of states of affairs.
- 2.0 Objects are things.
- 2.01 A state of affairs (a state of things) is a combination of objects (things).
- 2.010 In logic nothing is accidental: if a thing can occur in a state of affairs, the possibility of the state of affairs is essential and original to the thing itself. Logic concerns every possibility and all possibilities are its facts.
2.011 Each thing is in an infinite space of possible states of affairs.
2.012 Things are independent in so far as they can occur in all possible situations, but this form of independence is a form of connection with states of affairs, a form of dependence.
2.013 Objects contain the possibility of all situations. The possibility of its occurring in states of affairs is the form of an object.
2.02 Objects are simple.
- 2.020 Every statement about complexes can be resolved into a statement about their constituents and into the propositions that describe the complexes completely.
2.021 Objects make up the substance of the world. That is why they cannot be composite.
2.022 It is obvious that an imagined world, however different it may be from the real one, must have something—a form—in common with it.
2.022 Objects are just what constitute this unalterable form.
2.023 The substance of the world can only determine a form, and not any material properties. For it is only by means of propositions that material properties are represented—only by the configuration of objects that they are produced.
2.024 The substance is what subsists independently of what is the case. It is form and content.
2.025 Space, time, and colour are forms of objects.
2.026 There must be objects, if the world is to have unalterable form.
2.027 Objects, the unalterable, and the subsistent are one and the same.
- 2.0271 Objects are what is unalterable and subsistent; their configuration is what is changing and unstable.
2.0272 The configuration of objects produces states of affairs.
2.03 In a state of affairs, objects fit into each other like the links of a chain.
- 2.031 In a state of affairs objects stand in a determinate relation to one another.
2.032 The determinate way in which objects are connected in a state of affairs is the structure of the state of affairs.
2.033 Form is the possibility of structure.
2.034 The structure of a fact consists of the structures of states of affairs.
2.04 The totality of existing states of affairs is the world.
2.05 The totality of existing states of affairs also determines which states of affairs do not exist.
2.06 The existence (positive facts) and non-existence (negative facts) of states of affairs is reality.
- 2.061 States of affairs are independent of one another.
2.062 From the existence or non-existence of one state of affairs it is impossible to infer the existence or non-existence of another.
2.063 The sum-total of reality is the world.
2.1 We picture facts to ourselves.
- 2.11 A picture presents a situation in logical space, the existence and non-existence of states of affairs.
2.12 A picture is a model of reality.
2.13 In a picture, objects have the elements of the picture corresponding to them. The elements of the picture are the representatives of objects.
2.14 What constitutes a picture is that its elements are related to one another in a determinate way. A picture is a fact.
2.15 The fact that the elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way. Let us call this connexion of its elements the structure of the picture, and let us call the possibility of this structure the pictorial form of the picture.
- 2.151 Pictorial form is the possibility that things are related to one another in the same way as the elements of the picture.
- 2.1511 That is how a picture is attached to reality; it reaches right out to it.
2.1512 It is laid against reality like a measure. Only the end-points of the graduating lines actually touch the object that is to be measured.
2.1513 So a picture, conceived in this way, also includes the pictorial relation, which makes it into a picture.
2.1515 These correlations are, as it were, the antenna of the picture's elements, with which the picture touches reality.
2.16 If a fact is to be a picture, it must have something in common with what it depicts. There must be something identical in a picture and what it depicts, to enable the one to be a picture of the other at all.
2.17 What a picture must have in common with reality, in order to be able to depict it—correctly or incorrectly—in the way that it does, is its pictorial form.
- 2.171 A picture can depict any reality whose form it has. A spatial picture can depict anything spatial, a coloured one anything coloured, etc.
2.172 A picture cannot, however, depict its pictorial form: it displays it.
2.173 A picture represents its subject from a position outside it. (Its standpoint is its representational form.) That is why a picture represents its subject correctly or incorrectly.
