Conjunction introduction: Difference between revisions
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{{Short description|Rule of inference in propositional logic}} | |||
{{Infobox mathematical statement | {{Infobox mathematical statement | ||
| name = Conjunction introduction | | name = Conjunction introduction | ||
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: <math>P, Q \vdash P \land Q</math> | : <math>P, Q \vdash P \land Q</math> | ||
where <math>P</math> and <math>Q</math> are propositions expressed in some [[formal system]], and <math>\vdash</math> is a [[metalogic]]al [[Symbol (formal)|symbol]] meaning that <math>P \land Q</math> is a [[logical consequence|syntactic consequence]] if <math>P</math> and <math>Q</math> are each on lines of a proof in some [[formal system|logical system]] | where <math>P</math> and <math>Q</math> are propositions expressed in some [[formal system]], and <math>\vdash</math> is a [[metalogic]]al [[Symbol (formal)|symbol]] meaning that <math>P \land Q</math> is a [[logical consequence|syntactic consequence]] if <math>P</math> and <math>Q</math> are each on lines of a proof in some [[formal system|logical system]]. | ||
==References== | ==References== | ||
Latest revision as of 17:32, 14 October 2025
Template:Infobox mathematical statement Template:Transformation rules
Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or adjunction)[1][2][3] is a valid rule of inference of propositional logic. The rule makes it possible to introduce a conjunction into a logical proof. It is the inference that if the proposition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is true, and the proposition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} is true, then the logical conjunction of the two propositions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} is true. For example, if it is true that "it is raining", and it is true that "the cat is inside", then it is true that "it is raining and the cat is inside". The rule can be stated:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{P,Q}{\therefore P \land Q}}
where the rule is that wherever an instance of "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} " and "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} " appear on lines of a proof, a "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \land Q} " can be placed on a subsequent line.
Formal notation
The conjunction introduction rule may be written in sequent notation:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P, Q \vdash P \land Q}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} are propositions expressed in some formal system, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vdash} is a metalogical symbol meaning that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \land Q} is a syntactic consequence if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} are each on lines of a proof in some logical system.
References
- ↑ Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing. pp. 346–51.
- ↑ Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2014). Introduction to Logic (14th ed.). Pearson. pp. 370, 620. ISBN 978-1-292-02482-0.
- ↑ Moore, Brooke Noel; Parker, Richard (2015). "Deductive Arguments II Truth-Functional Logic". Critical Thinking (11th ed.). New York: McGraw Hill. p. 311. ISBN 978-0-07-811914-9.