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List of axiomatic systems in logic

List of axiomatic systems in logic is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand List of axiomatic systems in logic rather than just read about it. In short: This article contains a list of sample Hilbert-style deductive systems for propositional logics. Classical propositional calculus systems Classical propositional calculus is the standard propositional logic.

Key takeaways

  • List of axiomatic systems in logic belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect List of axiomatic systems in logic to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of List of axiomatic systems in logic from memory before moving on to harder problems.

Reference excerpt

This article contains a list of sample Hilbert-style deductive systems for propositional logics.

Classical propositional calculus systems Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent and its main property is that it is strongly complete, otherwise said that whenever a formula semantically follows from a set of premises, it also follows from that set syntactically. Many different equivalent complete axiom systems have been formulated. They differ in the choice of basic connectives used, which in all cases have to be functionally complete (i.e. able to express by composition all n-ary truth tables), and in the exact complete choice of axioms over the chosen basis of connectives.

Implication and negation The formulations here use implication and negation { → , ¬ } {\displaystyle \{\to ,\neg \}} as functionally complete set of basic connectives. Every logic system requires at least one non-nullary rule of inference. Classical propositional calculus typically uses the rule of modus ponens:

A , A → B B . {\displaystyle {\frac {A,A\to B}{B}}.}

We assume this rule is included in all systems below unless stated otherwise. Frege's axiom system:

A → ( B → A ) {\displaystyle A\to (B\to A)}

( A → ( B → C ) ) → ( ( A → B ) → ( A → C ) ) {\displaystyle (A\to (B\to C))\to ((A\to B)\to (A\to C))}

( A → ( B → C ) ) → ( B → ( A → C ) ) {\displaystyle (A\to (B\to C))\to (B\to (A\to C))}

( A → B ) → ( ¬ B → ¬ A ) {\displaystyle (A\to B)\to (\neg B\to \neg A)}

¬ ¬ A → A {\displaystyle \neg \neg A\to A}

A → ¬ ¬ A {\displaystyle A\to \neg \neg A}

Hilbert's axiom system:

A → ( B → A ) {\displaystyle A\to (B\to A)}

( A → ( B → C ) ) → ( B → ( A → C ) ) {\displaystyle (A\to (B\to C))\to (B\to (A\to C))}

( B → C ) → ( ( A → B ) → ( A → C ) ) {\displaystyle (B\to C)\to ((A\to B)\to (A\to C))}

A → ( ¬ A → B ) {\displaystyle A\to (\neg A\to B)}

( A → B ) → ( ( ¬ A → B ) → B ) {\displaystyle (A\to B)\to ((\neg A\to B)\to B)}

Łukasiewicz's axiom systems:

First:

( A → B ) → ( ( B → C ) → ( A → C ) ) {\displaystyle (A\to B)\to ((B\to C)\to (A\to C))}

( ¬ A → A ) → A {\displaystyle (\neg A\to A)\to A}

A → ( ¬ A → B ) {\displaystyle A\to (\neg A\to B)}

Second:

( ( A → B ) → C ) → ( ¬ A → C ) {\displaystyle ((A\to B)\to C)\to (\neg A\to C)}

( ( A → B ) → C ) → ( B → C ) {\displaystyle ((A\to B)\to C)\to (B\to C)}

( ¬ A → C ) → ( ( B → C ) → ( ( A → B ) → C ) ) {\displaystyle (\neg A\to C)\to ((B\to C)\to ((A\to B)\to C))}

Third:

A → ( B → A ) {\displaystyle A\to (B\to A)}

( A → ( B → C ) ) → ( ( A → B ) → ( A → C ) ) {\displaystyle (A\to (B\to C))\to ((A\to B)\to (A\to C))}

( ¬ A → ¬ B ) → ( B → A ) {\displaystyle (\neg A\to \neg B)\to (B\to A)}

Arai's axiom system:

( A → B ) → ( ( B → C ) → ( A → C ) ) {\displaystyle (A\to B)\to ((B\to C)\to (A\to C))}

A → ( ¬ A → B ) {\displaystyle A\to (\neg A\to B)}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with List of axiomatic systems in logic

Start with the simplest possible case. Write down what List of axiomatic systems in logic claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to List of axiomatic systems in logic before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about List of axiomatic systems in logic ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of List of axiomatic systems in logic

In research
List of axiomatic systems in logic appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses List of axiomatic systems in logic in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
List of axiomatic systems in logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Logic-related lists, Logical calculi, Mathematical logic, so understanding it makes those chapters shorter.
In everyday life
Look for List of axiomatic systems in logic outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study List of axiomatic systems in logic in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what List of axiomatic systems in logic means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain List of axiomatic systems in logic out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is List of axiomatic systems in logic in simple terms?

This article contains a list of sample Hilbert-style deductive systems for propositional logics. Classical propositional calculus systems Classical propositional calculus is the standard propositional logic.

Why does List of axiomatic systems in logic matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study List of axiomatic systems in logic?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on List of axiomatic systems in logic.

Tags

  • Logic-related lists
  • Logical calculi
  • Mathematical logic
  • Propositional calculus
  • Systems of formal logic

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