2.174 A picture cannot, however, place itself outside its representational form.
2.18 What any picture, of whatever form, must have in common with reality, in order to be able to depict it—correctly or incorrectly—in any way at all, is logical form, i.e. the form of reality.
- 2.181 A picture whose pictorial form is logical form is called a logical picture.
2.182 Every picture is at the same time a logical one. (On the other hand, not every picture is, for example, a spatial one.)
2.19 Logical pictures can depict the world.
2.2 A picture has logical form in common with what it depicts.
- 2.21 A picture depicts reality by representing a possibility of existence and non-existence of states of affairs. It contains the possibility of the situation that it represents. It agrees with reality or fails to agree; it is correct or incorrect, true or false.
2.22 What a picture represents it represents independently of its verity or falsity, by means of its pictorial form.
- 2.221 What a picture represents is its sense.
2.222 The agreement or disagreement or its sense with reality constitutes its verity or falsity.
2.223 In order to tell whether a picture is true or false we must compare it with reality.
2.224 It is impossible to tell from the picture alone whether it is true or false.
2.225 There are no pictures that are true a priori.
3. A logical picture of facts is a thought.
- 3.0 A state of affairs is thinkable, which means we can picture it to ourselves.
- 3.01 The totality of true thoughts is a picture of the world.
3.02 A thought contains the possibility of the situation of which it is the thought. What is thinkable is possible. Thought can never be of anything illogical.
3.03 Linguistic expression is a form of projection in geometry that is represented by coordinates in space.
- 3.031 The representation of language is the form of projection.
3.032 The logical structure of the expression is the geometric relation of space.
3.033 It is impossible to represent in language anything that 'contradicts logic' as it is in geometry to represent by its coordinates a point or figure that does not exist in space.
3.04 If a thought were correct a priori, it would be a thought whose possibility ensured its truth.
3.05 A priori knowledge that a thought was true would be possible only if its truth were recognisable from the thought itself (without anything a to compare it with).
3.1 In a proposition, an expression of a thought is perceived by the senses.
- 3.11 We use the perceptible sign of a proposition (spoken or written, etc.) as a projection of a possible situation. The method of projection is to think of the sense of the proposition.
3.12 I call the sign with which we express a thought a propositional sign. And a proposition (a significant sentence) is a propositional sign (uninterpreted sentence) in its projective relation to the world.
3.13 A proposition, therefore, does not actually contain its sense, but does contain the possibility of expressing it. A proposition contains the form, but not the content, of its sense.
3.14 What constitutes a propositional sign is that in its elements (the words) stand in a determinate relation to one another. A propositional sign is a fact.
3.2 In a proposition, a thought can be expressed in such a way that elements of the propositional sign correspond to the objects of the thought.
- 3.21 The configuration of objects in a situation corresponds to the configuration of these elements (simple signs) in the propositional sign.
3.22 Objects can only be named as simple signs of propositions. Signs are their representatives.
3.23 A name means an object. The object is its meaning.
3.24 Names are like points; propositions like arrows—they have sense. An arrow is bipolar because it has two directions. Its sense is asymmetrical because it indicates (its index points in) one direction.
3.25 The requirement that simple signs be possible is the requirement that sense be determinate.
3.3 Only the proposition has sense; only in the context of a proposition has a name meaning.
- 3.31 Any part of a proposition (itself an expression) that characterises its sense a a symbol. Everything essential to their sense that propositions can have in common with one another is a symbol. A symbol is the mark of a form and a content.
3.32 A sign is what can be perceived of a symbol.
- 3.321 In order to recognise a symbol by its sign, it must be observed how it is used with a sense.
3.322 If a sign is not necessary then it is meaningless. That is the meaning of Occam's razor. (If everything behaves as if a sign had meaning, then it does have meaning.)
3.33 In logical syntax (grammar), the meaning of a sign should never play a role. It must be possible to establish logical syntax without mentioning the meaning of a sign: only the description of expressions may be presupposed.
3.34 A proposition possesses essential and accidental traits. Accidental traits are those that result from the particular way in which the propositional sign is produced. Essential traits are those without which the proposition could not express its sense.
