The world constitutes a comprehensive sum of facts, not objects. These facts, in their entirety, shape the world by defining what exists and what does not. The arrangement of these facts within logical space forms the world's structure. Each fact has a binary nature: it can either occur or not, independent of other facts.
Atomic facts, composed of objects, are fundamental to existence. Objects inherently possess the potential to form atomic facts. In logic, this potential is intrinsic, not accidental. Objects are interconnected within atomic facts, mirroring how spatial or temporal objects cannot be conceived without space or time. Understanding an object includes knowing all its potential occurrences in atomic facts. Objects are simple and form the world's substance, implying the world would be indefinable without them. They are devoid of material properties, distinguished only by their logical form. Substance exists independently of facts, comprising both form and content. The structure of the atomic fact, defined by the arrangement of objects, forms the fact's structure. The world is the totality of all existing atomic facts, and this totality also defines non-existing atomic facts. Atomic facts are independent and their existence or absence constitutes reality. The overall reality, encompassing all atomic facts, defines the world.
We create representations of facts through pictures. These pictures model reality by reflecting the existence and non-existence of atomic facts within logical space. In a picture, each element corresponds to and represents an object. The structure of the picture, defined by the specific arrangement of its elements, mirrors the arrangement of objects in reality. This structural similarity is the picture's form of representation, establishing its connection to reality. The picture, as a fact itself, touches reality through this form of representation, depicting reality either accurately or inaccurately. To be a representation, a picture must share a logical form with what it depicts. It can represent any reality that matches its form – a spatial picture represents spatial reality, a colored one represents colored reality, and so on. However, a picture cannot represent its own form of representation but only exhibit it. The essential commonality between a picture and reality for effective representation is this logical form. Every picture is inherently a logical picture, capable of depicting the world.
A picture shares the logical form of representation with what it depicts. It portrays reality by illustrating a potential arrangement of atomic facts, representing a conceivable situation in logical space. The picture inherently includes the possibility of the state of affairs it represents. Its correspondence with reality determines its accuracy or inaccuracy; it is either right or wrong, true or false. The representation in a picture is consistent, independent of its truthfulness, due to its form of representation. The content of what the picture represents constitutes its sense. The truth or falsity of a picture is determined by how its sense aligns or misaligns with reality. To assess the truthfulness of a picture, it must be compared with reality; this cannot be deduced from the picture alone. No picture is inherently true without such a comparison.
The logical representation of facts in our minds is termed as thought. Thinking an atomic fact is possible if we can imagine it. The collection of all true thoughts forms a conceptual image of the world. A thought encompasses the potentiality of the state of affairs it contemplates, asserting that what is conceivable is also feasible. Logical consistency is inherent in thought; thinking illogically is impossible. Comparatively, it's akin to the impossibility of presenting in geometry a figure that defies spatial laws, or in language, expressing something that contradicts logic. While it's possible to spatially represent an atomic fact that violates physical laws, doing so against geometric laws is not. A thought that could be inherently true would be one whose possibility assures its truth. Knowledge of a thought's truth, a priori, would require that its truth be evident within the thought itself, without needing external reference.
In a proposition, thought is conveyed perceptibly through sensory signs like sounds or written symbols. The proposition uses these signs as a projection of a possible state of affairs. This projection method, the contemplation of the proposition's meaning, is the thought process. The sign expressing the thought is the propositional sign, and the proposition itself is this sign in its projective relationship with the world.
The proposition encompasses all that is part of the projection but not the actual content being projected. It contains the potential to express its sense, but not the sense itself. The proposition includes the form of its sense but not the content. The propositional sign is a fact, constituted by the specific arrangement of its elements – the words.
A proposition is not merely a combination of words, just as a musical theme is not just a mix of tones. It is structured and articulate. Only facts, not a mere collection of names, can convey a sense. The factual nature of the propositional sign is often obscured in its written or printed form. For clarity, one could imagine the propositional sign made up of physical objects like tables or chairs, where their spatial arrangement conveys the proposition's sense.
