diff --git a/content/list-of-definitions/valid-inference/index.md b/content/list-of-definitions/valid-inference/index.md new file mode 100644 index 00000000..67a5d4a8 --- /dev/null +++ b/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. | diff --git a/content/list-of-definitions/valid-inference/info.md b/content/list-of-definitions/valid-inference/info.md new file mode 100644 index 00000000..51c1da76 --- /dev/null +++ b/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. \ No newline at end of file diff --git a/content/textbook/logic-and-ai/index.md b/content/textbook/logic-and-ai/index.md index abeed108..50aa0061 100644 --- a/content/textbook/logic-and-ai/index.md +++ b/content/textbook/logic-and-ai/index.md @@ -7,7 +7,7 @@ resources: name: inferences params: date: 31/08/2024 - last_edited: 09/05/2025 + last_edited: 01/09/2024 id: txt-laa math: true --- @@ -32,7 +32,7 @@ Then, you'll learn about three ways in which logic is relevant for AI: On the most general level, **logic** is the study of _valid inference_. To understand this definition better, let's discuss inferences and validity in -turn. +turn. ### Inference @@ -44,7 +44,7 @@ _Philosopher's Walk_ or in the study, and she's not in the study, she therefore must be on the _Philosopher's Walk_. 2) If [Alan](https://en.wikipedia.org/wiki/Alan_Turing) can’t crack the code, -then nobody else can. Alan can crack the code. So nobody else can. +then nobody else can. Alan can crack the code. So nobody else can. [**AdH_comment**: I understand that this example is used to illustrate an invalid argument. However, it might be better to introduce this chapter with examples of valid reasoning first. Later on, the original example could be modified into an invalid argument in order to clarify the difference in meaning between valid and invalid arguments. It feels unnatural to start the second example with a falsity. Another option is to introduce this set of examples with a warning that no all reasonings are _right_. Instead of this example, I would use: "If Alan can't crack the code, then nobody else can. Alan can't crack the code. So nobody else can." or: "If Alan can crack the code, then nobody else can. Alan can crack the code. So nobody else can."] 3) [Blondie24](https://en.wikipedia.org/wiki/Blondie24) is a neural network-based AI system that struggled to reach world-class checkers @@ -53,7 +53,13 @@ Blue](https://en.wikipedia.org/wiki/Deep_Blue_(chess_computer)), instead, is a logic-based AI system that beat the [world champion](https://en.wikipedia.org/wiki/Garry_Kasparov) in chess. This shows that logic-based AI systems are inherently better than neural network-based -systems at games. +systems at games. [**AdH_comment:** Maybe this is an example of a hasty generalization, as the argument draws a general conclusion about AI system types based on two unrelated cases: one concerning checkers (Blondie24) and the other chess (Deep Blue). The two premises focus on different games and contexts, and therefore do not provide a logical basis for the conclusion that logic-based AI systems are inherently better than neural network-based systems at games. A different or more consistent set of premises and conclusion would better support the theory, for example: "[Blondie24](https://en.wikipedia.org/wiki/Blondie24) is a neural +network-based AI system that struggled to reach world-class checkers +performance. [Deep +Blue](https://en.wikipedia.org/wiki/Deep_Blue_(chess_computer)), instead, is a +logic-based AI system that beat the [world +champion](https://en.wikipedia.org/wiki/Garry_Kasparov) in chess. This shows +that logic-based AI systems are inherently better at winning a chess game than neural network-based systems are at checkers."]. 4) Since [Watson](https://en.wikipedia.org/wiki/IBM_Watson) is a logic-based AI system that beat [_Jeopardy!_](https://en.wikipedia.org/wiki/Jeopardy!), we can @@ -66,18 +72,58 @@ conclude that some logic-based AI systems are capable of beating game shows. relevance. Therefore, the next generation of GPT models will further improve in this respect. +[**AdH_comment:** The original structure of this section is: inference indicators related to the conclusion, an explanation of the conclusion, an explanation of the premises, premise indicators and general explanation of inferences. However, this order seems a bit disorganized to me. A clearer structure of this part would be: explanation of inferences, explanation of the conclusions and premises, then the indicators (both inference and premise), because they structure the argument, but they aren't the main part of an argument. Below is an example version of the revised text.] + +[**AdH_addtext:** + +**Caveat**: Our topic is _not_ how people