3.4 A proposition determines a place in logical space. The existence of possibility (the propositional sign with logical coordinates) is guaranteed by the mere existence of the constituents—by the existence of the proposition with a sense.
3.5 A propositional sign, applied and thought out, is a thought.
4. A thought is a proposition with a sense.
- 4.0 The totality of propositions is language.
- 4.00 Most of the propositions and questions to be found in philosophical works are not false but nonsensical. Consequently we cannot give any answer to questions of this kind, but can only point out that they are nonsensical. Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language. (They belong to the same class as the question whether the good is more or less identical than the beautiful.) And it is not surprising that the deepest problems are in fact not problems at all.
4.01 A proposition is a picture of reality. A proposition is a model of reality as we imagine it.
4.02 We can see this from the fact that we understand the sense of a propositional sign without its having been explained.
- 4.021 A proposition is a picture of reality: for if I understand a proposition, I know the situation that it represents. And I understand the proposition without having had its sense explained to me.
4.022 A proposition shows its sense. A proposition shows how things stand if it is true. And it says that they do so stand.
4.023 A proposition must restrict reality to two alternatives: yes or no. In order to do that, it must describe reality completely. A proposition is a description of a state of affairs. Just as a description of an object describes it by giving its external properties, so a proposition describes reality by its internal properties. A proposition constructs a world with the help of a logical scaffolding, so that one can actually see from the proposition how everything stands logically if it is true. One can draw inferences from a false proposition.
4.024 To understand a proposition means to know what is the case if it is true. It is understood by anyone who understands its constituents.
4.025 It belongs to the essence of a proposition that it should be able to communicate a new sense to us.
4.03 A proposition must use old expressions to communicate a new sense. A proposition communicates a situation, and so it must be essentially connected with the situation. And the connection is precisely that it is its logical picture. A proposition states something only in so far as it is a picture.
- 4.031 In a proposition, a situation is constructed and represented by way of experiment.
4.032 One name stands for one thing, another for another thing, and they are combined with one another. In this way the whole (entire) group—a living picture or vivid representation—presents a state of affairs.
4.033 The possibility of propositions is based on the principle that objects have signs as their representatives. The fundamental idea is that the 'logical constants' (relations of elementary logic and symbols of quantification) are not representatives; that there can be no representatives of the logic of facts.
4.034 Propositions about states of affairs of the world do not symbolise (project on to) objects in reality. The logical articulation of a proposition is a picture of a situation.
4.04 In a proposition there must be exactly as many distinguishable parts as in the situation that it represents. The two must possess the same logical (mathematical) multiplicity.
4.05 Reality is compared with propositions.
4.06 A proposition can be true or false only in virtue of being a picture of reality.
4.1 Propositions represent the existence and non-existence of states of affairs.
- 4.11 The totality of true propositions is the whole corpus of natural sciences.
- 4.111 Philosophy is not one of the natural sciences. (The word "philosophy" must mean something whose place is above or below the natural sciences, not beside them.)
4.112 Philosophy aims at the logical clarification of thoughts. Philosophy is not a body of doctrine but an activity. A philosophical work consists essentially of elucidations. Philosophy does not result in "philosophical propositions", but rather in the clarification of propositions. Without philosophy thoughts are, as it were, cloudy and indistinct: its task is to make them clear and to give them sharp boundaries.
4.113 Philosophy sets limits to the much disputed sphere of natural science.
4.114 It must set limits to what can be thought; and, in doing so, to what cannot be thought. It must set limits to what cannot be thought by working outwards through what can be thought.
4.115 It will signify what cannot be said, by presenting clearly what can be said.
4.12 Propositions can represent the whole of reality, but they cannot represent what they must have in common with reality in order to be able to represent it—logical form. In order to be able to represent logical form, we should have to be able to station ourselves with propositions somewhere outside logic, that is to say outside the world.
