Logic is the study of correct reasoning. It includes both formal and informal logic.
Mathematical logic is the study of formal logic within mathematics. Major sub-areas include model theory, proof theory, set theory, and recursion theory.
Formal logic uses a formal approach to study reasoning: it replaces concrete expressions with abstract symbols to examine the logical form of arguments independent of their concrete content.
Modus Ponens is a common structure in formal logic:
If P, then Q.
P is true.
Therefore, Q is true.
Symbolically:
P ⇒ Q (If P, then Q)P (P is true)∴ Q (Q is true)
Usually, only the syntax of a formal language is considered via the notion of a formal grammar.
Mathematical induction is technically a form of deductive reasoning. Although the process of mathematical induction may seem like it involves establishing something step by step, it is based on a logical framework that ensures the truth of a statement for all cases, which is characteristic of deduction.
Informal logic deals with everyday reasoning and focuses on the content of the arguments, considering context, plausibility, and relevance. It is less concerned with strict symbolic representation and more with how arguments are presented in natural language. Informal logic often addresses fallacies and reasoning errors that people make in everyday discourse.
An ad hominem fallacy occurs when someone attacks the person making an argument rather than addressing the argument itself.
Example:
Person A: "We should reduce carbon emissions to combat climate change."
Person B: "Why should we listen to you? You failed science in high school."
Unter einer Aussage verstehen wir in dem Kontext einfach generell irgendeine Behauptung, von der man klar sagen kann, ob sie zutrifft oder nicht. Eine Aussage (zu Englisch: „a proposition“) sei die Beschreibung einer Situation, die wir eindeutig entweder als „wahr“ oder als „falsch“ klassifizieren können.
Wahrheitstafel(truth table): stellt dir zu einer aussagenlogischen Formel alle möglichen Wahrheitswerte auf
Junktor ist eine Logische Verknüpfung zwischen Aussagen. Eine Operation, die aus diesen Aussagen eine neue Aussage erzeugt, indem sie für jede mögliche Variante der Wahrheitswerte der gegebenen Aussagen einen Wahrheitswert festlegt, nennen wir Junktor (zu Englisch: „logic operator“).
Logische Operator ist eine Funktion die einen Wahrheitswert liefert. Eine aussagenlogische Formel (Kombination von Aussagenvariablen) ist eine Zeichenkette bestehend aus Aussagenvariablen, also Buchstaben (wir wollen „p“, „q“ und „r“ dafür verwenden), die als Platzhalter für beliebige Aussagen stehen, und den logischen Operatoren ㄱ, ㅅ, v, ⇒ und ⇔.
Tautologie, Kontradiktion: 1. Aussage die immer wahr hist 2. Aussage die immer falsch ist
Predicate logic extends propositional logic by incorporating quantifiers and variables, allowing for more complex statements about objects and their properties, relationships, or functions.
Example 1: Universal Quantifier
Statement: "All humans are mortal."
∀x(H(x) ⇒ M(x))
Example 2: Existential Quantifier
Statement: "There is someone who is a mathematician."
∃x(M(x))
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.
The modern study of set theory was initiated by German mathematicians Richard Dedekind and Georg Cantor in the 1870s. George Cantor is considered the founder of set theory.
At the time, set theory was referred to as naive set theory, which led to several paradoxes (e.g., Russell’s Paradox, Barber’s Paradox). This was resolved with the introduction of axiomatic set theory, the most famous of which is ZFC (Zermelo–Fraenkel set theory).
The barber is the "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself?
Zwei Mengen X und Y sind gleich, wenn sie dieselben Elemente besitzen, also:
∀X∀Y(X=Y⟺∀z(z∈X⟺z∈Y))
Seien X, Y Zwei Mengen. Dann gilt:
X = Y ⟺ ((X ⊆ Y) ∧ ( Y ⊆ X))
∃X∀x(x∉X)
A set is considered countable if its elements can be placed in one-to-one correspondence with the set of natural numbers. Countable sets include N, Z, and Q. Uncountable sets, such as irrational numbers, include ℝ.
Seien X und Y zwei Mengen. Wir definieren das kartesische Produkt X×Y („X kreuz Y“) von X und Y durch:
X×Y := {(x,y)∣x∈X,y∈Y}.
Das kartesische Produkt ist nicht kommutativ, da die geordneten Paare (x, y) und (y, x) im Allgemeinen nicht gleich sind. Wir haben also X × Y ≠ Y × X.