actually reason (psychology of +reasoning), how to use arguments to convince others (rhetoric), or anything of +that sort. These things are good to know, of course, but they are not our main +interest. As logicians, we are interested in the structure of inferences. + +**Inferences** are the primary subject of logical theory. Note that in logic, +"inference" is a **technical term**, which does not necessarily have its +everyday meaning. An inference is a linguistic entity, consisting of premises +and a conclusion---_and nothing else_. It is not, for example, the psychological +process of drawing a conclusion from premises. + +Before we go ahead and look more closely at the quality of these inferences, +let's introduce some important terminology. + +These inferences, arguments, consist of a **conclusion** that is supported by the **premises** and indicators to dituiguish the conclusions and premises. + +The **conclusion**[^conclusion] of an inference is what's being inferred or +established based on the preceding arguments. It's the statement that logically follows from a set of sentences that support the conclusion. In 4., for example, the conclusion is that some logic-based AI systems are capable of beating game shows. The examples above all end with the conclusion. However, the conclusion can occur in any position. For example: + +    1*. Ada is not in the study. Thus she must be on the _Philosopher's Walk_, because she is either there or in the study. + +The **premises** of an inference are its assumptions or hypotheses, they are what the conclusion is based on. These statements provide support for the conclusion. They serve as the basis or evidence from whcih the conclusion is logically derived. Thus, the conclusion is drawn from the premises. In 4., for example, the premise is that Watson is a logic-based AI system that beat _Jeopardy!_ + +An inference can have any number of premises. While inference 4. has only one premise, inference 1. has _two_: that Ada is either on the _Philosopher's Walk_ or in the study, and that she's not in the study. In logical theory, we also consider the limit cases of having _no_ premises and of having _infinitely many_ premises. More about that later. + +Besides, **indicators** are used to structure the argument. They are used to clarify the structure of an argument by distinguishing conclusions from the supporting statements, i.e., the premises. + +We call phrases like "therefore", "so", and "we can conclude that" **inference indicators**, because they introduce the conclusion. In 5., the inference indicator "therefore" introduces the conclusion _the next generation of GPT models will further improve in this respect_. + +**Premise indicators** signalate that the following sentence is a premise. Examples of premise indicators are "since", "because" and "given that". These phrases show the parts of the inference that provide the support of the conclusion. + + +Logical theory is interested in whether an conclusion is supported by the premises. Inferences come with the expectation that the conclusion does, in fact, _follow from_ the premises, that the premises _support_ the conclusion in this way. In logical terminology, we want our inferences to be **valid**.[^valid] We'll turn to what that means in the next section. + +**Original text below:**] + Before we go ahead and look more closely at the quality of these inferences, let's introduce some important terminology. We call phrases like "therefore", "so", and "we can conclude that" **inference -indicators**. +indicators**. The **conclusion**[^conclusion] of an inference is what's being inferred or established. In 4., for example, the conclusion is that some logic-based AI systems are capable of beating game shows. The conclusion often follows the inference indicator, but it can also be the -other way around: +other way around. [**AdH_comment:** I don't think this is correct, regarding the example. The function of the indicator "since" in this example is to indicate the premise _Watson is a logic-based AI system that beat Jeopardy!_, instead of the conclusion as mentioned here. The sentence is a reason to accept the conclusion _We know some logic-based AI systems are capable of beating game shows_. I think the explanation (The conclusion ... way around) is meant to explain that the conclusion not always closes the inference. However, this is a bit unclear. See "AdH_addtext" for a changed explanation.] 