It's important to differentiate between stating that the sign 'aRb' represents 'a is in relation R to b' and recognizing that the positioning of 'a' and 'b' signifies that aRb. States of affairs can be described by propositions but cannot be directly named. While names are akin to points, propositions are like arrows, possessing direction and sense.
In propositions, thoughts are expressed such that the elements of the propositional sign correspond to the objects of the thoughts. These elements are termed "simple signs," and a fully dissected proposition is deemed "completely analysed." The simple signs in propositions are called names, where a name directly signifies its object, and the object is the meaning of the name.
In a proposition, the arrangement of simple signs mirrors the configuration of objects in the state of affairs. Names represent objects in propositions, but they can only be named and spoken of, not asserted; a proposition describes how things are, not what they are. The use of simple signs necessitates the determinateness of a proposition's sense.
Propositions about a complex entity are internally related to propositions about its constituent parts. A complex is defined by its description, making a proposition about it either correct or incorrect. The existence of a complex within a proposition is indicated by an element's indeterminateness; the proposition doesn't determine everything about the complex. The combination of symbols representing a complex can be outlined in a definition.
Each proposition has one unique, complete analysis. It articulates its content in a specific and identifiable manner. Names, as primitive signs, cannot be further broken down by definitions. Defined signs convey meaning through the signs that define them, showing the path of their meaning. Primitive signs and those defined by primitive signs cannot signify in the same manner. The unexpressed elements of a sign are revealed through its application, and the meanings of primitive signs can be clarified through elucidations, which are propositions containing these primitive signs. These elucidations can only be understood once the meanings of the primitive signs are known.
Sensibility in propositions allows names to have meaning only within their context. Parts of a proposition that characterize its sense are expressions or symbols, which include the proposition itself. These expressions are essential for the sense and characterize both a form and a content.
Expressions presuppose the forms of all propositions in which they can occur, acting as a common characteristic mark of a class of propositions. They are constant in their general form while other aspects vary. Expressions are represented by variables, whose values are the propositions containing the expression. In some cases, these variables become constants, making the expression a proposition.
A name has meaning only within a proposition, and every variable can be considered a propositional variable. Altering a part of a proposition into a variable creates a class of propositions, all values of the resultant propositional variable. The class of propositions is determined by the nature of the proposition itself, corresponding to a logical form or prototype.
The determination of values for a propositional variable is achieved by describing the propositions it characterizes, focusing on symbols rather than their meaning. This approach to description is essential to avoid ambiguity and misinterpretation.
Propositions are seen as functions of the expressions they contain. The perceptible part of a symbol is the sign, and different symbols can share a sign but signify in different ways. The method of symbolizing is arbitrary, and misunderstandings often arise in everyday language when the same word signifies in different ways.
To avoid such errors, a logical grammar or syntax is necessary, where the same sign is not used in different symbols and signs that signify differently are not applied in the same way. Recognizing a symbol requires understanding its significant use, and a sign determines a logical form only when used in a logical syntactic application.
A proposition has both essential and accidental features. The essential features enable it to express its sense, while accidental features are due to the particular method of producing the propositional sign. The essence of a symbol lies in what is common among all symbols that can serve the same function. Definitions serve as rules for translating between languages, and every correct symbolism should be translatable into another. What signifies in a symbol is what is common across all symbols that can replace it according to logical syntax rules. The sign of a complex is not arbitrarily resolved in analysis, but is consistent across different propositional structures.
A proposition defines a specific location within logical space. This location, or logical place, is affirmed solely by the existence of its components, which are the significant parts of the proposition. The propositional sign, along with its logical coordinates, constitutes this logical place.
Both geometrical and logical places share a common characteristic: they represent the potentiality of an existence. Despite a proposition delineating only a singular point in logical space, it implicitly necessitates the pre-existence of the entire logical space. Without this, logical operations such as denial, conjunction, and disjunction would continuously introduce new elements in coordination.
