CHARA'S DATABASE

Welcome to the Database.

Opening Notes

Epistemological analysis entailed searching for an identifiable base to anchor my pillars of intellect, while my introduction seeked comfort in sophist relativism, structure has undeniably proved its primacy and indispensability. Logic insists upon systematic progression within its sets, sequences in its algorithm, and syllogistic schemata as the guiding architecture through which truth is deduced, error is unveiled, and rational thought is disciplined into precision. Science both descriptivist and prescriptivist in synergy concluded me with a truth, the approximation of absolute truth, but never absolute reach. An absolute truth exists, but it is out of reach, at least to the finite mind in empirical dependency. Mind lacked definition, then lacked structure, thus it gave birth to an impossible existence, or more precisely a simulation of it, to satisfy the scope of imperfection within absolute perfection, an infinite decimal, a fraction to a whole, to an integer.

Logic

Classical logic

Classical logic serves as the foundation of philosophical and mathematical reasoning, its structural system entails the use of different components to study its reasoning. The three main parts of a Logical system contain language, deductive system, and model-theoretic semantics. "Classical logic is defined as a set of principles and methods of inference or valid reasoning (Mortari, 2016, p. 435)."

language and semantics

Logicians and mathematicians often operate through formal language, which in return is extracted from natural languages; in analysis of the two, natural language is said to have underlying logical forms and that these forms are displayed by formulas of a formal language. Grammar, prescriptivist in nature, follows a strict logical structure; however, reliance on natural language for an impeccable faultless reasoning invites for insecurities within its flow; this instability attributes to the misaccount for the relationship between thought, semantics and its underlying structure that grammar conceals, challenging notions on the cognitive nature of linguistics. Natural language raises significant complications in this endeavor, for instance, inferences are not with guaranteed reasoned security, e.g: "If it is raining, the streets will be wet." > "The streets are wet, so it must be raining." A logical fallacy is committed affirming a false consequence. Several syntactic errors occur within natural language such as Amphibology, False Equivocation, Composition Fallacy, Division Fallacy, False Dilemma (Forcing Binary Choice), Misleading Implication, Scope Ambiguity in Negation, etc.. When Translating it into formal language however, space for false inferences or ambiguity is eliminated as the form derives the idea to its purest essence. · R → W (If it rains, the streets are wet). · W does not necessarily mean R (There could be another cause).

To draw a comparison between the formal and natural, NL contains higher levels of syntactical ambiguity for its rich lexicology, while FL are low due to the symbols and abstract forms being precise and restricted to their meaning. Example «Chara is a star» here, context sensitivity and the multitude of meaning creates ambiguity, whether star means the celestial body, a principal performer of a film/play, or a conventional five pointed shape, it is unnecessary and ubiquitously hindering of the pragmatic flow of logic. Formal language proves superior for its radical, quintessential precision. Formal language in logic and mathematics uses specific notations to represent its structure. These notations define the syntax and semantics of the language. It is important to familiarize with the key notations used in formal languages:

  • 1. Alphabet (Σ)
  • A formal language consists of a finite set of symbols (letters, variables, operators, etc.).
  • Example:
  • Σ = {a, b, c, 0, 1, +, →, ∧, ∨} (Set of allowed symbols)
  • Natural numbers: Σ = {0, 1, 2, 3, ...}
  • Logical symbols: Σ = {¬, ∧, ∨, →, ∀, ∃}
    • Logical operators
  • ¬ (Negation) → "not"
  • ∧ (Conjunction) → "and"
  • ∨ (Disjunction) → "or"
  • → (Implication) → "if...then"
  • ↔ (Biconditional) → "if and only if"
  • Example: P → Q (If P, then Q) ----- (P ∨ Q) → R (If P or Q, then R)
    • Quantifiers (For Predicate Logic)
  • ∀x (Universal Quantifier) → "For all x..."
  • ∀x (P(x) → Q(x)) ("For all x, if P(x) then Q(x)")
  • x (Existential Quantifier) → "There exists an x..."
  • ∃x P(x) ("There exists an x such that P(x) is true")

Deductive SYSTEMS

A deductive system is the formal backbone through which logical consequence is operationalized. It contains a set of axioms (statements accepted as true without proof) and inference rules that dictate how new truths may be derived from existing ones. This system does not seek truth in the empirical or ontological sense; rather, it enforces internal consistency, truth becomes truth within a system, defined by what follows from its rules. In classical logic, a valid argument is one in which the conclusion necessarily follows from the premises; that is, if the premises are true, the conclusion cannot be false. This conditional necessity is captured through syntactic derivability. For instance, if we denote a set of premises as Γ and a conclusion as ϕ, the notation Γ ⊢ ϕ states that ϕ is derivable from Γ under the rules of the system. Common inference rules include:

    Modus Ponens (→ Elimination):

    If P → Q and P are both true, then Q must be true. P → Q P ∴ Q

    Modus Tollens:

    If P → Q and ¬Q are true, then ¬P follows. P → Q ¬Q ∴ ¬P

    Hypothetical Syllogism

    If P → Q and Q → R, then P → R. These rules are not arbitrarily chose, they are grounded in our fundamental intuitions about consistency, contradiction, and entailment. Yet the scope of their validity rests solely within the axiomatic framework in which they operate. Should one alter the axioms or the inference rules, an entirely new logical landscape emerges (non-classical logics, intuitionistic systems, paraconsistent frameworks). Thus, the deductive system forms a closed universe, a linguistic machine producing truth-values not because of their correspondence to the world, but because of their coherence within a system.