6) We know that some logic-based AI systems are capable of beating game shows, since [Watson](https://en.wikipedia.org/wiki/IBM_Watson) is a logic-based AI @@ -99,7 +145,7 @@ Inferences are the primary subject of logical theory. Note that in logic, "inference" is a **technical term**, which does not necessarily have its everyday meaning. An inference is a linguistic entity, consisting of premises and a conclusion---_and nothing else_. It is not, for example, the psychological -process of drawing a conclusion from premises. +process of drawing a conclusion from premises. [**AdH_comment**: I think it is better to start the section with this paragraph, beacuse it explains why it is important to understand this section.] Inferences come with the expectation that the conclusion does, in fact, _follow from_ the premises, that the premises _support_ the conclusion in this way. In @@ -112,6 +158,7 @@ that sort. These things are good to know, of course, but they are not our main interest. ### Validity +[**AdH_addtext:** In the previous section, we have looked at the basic concepts of an inference, i.e., arguments as a collection of premises, by which the conclusion is derived. This suggets that Validity is one of the key concepts within logic. Finish this ] Consider inference 1) again: @@ -213,6 +260,12 @@ conclusions, + and a **proof theory**, which is a model of _stepwise valid inference_. +[**AdH_comment**: Maybe it is an idea to introduce the Chinese room experiment of Searle here. The Chines room makes clear what the difference between syntx and semantics is. For example: + +To understand the difference between syntax and semantic better, we look at the [Chinese Room](https://en.wikipedia.org/wiki/Chinese_room) argument of the American philosopher John Searle. According to Searle, computers can't think properly, because a computer only has syntax, instead of semantics too. Suppose a closed room. In the room, there is a person (or a computer) surrounded by multiple books. These books consist of translation rules. The person in the room recieves cards with Chinese questions. However, the person doesn't master Chinese. With the books the person is capable to manipualte the signs on the cards to answer the questions in Chinese, without understanding the meaning of either the questions and the answers. The person outside of the room, receives the manipulated answers in perfect Chinese. This person would think the person inside the room is a native Chinese speaker. This is the same case as with a computer. The computer's syntax, manipulation of symbols by applying rules, is like what happens inside the room. The semantics is the understanding of the symbols' meaning, by the one outside the room. According to Searle, the person inside the room or the computer is not capable of semantics, meaning, but only of syntax. + +] + Together, these three components provide a mathematical model of valid inference. Throughout the course, you'll learn more about syntax, semantics, and proof theory by studying how they are used in different AI applications. By the @@ -312,6 +365,8 @@ Now you have a first idea of what the science of logic is all about.[^logic] But before we can talk about the role of logic in AI, we need a working definition of AI. +{{< definition term="AI" id="ai-160620250933" >}}{{< /definition >}} + In this course, we take **AI** to be the study of the models and replication of _intelligent behavior_. Here we're understand "intelligent behavior" in a rather inclusive way, counting such diverse activities as behavior of [switching @@ -325,8 +380,7 @@ in this way. We'll go through them in turn. ### As a foundation -The first way in which logic is relevant to AI is the most direct one: valid -inference simply _is_ paradigmatic intelligent behavior. So, logical systems +The first way in which logic is relevant to AI is the most direct one: valid inference simply _is_ paradigmatic intelligent behavior. So, logical systems directly target what we're trying to model in AI---logical systems are models of intelligent behavior. So, by our definition, logical systems are part of AI. This makes logic a **subdiscipline** of AI. @@ -419,6 +473,8 @@ The expert information typically takes the form of **if-then rules**. The expert information in the KB of an expert system for medical diagnosis, for example, could include the following: +[**AdH_comment:** maybe the difference between antecedent and consequent could be mentioned shortly.] + + _If_ the patient has a runny nose, a sore throat, and a mild fever, _then_ the patient likely has the common cold. @@ -628,6 +684,7 @@ World. Penguin](https://en.wikipedia.org/wiki/The_Master_Algorithm). In general, I recommend to use the internet to keep up to date on logic and AI developments. Read, Learn, Improve! + **Notes:** [^inference]: Another common term for the same concept is "argument". diff --git a/data/definitions.yaml b/data/definitions.yaml new file mode 100644 index 00000000..3f6e6b64 --- /dev/null +++ b/data/definitions.yaml @@ -0,0 +1,442 @@ +main: + - term: AI + long: | + "The study of the models and replication of _intelligent behavior_ in machines or systems, the field focused on creating machines or systems that exhibit behaviors deemed intelligent by human standards." + short: | + "the study of the models and replication of _intelligent behavior_ in machines or systems" + chapter: logic-and-ai + id: ai-160620250933 + + - term: Affirming-the-consequent + long: | + "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." + short: | + "A formal fallacy where one infers the antecedent from the consequent of a conditional" + chapter: valid-inference + id: affriming-the-consequent-010720251418 + + - term: Axiomaic-method + long: | + "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." + short: | + "A method of reasoning in formal systems that begins with a set of axioms (assumed truths) and derives theorems using specified rules of inference." + chapter: valid-inference + id: axiomatic-method-010720251419 + + - term: Belief-modulation + long: | + "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." + short: | + "The process of updating degrees of belief in response to new evidence." + chapter: valid-inference + id: belief-modulation-010720251424 + + - term: Classical-logic + long: | + "The standard formal system of logic that assumes every proposition is either true or false (principle of bivalence) and that contradictions cannot be true (principle of non-contradiction). It forms the basis of much of modern mathematics and deductive reasoning." + short: | + "A standard formal system based on bivalent truth values and principles of non-contradiction." + chapter: logic-and-ai + id: classical-logic-160620251026 + + - term: Conditional-premise + long: | + "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." + short: | + "A proposition of the form “If A, then B” (A → B)" + chapter: valid-inference + id: conditional-premise-010720251425 + + - term: Conditional-probabilities + long: | + "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." + short: | + "The probability that a statement B is true given that another statement A is true" + chapter: valid-inference + id: conditional-probabilities-010720251427 + + - term: Conclusion + long: | + "A proposition that follows logically from premises within an argument. The conclusion of an inference is what's being inferred or established. It is the statement whose truth is supported or entailed by the given propositions in an argument." + short: | + "The proposition that logically follows from one or more premises." + chapter: logic-and-ai + id: conclusion-160620251028 + + - term: Correctness + long: | + "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)." + short: | + "A general property of inferences where conclusions are appropriate or justified given the premises." + chapter: valid-inference + id: correctness-010720251428 + + - term: Deduction + long: | + "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." + short: | + "A form of reasoning in which the conclusion follows necessarily from the premises." + chapter: valid-inference + id: deduction-010720251429 + + - term: Deductively-valid-inference + long: | + "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." + short: | + "An inference is deductively valid if it preserves truth under all interpretations" + chapter: valid-inference + id: deductively-valid-inference-010720251430 + + + - term: Formal-derivations + long: | + "Model of stepwise valid inference based on logical rules. It is a finite sequence of formulas in a formal system where each formula is either an axiom or follows from earlier formulas by rules of inference. It demonstrates, step by step, how a conclusion can be logically deduced from a set of premises using only syntactic rules, independent of any interpretation or meaning." + short: | + "Model of stepwise valid inference based on logical rules." + chapter: logic-and-ai + id: formal-derivations-160620251030 + + - term: Formalization + long: | + "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." + short: | + "The process of translating informal, natural language arguments into a precise formal language" + chapter: valid-inference + id: formalization-010720251431 + + - term: Formal-language + long: | + "A set of strings constructed from a finite alphabet according to specific syntactic rules (grammar). It is abstract and independent of meaning, designed to capture the form of expressions rather than their content." + short: | + "A set of strings constructed from an alphabet according to specific syntactic rules." + chapter: logic-and-ai + id: formal-languages-160620251032 + + - term: Formal models + long: | + "Structure that interprets the symbols and