The logical framework surrounding the proposition establishes and defines the boundaries of the logical space. Therefore, a proposition extends throughout the entire logical space, influenced by and influencing the structure of this space.
The applied propositional sign, when thought, embodies the thought itself.
A thought, when articulated, becomes a significant proposition. The entirety of propositions constitutes language. Humans have the ability to construct languages capable of expressing any sense, akin to speaking without a conscious understanding of sound production. Colloquial language, complex and intertwined with human nature, obscures the logic of language. Language's external form disguises thought, making it challenging to deduce the thought's form from the language's appearance.
Many philosophical propositions and questions are not false but senseless, arising from a misunderstanding of language logic. All philosophy, in a sense, is a critique of language. A proposition is a model of reality as conceived in thought. It might not resemble a direct picture of reality, just as a musical score or phonetic spelling doesn't immediately appear as a representation of music or speech. However, these symbolisms do depict what they represent.
The proposition is a picture in the way it represents a state of affairs. It shows its sense and indicates how things stand if true. A proposition simplifies reality into a format where affirmation or negation aligns it with reality. It describes a fact and, like a description of an object, portrays internal properties of reality. Understanding a proposition means understanding the state of affairs it presents if true.
Translation between languages involves translating the constituent parts of propositions. The meanings of simple signs (words) need explaining for comprehension. Propositions can impart new senses using established words. They are logical pictures of states of affairs, assembled experimentally. The principle of representation underlies the possibility of propositions, where logical constants don't represent, but the logic of facts cannot be represented.
In a proposition, there must be as many distinguishable elements as in the state of affairs it represents, sharing the same logical multiplicity. This multiplicity cannot be represented outside itself. Reality is compared to the proposition, which can only be true or false by being a picture of reality. The concept of truth in propositions is not about sign-to-object relations, but about representing the state of affairs accurately. The proposition must already have a sense; assertion or denial does not impart sense but only affirms the sense already present. Every proposition determines a logical place, and denial indicates a different logical place than the proposition it denies.
A proposition represents the existence or non-existence of atomic facts. The sum of all true propositions constitutes the entirety of natural science. Philosophy, distinct from natural sciences, aims at the logical clarification of thoughts. It is an activity focused on elucidations, seeking to make propositions clear and distinct, rather than generating philosophical propositions.
Psychology, like other natural sciences, is not closely related to philosophy. Philosophy of psychology, or theory of knowledge, is more aligned with philosophy. Philosophical inquiries into language and thought, however, should avoid unnecessary psychological investigations. Similarly, theories like Darwinism, while significant in natural science, do not intersect directly with philosophical inquiry.
Philosophy's role is to delineate the limits of natural science and to clarify what can and cannot be thought. It seeks to illuminate the speakable, thereby indirectly indicating the unspeakable. Clear and distinct thinking is possible for all conceivable thoughts, and clear expression is achievable for all that can be said.
Propositions can represent the entirety of reality but cannot represent their own logical form, which is intrinsic to them and manifests in their structure. This logical form cannot be expressed through language but is shown in the way propositions are structured. The logical form is evident in the relations and structures within propositions, revealing internal properties and relations, which are not explicitly stated but are inherent in the propositions.
Formal properties and concepts in logic are not about the content but about the form. They cannot be explicitly stated in propositions but are implied in the structure and use of logical symbols. Formal concepts are represented by propositional variables in logical symbolism. The existence of formal concepts is not a matter of assertion but of logical structure.
Logical forms are beyond numerical categorization, negating any philosophical preference for monism, dualism, or other numerical metaphysics. In essence, the nature of logic and its representation in language is about form rather than content, about the structure of thought rather than its specific subject matter.
The sense of a proposition is determined by its alignment or misalignment with the possibilities of existence or non-existence of atomic facts. The most fundamental form of proposition, the elementary proposition, directly asserts the existence of an atomic fact. A key characteristic of an elementary proposition is that it cannot be contradicted by another elementary proposition.