- 4.121 Propositions cannot represent logical form: it is mirrored in them. What finds its reflection in language, language cannot represent. What expresses itself in language, we cannot express by means of language. Propositions show the logical form of reality. They display it.
4.122 Objects and states of affairs have formal properties and relations. Facts have structural properties and relations.
4.123 The propositional variable signifies the formal concept, and its values signify the objects that fall under the concept. Every variable is the sign for a formal concept. For every variable represents a constant form that all its values possess, and this can be regarded as a formal property of those values.
4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs.
- 4.21 The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs. As all states of affairs are independent of all other states of affairs, all elementary propositions are independent of all other elementary propositions.
4.22 An elementary proposition consists of names. It is a nexus, a concatenation, of names.
4.23 Even if the world is infinitely complex, so that every fact consists of infinitely many states of affairs and every state of affairs is composed of infinitely many objects, there would still have to be objects and states of affairs.
4.24 It is only in the nexus of an elementary proposition that a name occurs in a proposition.
4.25 Names are the simple symbols.
4.26 If an elementary proposition is true, the state of affairs exists: if an elementary proposition is false, the state of affairs does not exist.
4.27 If all true elementary propositions are given, the result is a complete description of the world. The world is completely described by giving all elementary propositions, and adding which of them are true and which false. For n states of affairs, there are possibilities of existence and non-existence. Of these states of affairs any combination can exist and the remainder not exist.
4.28 There correspond to these combinations the same number of possibilities of verity—and falsity—for n elementary propositions.
4.3 Truth possibilities of elementary propositions mean possibilities of existence and non-existence of states of affairs. We can represent truth-possibilities by schemata of the following kind (tables where 'T' means 'true' and 'F' means 'false', with rows of Ts and Fs under the row of elementary propositions symbolise their truth-possibilities in a way that can easily be understood).
4.4 A proposition is an expression of agreement and disagreement with truth-possibilities of elementary propositions.
- 4.41 Truth-possibilities of elementary propositions are the conditions of the verity and falsity of propositions.
4.42 For n elementary propositions there are ways in which a proposition can agree and disagree with their truth possibilities.
4.43 We can express agreement with truth-possibilities by correlating the mark 'T' (true) with them in the schema. The absence of this mark ('F' or false) means disagreement. The expression of agreement and disagreement with the truth possibilities of elementary propositions expresses the truth-conditions of a proposition. A proposition is the expression of its truth-conditions.
4.44 The sign that results from correlating the mark 'T' with truth-possibilities is a propositional sign.
4.45 For n elementary propositions there are Ln possible groups of truth-conditions. The groups of truth-conditions that are obtainable from the truth-possibilities of a given number of elementary propositions can be arranged in a series.
4.46 Among the possible groups of truth-conditions there are two extreme cases. In one of these cases the proposition is true for all the truth-possibilities of the elementary propositions. Truth-conditions are tautological. In the second case the proposition is false for all the truth-possibilities: the truth-conditions are contradictory. In the first case we call the proposition a tautology; in the second, a contradiction. A tautology has no truth-conditions, since it is unconditionally true. A contradiction is true on no condition.
- 4.461 Tautologies and contradictions are not nonsensical.
4.462 Tautologies and contradictions are not pictures of reality. They do not represent any possible situations. For the former admit all possible situations, and latter none. In a tautology the conditions of agreement with the world—the representational relations—cancel one another, so that it does not stand in any representational relation to reality.
4.463 The truth-conditions of a proposition determine the range that it leaves open to the facts. A tautology opens to reality the whole—the infinite whole—of logical space. A contradiction restricts the whole of logical space leaving no point of it for reality. Thus neither of them can determine reality in any way.
4.464 The truth of a tautology is certain, a proposition possible, a contradiction impossible. (Certain, possible, impossible: here we have the first indication of the scale that we need in the theory of probability.)
4.5 The existence of a general propositional form is proved by the fact that there cannot be a proposition (a picture or a model) whose form could not have been foreseen (i.e., constructed). The general form of a proposition is: This is how things stand.