formulas of a formal language in a way that allows one to evaluate their truth or falsity." + short: | + "Structure that interprets the symbols and formulas of a formal language in a way that allows one to evaluate their truth or falsity." + chapter: logic-and-ai + id: formal-models-160620251035 + + + - term: (Frist) incompleteness theorem + long: | + "Gödel’s result showing that any sufficiently expressive, consistent formal system cannot prove all true statements expressible within its own language. Every logical system that is free of internal contradictions and models basic mathematical reasoning, has a mathematical statement in it that is undecidable in the system." + short: | + "the fundamental limitation of formal systems that any consistent formal theory of arithmetic is incomplete." + chapter: logic-and-ai + id: first-incompleteness-theorem-160620251037 + + + - term: If-then rules + long: | + "Symbolic representations of conditional statements used in rule-based systems: if condition A, then consequence B. The words _if_ and _then_ connect the antecedent to the consequent." + short: | + "A logical or computational construct that expresses a dependency between the antecedent and the consequent." + chapter: logic-and-ai + id: if-then-rules-160620251040 + + + - term: Indicators + long: | + "Logical cues that signal the role of statements, i.e., the premises or the inference." + short: | + "Logical cues that signal the role of statements, i.e., the premises or the inference." + chapter: logic-and-ai + id: indicators-160620251041 + + - term: Induction + long: | + "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." + short: | + "A mode of reasoning that infers general rules or future cases from specific observations." + chapter: valid-inference + id: induction-010720251431 + + - term: Inductively-valid-inference + long: | + "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." + short: | + "A probabilistically strong inference where the truth of the premises significantly raises the probability of the conclusion." + chapter: valid-inference + id: inductively-valid-inference-010720251432 + + - term: Inference + long: | + "The process of deriving new propositions (conclusions) from a set of given propositions (premises) according to rules of logic. It formalizes how knowledge propagates under constraints of logical validity. Types of inference: deductive inference and inductive inference." + short: | + "The process of deriving logical consequences from premises via valid rules." + chapter: logic-and-ai + id: inference-160620251042 + + + - term: Intuitionistic Logic + long: | + "Assumes that truth needs to be constructed. A form of logic rejecting the law of the excluded middle, emphasizing constructivist proof over truth-as-correspondence." + short: | + "A refinement of classical logic that a proposition is true only if we can constructively prove it." + chapter: logic-and-ai + id: intuitionistic-logic-160620251044 + + + - term: Invalid-inference + long: | + "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." + short: | + "An inference where, even if the premises are true, the conclusion may still be false." + chapter: valid-inference + id: invalid-inference-010720251433 + + - term: Knowledge representation + long: | + "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." + short: | + "The encoding of information into a formal structure that supports automated reasoning in AI and cognitive systems." + chapter: valid-inference + id: knowledge-representation-010720251434 + + - term: Logical-conjunction + long: | + "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." + short: | + "A compound proposition of the form A ∧ B." + chapter: valid-inference + id: logical-conjunction-010720251435 + + - term: Logical-form + long: | + "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." + short: | + "The underlying structure of an argument abstracted from its content." + chapter: valid-inference + id: logical-form-010720251436 + + - term: Logical formulas + long: | + "A well-formed expression constructed from symbols of a formal language of logic that represents a proposition or a relationship between propositions. Logical formulas are the primary vehicles for expressing statements, arguments, and proofs in formal logic." + short: | + "Well-formed expressions in a formal language, typically constructed from variables, connectives, and quantifiers." + chapter: logic-and-ai + id: logical-formulas-160620251045 + + + - term: Logical laws + long: | + "Universally valid principles or tautologies within a given logical system. They describe invariant relationships between logical formulas and form the basis of formal reasoning. In classical logic, these laws hold