Elementary propositions are composed of names, essentially being a connection or concatenation of these names. In the analysis of propositions, we inevitably reach elementary propositions, which are immediate combinations of names. Even in a hypothetical world of infinite complexity, there must still be distinguishable objects and atomic facts.
Names appear in propositions specifically within the context of elementary propositions. These names are simple symbols, represented by individual letters (x, y, z), and the elementary proposition is expressed as a function of these names, like "fx", "ϕ(x, y)", etc., or indicated by letters like p, q, r.
When two signs are used with the same meaning, this is expressed by the equality sign "=". For instance, "a = b" signifies that the sign "a" can be replaced by "b". This is a mere expedient in presentation and doesn't assert anything about the actual meaning of the signs "a" and "b".
Understanding whether two names signify the same or different things is crucial for understanding a proposition containing them. Knowing that two words in different languages are synonymous inherently means knowing they are interchangeable in translation.
The truth of an elementary proposition signifies the existence of the corresponding atomic fact, and its falsity signifies the non-existence of that atomic fact. The complete description of the world is achieved by specifying all elementary propositions and indicating which are true and which are false.
Regarding the existence of 'n' atomic facts, there are 'K_n' possibilities, representing all combinations of these atomic facts existing or not existing. Correspondingly, there are an equal number of possibilities for the truth or falsehood of 'n' elementary propositions. This framework underlines the logical structure of reality as represented by propositions and the foundational role of elementary propositions in this structure.
The truth-possibilities of elementary propositions correspond to the various combinations of the existence and non-existence of atomic facts. These possibilities can be illustrated using truth tables, where "T" represents "true" and "F" represents "false". Each row in the truth table signifies a distinct combination of truth-values for a set of elementary propositions.
For instance, considering three elementary propositions p, q, and r, a truth table can be constructed to display all their possible truth combinations:
p | q | r
T | T | T
F | T | T
T | F | T
T | T | F
F | F | T
F | T | F
T | F | F
F | F | F
In this table:
• Each column represents an elementary proposition. • Each row represents a possible scenario of truth-values for these propositions. • "T" in a cell indicates the corresponding atomic fact exists (the proposition is true). • "F" indicates the atomic fact does not exist (the proposition is false).
This schematic representation allows for a comprehensive understanding of all potential combinations of truth-values for a given set of elementary propositions, illuminating the logical structure underlying the propositions.
A proposition expresses agreement or disagreement with the truth-possibilities of elementary propositions. The truth-possibilities of these elementary propositions form the basis for the truth or falsehood of more complex propositions. Understanding general propositions depends fundamentally on understanding elementary ones.
Each proposition aligns with certain truth-possibilities of elementary propositions. This alignment can be represented symbolically, where "T" (true) indicates agreement with a truth-possibility, and the absence of "T" indicates disagreement. Thus, the proposition expresses its truth-conditions.
For example, a propositional sign can be formed by coordinating "T" with various truth-possibilities. However, signs like "F" and "T" don't correspond to any object or complex of objects; there are no "logical objects." The combination of these signs in a schema represents the proposition's truth-conditions. For instance, a schema showing different combinations of "T" and "F" for two elementary propositions 'p' and 'q' can be a propositional sign.
There are extreme cases in these groupings of truth-conditions: tautologies and contradictions. A tautology is unconditionally true, true under all truth-possibilities, while a contradiction is never true under any circumstance. They are, in a sense, senseless, as they don't present any possible state of affairs. Tautology allows every possible state of affairs, whereas contradiction allows none. They do not limit or define reality in any way.
The truth of a tautology is certain, that of a contradiction is impossible, and that of other propositions is possible. This concept plays a role in the theory of probability. The logical product of a tautology and a proposition is identical to the proposition, as the essence of the symbol cannot change without altering its sense.