5. A proposition is a truth-function of elementary propositions.
- 5.0 Elementary propositions are the truth-arguments of propositions.
5.1 Truth-functions can be arranged in series. That is the foundation of the theory of probability. In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way.
5.2 The structures of propositions stand in internal relations to one another.
5.3 All propositions are results of truth-operations on elementary propositions. A truth-operation is the way in which a truth-function is produced out of elementary propositions. It is of the essence of truth-operations that, just as elementary propositions yield a truth-function of themselves, so too in the same way truth-functions yield a further truth-function. When a truth-operation is applied to truth-functions of elementary propositions, it always generates another truth-function of elementary propositions, another proposition. When a truth-operation is applied to the results of truth-operations on elementary propositions, there is always a single operation on elementary propositions that has the same result. Every proposition is the result of truth-operations on elementary propositions. All truth-functions are results of successive applications to elementary propositions of a finite number of truth-operations.
5.4 At this point it becomes manifest that there are no 'logical objects' or 'logical constants'.
- 5.41 The general propositional form is the essence of a proposition.
5.42 To give the essence of a proposition means to give the essence of all description, and thus the essence of the world.
5.5 Every truth-function is a result of successive applications to elementary propositions of the joint negation (a conjunctive negation symbolised as N) operation.
- 5.51 If ξ (a variable which has propositions as its value) has only one value, then N(ξ) = ¬p (not p); if it has two values such that ξ = (p, q), then N(ξ) = ¬p∧¬q (neither p nor q).
5.52 Consider the following traductions: N[N(p), N(q)] = p∧q; N[N(p, q)] = p∨q.
5.6 The limits of my language mean the limits of my world.
- 5.61 Logic pervades the world: the limits of the world are also its limits.
5.62 This remark provides the key to the problem, how much truth there is in solipsism. For what the solipsist means is quite correct; only it cannot be said, but makes itself manifest. The world is my world: this is manifest in the fact that the limits of language (of that language which alone I understand) mean the limits of my world. The world and life are one.
5.63 I am my world. (The microcosm of the macrocosm.)
5.64 The philosophical self is not the human being, not the human body, human mind or the human soul but rather the metaphysical subject, the limit of the world—not a part of it.
6. The general form of a truth-function is [p, ξ, N(ξ)].
- 6.0 This is the general form of a proposition. Every proposition ξ is a result of successive applications to elementary propositions p of the operation N(ξ).
6.1 The propositions of logic are tautologies.
- 6.11 Therefore the propositions of logic say nothing.
6.12 The fact that the propositions of logic are tautologies shows the formal—logical—properties of language and the world.
- 6.121 The propositions of logic demonstrate the logical properties of propositions by combining them so as to form propositions that say nothing.
6.123 The general validity of logic might be called essential, in contrast with the accidental general validity of such propositions as 'All humans are mortal'.
6.124 The propositions of logic describe the scaffolding of the world, or rather they represent it.
6.125 One can calculate whether a proposition belongs to logic, by calculating the logical properties of the symbol. And this is what we do when we 'prove' a logical proposition. For, without bothering about sense or meaning, we construct the logical proposition out of others using only rules that deal with signs. The proof of logical propositions consists in the following process: we produce them out of other logical propositions by successively applying certain operations that always generate further tautologies out of the initial ones. (And in fact only tautologies follow from a tautology.) Of course this way of showing that the propositions of logic are tautologies is not at all essential to logic, if only because the propositions from which the proof starts must show without any proof that they are tautologies.
6.126 All the propositions of logic are of equal status.
6.13 Logic is not a body of doctrine, but a mirror-image of the world. Logic is transcendental.
6.2 Mathematics is a logical method. The propositions of mathematics are equations.
- 6.21 A proposition of mathematics does not express a thought.
6.22 The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics.
6.23 If two expressions are combined by means of the sign of equality, that means that they can be substituted for one another. But it must be manifest in the two expressions themselves whether this is the case or not. When two expressions can be substituted for one another, that characterises their logical form.
