Update definitions Valid inference

content/list-of-definitions/valid-inference/index.md

@@ -0,0 +1,41 @@
+---
+title: Valid inference
+subtitle: test
+author: Annefleur de Haan
+weight: 2
+params: 
+  id: txt-val
+  math: true
+---
+
+| Concept                    | Definition |
+|---------------------------|------------|
+| Affirming the consequent  | A formal fallacy where one infers the antecedent from the consequent of a conditional: from A → B and B, one erroneously concludes A. This is invalid because B might result from causes other than A. |
+| Axiomatic method          | A method of reasoning in formal systems that begins with a set of axioms (assumed truths) and derives theorems using specified rules of inference. This method underpins mathematical logic and formal sciences. |
+| Belief modulation         | The process of updating degrees of belief in response to new evidence, typically formalized using Bayesian updating. It captures the dynamics of rational inference in uncertain or evolving informational contexts. |
+| Conditional premise       | A proposition of the form "If A, then B" (A → B), where A is called the antecedent and B the consequent. It forms the basis for many logical inferences, such as modus ponens and modus tollens. |
+| Conditional probabilities | The probability that a statement B is true given that another statement A is true, formally written as $Pr(B\|A)$. Central to Bayesian reasoning, it quantifies dependence between propositions. |
+| Correctness               | A general property of inferences where conclusions are appropriate or justified given the premises. In logic, this often reduces to formal validity (deductive) or strength (inductive). |
+| Deduction                 | A form of reasoning in which the conclusion follows necessarily from the premises. If the premises are true and the reasoning is valid, the conclusion cannot be false. It contrasts with inductive and abductive reasoning. |
+| Deductively valid inference | An inference is deductively valid if it preserves truth under all interpretations: if the premises are true, the conclusion must be true. Validity depends only on logical form, not content. |
+| Formalization             | The process of translating informal, natural language arguments into a precise formal language (e.g. propositional or predicate logic) to make logical structure and validity explicit. |
+| Induction                 | A mode of reasoning that infers general rules or future cases from specific observations. Conclusions are not guaranteed, but probabilistically supported; hence, inductive inferences are ampliative but fallible. |
+| Inductively valid inference | A probabilistically strong inference where the truth of the premises significantly raises the probability of the conclusion, though it does not entail it. Inductive validity is gradient, not binary. |
+| Invalid inference         | An inference where, even if the premises are true, the conclusion may still be false. This indicates a failure in logical form or a misuse of inference rules. |
+| Knowledge representation  | The encoding of information (objects, facts, rules, relations) into a formal structure (often using logic or graph models) that supports automated reasoning in AI and cognitive systems. |
+| Logical conjunction       | A compound proposition of the form A ∧ B, which is true only when both A and B are true. It represents the intersection of truth conditions and is one of the basic connectives in propositional logic. |
+| Logical form              | The underlying structure of an argument abstracted from its content. Logical form reveals whether an inference is valid based solely on its syntax and connectives, irrespective of subject matter. |
+| Modus ponens              | A valid deductive argument form: from A → B and A, one infers B. This form preserves truth and is foundational to deductive systems. |
+| Modus tollens             | A valid deductive form: from A → B and ¬B (not B), one infers ¬A. It is often used to refute hypotheses by demonstrating that their consequences are false. |
+| Monotonicity              | A property of deductive logic wherein adding new premises to an argument cannot invalidate an already valid inference. Once a conclusion is validly drawn, it remains valid under additional premises. |
+| Non-monotonicity          | A feature of many inductive or default reasoning systems in which new information can retract previously justified conclusions. This models real-world reasoning more realistically than monotonic logic. |
+| Probabilities             | Quantitative measures (between 0 and 1) of how likely a proposition is to be true, given a defined probability space. Probabilistic reasoning formalizes uncertainty and is core to statistics, AI, and epistemology. |
+| Quantifier                | Logical operators that express generality or existence over a domain. The universal quantifier (∀) means "for all," while the existential quantifier (∃) means "there exists." Used in predicate logic to formulate general statements. |
+| Rhetoric                  | The art of persuasive communication, concerned with style, emotional appeal, and audience effects. Unlike logic, it does not require arguments to be valid or sound, only effective. |
+| Semantic model            | A formal interpretation of a logical language that assigns meanings (e.g., truth values) to expressions. It provides a way to evaluate whether statements are true within specific "possible worlds" or domains. |
+| Semantics                 | The study of meaning in logic and language, especially how truth conditions are assigned via models. It contrasts with syntax, which studies formal structure without regard to meaning. |
+| Strong inductive inference | An inductive inference where the conclusion's probability is significantly higher given the premises than it was a priori. Such inferences justify belief revision and are central to scientific reasoning. |
+| Subset                    | A set A is a subset of B (A ⊆ B) if every element of A is also in B. In logic and set theory, this relation is fundamental to hierarchical and inclusion structures. |
+| Truth-preservation        | A feature of valid deductive inferences: if the premises are true, the conclusion must also be true. This ensures that reasoning does not introduce falsehood. |
+| Valid inference           | An inference in which the conclusion follows necessarily from the premises. Validity is defined syntactically (by rules of inference) or semantically (truth-preservation across all models). |
+| Weak inductive inference  | An inference where the conclusion becomes only marginally more probable given the premises. Such inferences are low in inductive strength and often fail to justify confident belief. |

content/list-of-definitions/valid-inference/info.md

@@ -0,0 +1 @@
+Here you can find a list of important definitions. Note: these definitions are not sufficient to understand the concept in full, you should read the textbook, practice and attent the lectures and tutorials to gain real understanding. This list is meant to refresh some basics.