Logical combinations of signs correspond to specific logical combinations of their meanings. Arbitrary combinations only correspond to unconnected signs. Tautology and contradiction, being the limits of symbol combinations, essentially represent the dissolution of these combinations. In tautology and contradiction, the relations between signs are present but are not essential to the symbol's meaning.
The general form of a proposition represents a framework within which any possible sense can be expressed, given that the meanings of the names are suitably defined. This form is essentially a description that captures only the essential characteristics of a proposition, ensuring its status as the most general form. The existence of such a general form is evidenced by the fact that it's impossible to conceive a proposition whose form couldn't have been anticipated or constructed. The general form of a proposition can be simply stated as "Such and such is the case."
If all elementary propositions are provided, the task then becomes to determine what propositions can be constructed from these elements. These constructed propositions, derived from the combination of all elementary propositions, encompass all possible propositions. In this sense, all propositions can be viewed as generalizations or extensions of elementary propositions.
The general propositional form functions as a variable. This means it represents a structure that can take various specific forms, depending on the particular values (or meanings) assigned to its constituent parts. This variable nature allows it to encompass a wide range of specific propositions, each differing in content but sharing the same underlying logical structure.
Propositions function as truth-functions of elementary propositions, where each elementary proposition is a truth-function of itself. In this context, the elementary propositions serve as the truth-arguments for more complex propositions. This means that the truth-value of a proposition is determined by the truth-values of the elementary propositions that constitute it.
There can be a natural inclination to confuse the arguments of functions with the indices of names. The argument in a function is crucial to understanding the function's meaning, much like how an index can help discern the meaning of a sign. For instance, in Russell's notation "+c", "c" acts as an index signifying that the entire sign represents addition within the realm of cardinal numbers. Such a notation is based on arbitrary agreement, and a simpler sign could be used instead.
In contrast, in the expression "∼p", "p" is not an index but an argument. The meaning of "∼p" (the negation of p) cannot be comprehended without first understanding the meaning of "p". This distinction is essential because, unlike an index that might serve as a part of a description, an argument directly contributes to the function's operation and meaning.
This confusion between argument and index, as suggested, lies at the core of Frege's theory regarding the meaning of propositions and functions. Frege treated the propositions of logic as names, with their arguments acting as indices to these names. This perspective differs from understanding propositions as truth-functions, where arguments actively determine the truth-value of the proposition.
Propositions are truth-functions of elementary propositions, and this concept forms the foundation of the theory of probability. Every proposition can be represented as a truth-function, where its truth depends on the truth-conditions of the elementary propositions it comprises.
The truth-functions of elementary propositions can be systematically catalogued in a schema, where "T" (True) and "F" (False) are used to denote the various truth-possibilities. This schema outlines the logical relationships between different propositions, highlighting the conditions under which each proposition is true.
For example, the proposition "p or q" (notated as "p ∨ q") is true when at least one of "p" or "q" is true. This and other logical relationships can be clearly visualized in a truth-table format.
The truth-grounds of a proposition are those truth-possibilities of its arguments that verify the proposition. The concept of following or inference in logic is based on the relationship between the truth-grounds of different propositions. If the truth-grounds of one proposition are contained within another, the former follows from the latter.
This logical structure underpins the theory of probability. The probability of a proposition given another is determined by the ratio of shared truth-grounds to the total truth-grounds of the given proposition.
In this framework, independent propositions (those without shared truth-arguments) afford a probability of 1/2 to each other. However, if one proposition follows from another, it is given a probability of 1 by the preceding proposition.
Probability is not an inherent property of a proposition but a measure of our knowledge or lack thereof regarding the proposition's truth-conditions. It is a generalization based on what is known about the form of a fact, rather than the fact itself. Probability propositions are thus general descriptions or extracts of other propositions, applied in situations where certainty is not attainable.
The internal relations between the structures of propositions can be elucidated through operations. An operation transforms one proposition into another, revealing a relation between their structures based on their formal properties and internal similarities.