Metalogic

Truth conditions

Truth conditions are the circumstances under which a proposition is deemed true. They act as the bridge between language and the world, mapping expressions to the states of affairs that would make them true. In classical semantics, a sentence such as "Snow is white" is true if and only if snow is indeed white. This T-schema (Tarski's Convention T) reflects the core of correspondence theory. However, truth conditions are not always accessible, nor reducible to empirical states; they often exist as idealized metaphysical commitments. In formal logic, the truth value of a proposition is assigned relative to a model and an interpretation, meaning that truth is a function not of language alone, but of a system that interprets it. This demonstrates that truth, while often treated as binary, is conditional upon context and structure extending to possible worlds semantics, where a proposition's truth is evaluated across a multiplicity of hypothetical realities.

Every proposition bears a truth value, and its structure within formal logic relies on both syntactic arrangement and semantic interpretation. The correspondence theory of truth assumes that for a proposition to be true, it must correspond to a fact in the world. Yet, within formal systems, truth is not empirical, it is model-relative. A formula is said to be true within a model if the interpretation assigned to its variables and constants satisfies the formula under the system's axioms. Truth is a function assigning binary values (True/Flase) to each possible interpertation, a manifestation of such mechanism are expressed through truth tables governing the law of tautology.

Inference and Derivability

Inference within a deductive system follows a formal sequence: from premises to conclusion, through rules of inference such as modus ponens (If P → Q and P, then Q), modus tollens, hypothetical syllogism, disjunctive syllogism, etc. Validity is preserved through form, not content. A valid argument is one where the truth of the premises guarantees the truth of the conclusion. This moves logic from a descriptive to a prescriptive domain; it not only explains how we reason, but how we ought to. Formal systems construct these inferences through sequents (P1, P2, ..., Pn ⊢ C), meaning the conclusion C is derivable from the premises P1 to Pn.

Propositional Content

The propositional content of a statement is its informational core, what is being asserted independent of linguistic form. It abstracts from the manner of expression to the meaning that can be judged as true or false. For instance, "Chara loves Homura" and "Homura is loved by Chara" differ syntactically but share the same propositional content. This abstraction is essential in logic and philosophy of language, as it allows us to assess the logical form of arguments, identify entailments, and explore inferential relationships. Propositional content is central in intensional contexts, where substitution of co-referential terms might not preserve truth (e.g., belief reports). This points to the distinction between extension (what the term refers to) and intension (its conceptual content), both of which are critical in formal semantics.

Metalogic

Metalogic is the study of the properties of logical systems from a higher vantage point. While logic studies the validity of arguments within a system, metalogic examines the system itself, its consistency, completeness, soundness, and decidability. Godel's Incompleteness Theorems emerge here as monumental results: the first states that in any sufficiently expressive formal system, there exist true propositions that cannot be proven within the system (incompleteness); the second shows that such a system cannot prove its own consistency. This imposes a fundamental boundary on deductive reasoning, exposing the limits of formal systems and prompting philosophical reflection on the nature of mathematical truth. Metalogic also addresses syntactic vs semantic entailment, model theory, proof theory, and the relationship between syntax (form) and semantics (meaning). It is a meta-perspective that reveals both the power and the inherent constraints of logical inquiry.

The Finite Limit of Formal Systems

The pursuit of totalizing knowledge within formal systems encountered its most sobering revelation with the advent of Kurt Godel's incompleteness theorems. A formal system powerful enough to express arithmetic, when internally consistent, is inherently incomplete. That is, there exist propositions expressible within the system that are neither provable nor disprovable using the system's own axioms and inference rules. This shatters the naive vision of logic as an omnipotent architecture capable of encapsulating all mathematical truths. The first incompleteness theorem asserts that no such system can be both complete and consistent. The second theorem affirms that the system itself cannot demonstrate its own consistency without stepping outside its formal bounds. In essence, truth transcends formal proof; there exist truths the system can name but cannot grasp. This forms an epistemological boundary; there is always an exterior, an unreachable vantage from which consistency could be confirmed, but that vantage defies formalization within. That is Metalogic.

The Continuum Hypothesis and Unprovability

The Continuum Hypothesis (CH), posited by Cantor, concerns the possible sizes of infinite sets, particularly whether any set exists whose cardinality is strictly between that of the integers (ℕ) and the real numbers (ℝ). Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), the dominant foundation of modern mathematics, cannot prove nor disprove CH. What emerges from Godel and Cohen's work is that CH is independent of ZFC: both CH and ¬CH are logically compatible with the axioms. Therefore, CH becomes a candidate for axiomatic adoption or rejection, similar to Euclid's Fifth Postulate in geometry. In this sense, unprovable propositions inhabit an ambiguous ontological status: neither truths nor falsehoods, but formal suggestions awaiting elevation to axiomhood through philosophical or utilitarian justification. Analogous to the parallel postulate in geometry, if the 5th axiom doesn't hold and there are no parallel lines then you are playing in the universe of spherical geometry, if you choose one parallel line you switch to the realm of flat geometry, and if you choose more than 1 parallel line, then hyperbolic geometry it is. All of these geometries are correct, accepting the axiom remains contingent to what you desire to engineer. The axiom of choice, thus, functions the same in the analogical sense.

Axiom

An axiom is not a truth imposed by necessity, but rather a starting point chosen for the structure it enables. Within formal logic, axioms are the irreducible seeds of reason, the premises we do not prove, but from which all proof arises. Their power lies not in their truth alone, but in their generative capacity. Axioms are the foundational gestures of faith within reason, the assumptions that define the boundaries of the system. Thus, to call a proposition "axiomatic" is not to declare it self-evident, but to declare it indispensable for the system we desire to construct.

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