independently of the content of the formulas—they are purely a function of logical form." + short: | + "General principles governing valid inference and logical equivalences." + chapter: logic-and-ai + id: logical-laws-160620251046 + + - term: Logic + long: | + "The formal study of valid reasoning. It investigates the principles that distinguish correct from incorrect inferences, abstracting away from specific content to focus on structure. In both mathematics and philosophy, logic provides the foundational framework for evaluating arguments, constructing proofs, and modeling rational thought." + short: | + "The systematic study of the principles of valid inference and correct reasoning." + chapter: logic-and-ai + id: logic-160620251048 + + - term: Machine learning + long: | + "Subfield of artificial intelligence (AI) focused on developing algorithms that allow computers to learn patterns from data and make predictions or decisions without being explicitly programmed for each specific task." + short: | + "Subfield of AI where systems learn patterns from data rather than being explicitly programmed with symbolic rules." + chapter: logic-and-ai + id: machine-learning-160620251048 + + - term: Metalogic + long: | + "The study of the properties of formal logical systems themselves, rather than the content of those systems. While logic investigates what follows from what within a system (e.g., deriving conclusions from premises), metalogic steps outside the system to examine its structure, capabilities, and limitations." + short: | + "The study of properties of logical systems themselves, such as completeness, soundness, and decidability." + chapter: logic-and-ai + id: metalogic-160620251050 + + - term: Modus-ponens + long: | + "A valid deductive argument form: from A → B and A, one infers B. This form preserves truth and is foundational to deductive systems." + short: | + "A valid deductive argument form: from A → B and A, one infers B." + chapter: valid-inference + id: modus-ponens-010720251436 + + - term: Modus-tollens + long: | + "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." + short: | + "A valid deductive form: from A → B and ¬B (not B), one infers ¬A." + chapter: valid-inference + id: modus-tollens-010720251437 + + - term: Monotonicity + long: | + "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." + short: | + "A property of deductive logic wherein adding new premises to an argument cannot invalidate an already valid inference." + chapter: valid-inference + id: monotonicity-010720251438 + + - term: Non-monotonicity + long: | + "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." + short: | + "A feature of many inductive or default reasoning systems in which new information can retract previously justified conclusions." + chapter: valid-inference + id: non-monotonicity-010720251439 + + + - term: Premise + long: | + "A statement assumed to be true within an argument from which a conclusion is derived. The conclusion is based on the premises of the inferences." + short: | + "a propositional assumption or starting point in a logical argument or formal inference." + chapter: logic-and-ai + id: premise-160620251051 + + - term: Probabilities + long: | + "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." + short: | + "Quantitative measures of how likely a proposition is to be true." + chapter: valid-inference + id: probabilities-010720251440 + + - term: Proof theory + long: | + "A model of stepwise valid inference. The study of formal proofs as mathematical objects, often used to analyze the structure and derivability in logical systems." + short: | + "a branch of mathematical logic that studies the formal structure of proofs." + chapter: logic-and-ai + id: proof-theory-160620251052 + + - term: Quantifier + long: | + "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." + short: | + "Logical operators that express generality or existence over a domain." + chapter: valid-inference + id: quantifier-010720251441 + + - term: Rhetoric + long: | + "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." + short: | + "The art of persuasive communication, concerned with style." + chapter: valid-inference + id: rhetoric-010720251442 + + - term: Semantics + long: | + "A model of the meaning of the premises and conclusions. The branch of logic dealing with meaning and truth in formal languages, often via interpretations or models." + short: | + "The branch of logic that studies the meaning of expressions in a formal language." + chapter: logic-and-ai + id: semantics-160620251053 + + - term: Semantic-model + long: | + "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." + short: | + "A formal interpretation of a logical language that assigns meanings" + chapter: valid-inference + id: semantic-model-010720251443 + + - term: Strong-inductive-inference + long: | + "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." + short: | + "An inductive inference where the conclusion's probability is significantly higher given the premises than it was a priori." + chapter: valid-inference + id: strong-inductive-inference-010720251444 + + - term: Subset + long: | + "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." + short: | + "A set A is a subset of B (A ⊆ B) if every element of A is also in B." + chapter: valid-inference + id: subset-010720251445 + + - term: Subsymbolic AI + long: | + "AI approach that operate on distributed representations rather than explicit symbols. Subsymbolic AI prefers conditional probabilities and inductive inference over if-then rules and deductive inference." + short: | + "Approaches in AI that represent and process information without using explicit symbols or structured logic." + chapter: logic-and-ai + id: subsymbolic-ai-160620251054 + + - term: Symbolic AI + long: | + "An approach to AI in which knowledge and reasoning are represented using explicit symbols and formal rules. It models intelligence as a process of manipulating symbolic representations of concepts, objects, and relationships—often through logical inference." + short: | + "The rules and inference mechanism of an expert system are completely transparent, human-readable and symbolic." + chapter: logic-and-ai + id: symbolic-ai-160620251055 + + - term: Syntax + long: | + "The branch of logic that studies the formal structure of expressions in a logical language, without reference to their meaning. It specifies the rules that determine which combinations of symbols count as well-formed formulas in a given formal system. It includes an alphabet, formation rules, terms and objects." + short: | + "The formal rules governing the structure and formation of expressions in a formal language." + chapter: logic-and-ai + id: syntax-160620251056 + + - term: System 1 thinking + long: | + "Refers to the fast, automatic, intuitive, and effortless mode of cognition described in dual-process theories of the mind, particularly as articulated by Daniel Kahneman in Thinking, Fast and Slow. It contrasts with System 2, which is slower, deliberative, and analytic." + short: | + "Fast, automatic, intuitive mental processes, often modeled in AI via heuristics or neural networks." + chapter: logic-and-ai + id: system-1-thinking-160620251057 + + - term: System 2 thinking + long: | + "Refers to the slow, deliberate, effortful, and conscious mode of cognitive processing, as characterized in dual-process theories of reasoning—most notably in Daniel Kahneman’s Thinking, Fast and Slow. It operates in contrast to the fast, automatic, and intuitive System 1." + short: | + "Slow, deliberative, analytical reasoning, often modeled via rule-based or symbolic systems." + chapter: logic-and-ai + id: system-2-thinking-160620251058 + + + - term: Truth-preservation + long: | + "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." + short: | + "A feature of valid deductive inferences." + chapter: valid-inference + id: truth-preservation-010720251446 + + + - term: Undecidability theorem + long: | + "A theorem which establishes that there exist well-defined decision problems in formal logic and computation that cannot be solved by any algorithm. That is, there is no mechanical procedure that can always give a yes-or-no answer to every instance of the problem." + short: | + "A theorem demonstrating that certain problems cannot be resolved algorithmically within a given formal system." + chapter: logic-and-ai + id: undecidability-theorem-160620251100 + + - term: Validity + long: | + "A central semantic notion in logic. A formula or argument is valid if, under every interpretation (or in every model), the truth of its premises guarantees the truth of its conclusion. Validity is thus about truth preservation across all possible situations consistent with the formal semantics of the system." + short: | + "A property of arguments such that if the premises are true, the conclusion must necessarily be true." + chapter: logic-and-ai + id: validity-160620251101 + + - term: Valid-inference + long: | + "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)." + short: | + "An inference in which the conclusion follows necessarily from the premises." + chapter: valid-inference + id: valid-inference-010720251447 + + - term: Weak-inductive-inference + long: | + "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." + short: | + "An inference where the conclusion becomes only marginally more probable given the premises." + chapter: valid-inference + id: weak-inductive-inference-010720251448