• Operations express the transition from one form of proposition to another, highlighting the differences and similarities in their structures. • The operation itself doesn't characterize a specific form but the transformation between forms. • The result of an operation can serve as the basis for another operation, facilitating a progression in a formal series. This concept is key in logical hierarchies, such as those proposed by Russell and Whitehead. • Successive applications of an operation, where an operation is applied repeatedly to its own result, are also possible.
In a formal series, the general term can be represented as "[a, x, O′x]", where "a" is the initial term, "x" any arbitrary term in the series, and "O′x" the form of the term that immediately follows "x". This notation captures the concept of successive applications, akin to saying "and so on".
Operations can counteract each other or even nullify each other's effects. For instance, double negation (∼∼p) effectively cancels out, leaving the original proposition (p) unchanged. The interaction and cancellation of operations play a crucial role in the logical structure and analysis of propositions.
All propositions are formed as results of truth-operations on elementary propositions. A truth-operation is essentially the process by which a truth-function is derived from these elementary propositions. As truth-functions emerge from elementary propositions through truth-operations, new truth-functions can similarly be constructed from existing truth-functions.
• Every truth-operation applied to the results of previous truth-operations on elementary propositions creates another truth-function of elementary propositions, thus forming a new proposition. • The outcome of every truth-operation on the results of truth-operations on elementary propositions is equivalent to the result of a single truth-operation applied directly to elementary propositions. • Consequently, every proposition can be seen as the result of truth-operations performed on elementary propositions.
The significance of this process extends beyond scenarios where 'p', 'q', 'r', etc., are merely elementary propositions. Even when 'p' and 'q' are themselves truth-functions of elementary propositions, the propositional sign still represents a truth-function of these elementary propositions.
Truth-functions encompass all results obtained from applying a finite number of truth-operations to elementary propositions successively. This framework underlines the foundational role of elementary propositions and truth-operations in the construction and analysis of more complex propositions in logic.
The concept that all propositions result from truth-operations on elementary propositions is crucial in understanding the nature of logical statements. Truth-operations generate truth-functions from elementary propositions, and these operations can be applied successively to form complex propositions.
• Logical constants like ∨ (or), ⊃ (implies), and others are not relations in the traditional sense but are part of the logical structure that forms propositions. • The operation itself does not assert anything; it is the resultant proposition that holds meaning, depending on the operation's bases. • Operations can be repeated, reversed, or even nullified. For instance, double negation (∼∼p) effectively results in the original proposition (p). • The essence of logic lies in the form of the proposition, which is the general form of expression in logic. This form is the common feature of all propositions. • Logic must be self-sufficient; a sign in logic must be able to signify something. Logic allows everything that is possible within its structure, and logical mistakes are inherently prevented by the logical structure of language itself. • The principle of Occam's razor in logic implies that unnecessary elements in a symbolism are meaningless. Logical equivalence is determined by functionality, not by the presence of signs. • The complexity of logical operations and the number of necessary fundamental operations depend on the chosen notation system. Logic involves constructing a system of signs with a definite mathematical multiplicity.
In summary, logic is a self-contained system where propositions are formed through operations on elementary propositions. The nature of logical constants, the formation of complex propositions through operations, and the inherent prevention of logical errors are all fundamental aspects of this system. The essence of logic, and thus of all description and the world itself, lies in the general form of the proposition.
Every truth-function originates from the successive application of a specific operation, denoted as (— — — — —T)(ξ, ...), to elementary propositions. This operation negates all propositions within the right-hand bracket, and is termed the negation of these propositions.
• In logical notation, expressions in brackets with propositions as terms can be indicated by a variable, ξ, if the order of terms is irrelevant. The line over ξ signifies it represents all values within the brackets. • The method of describing the values of the variable ξ can vary, including direct enumeration, defining a function whose values for all x are the propositions to be described, or providing a formal law for constructing these propositions.