- 6.231 An equation merely marks the point of view from which I consider the two expressions: it marks their equivalence in meaning.
6.232 The question whether intuition is needed for the solution of mathematical problems must be given the answer that in this case language itself provides the necessary intuition.
6.233 The process of calculating serves to bring about that intuition. Calculation is not an experiment.
6.234 Mathematics is a method of logic.
6.3 Exploration of logic means the exploration of everything that is subject to law—outside of logic everything is accidental.
- 6.31 We cannot compare a process with 'the passage of time'—there is no such thing—but only with another process (such as the working of a chronometer). Hence we can describe the lapse of time only by relying on some other process. Something exactly analogous applies to space: e.g. when people say that neither of two events (which exclude one another) can occur, because there is nothing to cause the one to occur rather than the other, it is really a matter of our being unable to describe one of the two events unless there is some sort of asymmetry to be found. And if such an asymmetry is to be found, we can regard it as the cause of the occurrence of the one and the non-occurrence of the other.
6.32 The procedure of induction consists in accepting as true the most parsimonious hypothesis that can be reconciled with our experiences. This procedure, however, has no logical justification but only a psychological one. It is clear that there are no grounds for believing that the simplest eventuality will in fact be realised.
6.33 There is no compulsion making one thing happen because another has happened. The only necessity that exists is logical necessity. The only impossibility that exists is logical impossibility.
6.34 The whole modern conception of the world is founded on the illusion that the laws of nature are the explanations of natural phenomena and are inviolable (like God and Fate). The world is independent of my will.
6.4 All propositions are of equal value.
- 6.41 The sense of the world must lie outside the world. In the world no value exists. For all that happens and is the case is accidental.
6.42 It is impossible for there to be propositions of ethics and aesthetics, which are transcendental.
6.43 It is impossible to speak about the will insofar as it is the subject of ethical attributes.
6.44 If the good or bad exercise of the will does alter the world, it can alter only the limits of the world, not the facts—not what can be expressed by means of language. In short the effect must be that it becomes an altogether different world.
6.45 Death is not an event in life: humans do not live to experience death. If eternity means not infinite temporal duration but timelessness, then eternal life belongs to those who live in the present. Our life has no end in just the way in which our visual field has no limits.
6.46 There is no guarantee of the temporal immortality of the human soul, or its eternal survival after death. Is not this eternal life itself as much of a riddle as our present life? The solution of the enigma of life in space and time lies outside space and time.
6.47 How things are in the world is a matter of complete indifference for what is higher. God does not reveal themselves in the world.
6.48 The facts all contribute only to setting the problem, not to its solution.
6.49 It is not how things are in the world that is mystical, but that it exists. To view the world sub specie aeternitatis (under the aspect of eternity, i.e. without reference to or dependence upon temporal reality) is to view it as a whole—a limited whole. Feeling the world as a limited whole—it is this that is mystical.
6.5 When the answer cannot be put into words, neither can the question be put into words. The enigma does not exist. If a question can be framed at all, it is also possible to answer it.
- 6.51 Scepticism is not irrefutable, but obviously nonsensical, when it tries to raise doubts where no questions can be asked. For doubt can exist only where a question exists, a question only where an answer exists, and an answer only where something can be said.
6.52 We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched. Of course there are then no questions left, and this itself is the answer.
- 6.521 The solution of the problem of life is seen in the vanishing of the problem.
6.522 There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical.
6.53 The correct method of philosophy would really be the following: to say nothing except what can be said—i.e., propositions of natural science—and then, whenever someone else wanted to say something metaphysical, to demonstrate to him that he had failed to give a meaning to certain signs in his propositions.
6.54 My propositions serve as elucidations in the following way: anyone who understands me eventually recognises them as nonsensical, when he has used them to climb up beyond them. He must transcend these propositions, and then he will see the world aright.
7. Whereof one cannot speak, thereof one must be silent.