The operation N(ξ) represents the negation of all the values of the propositional variable ξ. This operation's effects can be explicitly defined, allowing for precise expression in logic.
The concept of logic mirrors the world through an infinitely fine network of truth-operations and truth-functions. Logical constants like ∨ (or) and ⊃ (implies) are not relations in the traditional sense but are part of this logical structure.
Truth-functions, including negation, do not assert anything themselves but are the outcomes of operations. They do not signify material functions; rather, they are the result of the application of logical operations.
The essence of logic lies in the elementary proposition, from which all logical operations derive. Logic's application must be consistent with its structure, and it cannot contradict its own application. Logic, therefore, must be self-sufficient and coherent within its own system.
In summary, the logic of propositions is based on the application of truth-operations to elementary propositions, creating a complex and interconnected system that reflects the logical structure of thought and language.
Wittgenstein's remarks delve into the intricate relationship between language, thought, and the limits of our world. He posits that the boundaries of language define the boundaries of our world, as we cannot conceive or articulate what lies beyond the scope of our language.
• Logic fills the world, and the limits of the world are also the limits of logic. We cannot speak of things existing or not existing in the world within the framework of logic, as this would imply stepping outside these limits. • The notion of solipsism, the idea that only one's own mind is certain to exist, aligns with these thoughts. Solipsism, correctly interpreted, manifests not in what is said but in what is shown: our understanding of the world is inherently bound to our language and perspective. • The world and life are inseparable, and each individual's world is a reflection of their own self – the microcosm. • The concept of the thinking or presenting subject does not have an objective existence in the world; it is rather a limit of the world. This is analogous to the way we cannot see our own eye within our field of vision. The subject, therefore, cannot be a part of the world but is a boundary of it. • This leads to the idea that there is no inherent, a priori order to the world; everything could be otherwise. What we perceive or describe could be different, negating a predetermined order of things. • Solipsism, when followed to its logical conclusion, aligns with pure realism. The 'I' in solipsism becomes a point without extension, leaving a reality that is correlated with it. • In philosophy, we can discuss a non-psychological 'I', which is not the individual human or the soul, but a metaphysical subject – a limit, not a component, of the world.
Wittgenstein's reflections here underscore a philosophy where the limits of our understanding and our world are shaped by the confines of our language and thought processes. This viewpoint bridges individual perception with a broader, more objective reality, grounding philosophical inquiry in the relationship between language, thought, and the fabric of our existence.
Every proposition results from applying the operation N'(ξ) to elementary propositions. Understanding a proposition's construction gives insight into how operations transform one proposition into another. The operation Ω'(η) transitions between propositions. Numbers are defined through a series of operations, exemplifying how they represent operation exponents. The concept of number encompasses all numbers, defining numerical equality. The cardinal number's form is [0, ξ, ξ + 1]. Class theory is unnecessary in mathematics due to the specific nature of mathematical generality.
Logic's propositions are tautologies, meaning they inherently state truths but don't convey new information. They highlight the formal-logical properties of language and the world, revealing inherent structures in propositions. Logical propositions are unique; their truth is evident from their structure, unlike non-logical propositions. The nature of logical propositions allows us to discern their truth without empirical evidence, solely through logical analysis. Logic, therefore, doesn't depend on worldly facts but reflects the underlying structure of the world and language. It's transcendental, not a theory per se, but a reflection of how the world is constructed.
Mathematics, as a logical method, uses equations that are pseudo-propositions expressing no thoughts. In practical life, mathematical propositions are not directly needed; instead, they serve to bridge non-mathematical propositions. Mathematics mirrors the world's logic through equations, much like logic uses tautologies. Equations signify the substitutability of expressions, revealing their logical form and equality in meaning. Mathematical correctness is evident within the propositions themselves, not through comparison with external facts. Mathematical equations don't require external intuition; the language and process of calculation provide necessary insight. The essence of mathematics lies in its self-evident nature and the method of substitution, leading from known equations to new ones. This process is exemplified in the proof of propositions like 2 × 2 = 4, where substitution according to definitional equations demonstrates the result.
Logical research explores all regularity, with anything beyond logic being accidental. The so-called law of induction is not a logical law as it's a significant proposition, and thus, not a priori. The law of causality is the form of a law, representing a class of laws like those in mechanics, such as the law of least action. Logical understanding doesn't a priori believe in laws like conservation but acknowledges the possibility of their logical form.
In science, propositions like causation or continuity are a priori intuitions of possible proposition forms. Mechanics, for instance, offers a unified form for describing the universe, akin to applying a network over a patterned surface. This doesn't assert anything about the pattern itself but about the way it's described. The simplicity of description in one mechanical system over another says something about the world.
Mechanics tries to construct all true propositions needed to describe the world in a unified plan. Through logical apparatus, physical laws speak about the world's objects, although these descriptions are general, not specifying particular entities.
Geometry, while able to describe the network (spatial relationships), says nothing about the actual form and position of the objects within it. Laws like causality concern the network's structure, not the objects it describes.
The process of induction, assuming the simplest law that harmonizes with experience, lacks a logical foundation, being purely psychological. There's no logical necessity that dictates one event must follow another, only logical possibilities and impossibilities. For instance, two colors can't occupy the same space in the visual field, which is a logical impossibility due to the structure of color perception.
The modern view mistakenly sees natural laws as explanations of phenomena, stopping at these laws as ultimate truths. However, these laws don't provide the ultimate explanation but are part of a logical structure used to describe observations. The world's workings are independent of human will, governed only by logical necessity and impossibility.
All propositions are of equal value in terms of their ability to convey facts. The meaning or value of the world must exist outside of it, as within the world everything is accidental and devoid of inherent value. Consequently, there can be no ethical propositions within the world, as ethics transcend worldly happenings.
Ethics, being transcendental, cannot be articulated through propositions. Ethical laws like "thou shalt..." don't pertain to consequences or events within the world. Ethical reward and punishment are inherent in the actions themselves, not in external consequences. Discussions about the will in an ethical context are not about its worldly manifestations, which are of psychological interest, but about its role as the bearer of ethics.
The impact of good or bad willing is not on worldly facts but on the world's limits—it alters the world's overall nature. The world experienced by a happy person differs from that of an unhappy person. In death, the world does not change but ceases for the individual. Eternal life, in the sense of timelessness, is living in the present, not endless temporal existence. The mystery of life and death lies outside space and time, not within them.
The way the world is structured is irrelevant to higher, transcendental aspects. Facts are part of the world's structure but not indicative of its ultimate nature. The mystical aspect of the world is not how it is structured or what happens within it, but the fact that it exists. Contemplating the world sub specie aeterni, as a limited whole, brings about the mystical feeling, acknowledging the world's existence as a finite entity.
Questions that cannot be answered don't truly exist as valid questions. This principle underlines the nature of skepticism; it's senseless to doubt where no question can be formulated. Doubt is only meaningful where there is a question and consequently an answer.
Even after answering all possible scientific questions, the fundamental problems of life remain untouched. The absence of questions in this context is itself the answer, suggesting that the solution to life's problem lies in realizing the problem itself is nonexistent. This could explain why individuals who find meaning in life often cannot articulate what that meaning is.
The inexpressible exists, manifesting as the mystical, which shows itself but cannot be articulated. The correct method in philosophy would be to restrict oneself to statements that can be made, like those in natural science, and to point out the lack of meaning in metaphysical statements. This approach may not satisfy those seeking traditional philosophical wisdom, but it aligns with a more rigorous, logical perspective.
The purpose of these propositions is to provide clarity. Once one understands and transcends them, they become senseless, like a ladder that is discarded after use. Surpassing these propositions allows one to see the world correctly, implying a transcendence beyond conventional understanding to a more profound comprehension of reality.
Whereof one cannot speak, thereof one must be silent.