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| {{Portal|Logic}}
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>Here is more info about [http://www.youtube.com/watch?v=90z1mmiwNS8 Dentists in DC] review our own web-page. |
| This is a list of [[Rule of inference|rules of inference]], logical laws that relate to mathematical formulae.
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| ==Introduction==
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| '''Rules of inference''' are syntactical '''transform''' rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.
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| ''Discharge rules'' permit inference from a subderivation based on a temporary assumption. Below, the notation
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| : <math>\varphi \vdash \psi\,\!</math>
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| indicates such a subderivation from the temporary assumption <math>\varphi\,\!</math> to <math>\psi\,\!</math>.
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| ==Rules for classical sentential calculus==
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| Sentential calculus is also known as [[propositional calculus]].
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| ===Rules for negations===
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| ;[[Reductio ad absurdum]] (or ''Negation Introduction''):
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| : <math>\varphi \vdash \psi\,\!</math>
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| : <math>\underline{\varphi \vdash \lnot \psi}\,\!</math>
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| : <math>\lnot \varphi\,\!</math>
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| ;Reductio ad absurdum (related to the law of [[excluded middle]]):
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| : <math>\lnot \varphi \vdash \psi\,\!</math>
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| : <math>\underline{\lnot \varphi \vdash \lnot \psi}\,\!</math>
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| : <math>\varphi\,\!</math>
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| ;[[Noncontradiction]] (or ''Negation Elimination''):
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| : <math>\varphi\,\!</math>
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| : <math>\underline{\lnot \varphi}\,\!</math>
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| : <math>\psi\,\!</math>
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| ;[[Double negative elimination|Double negation elimination]]:
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| : <math>\underline{\lnot \lnot \varphi}\,\!</math>
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| : <math> \varphi\,\!</math>
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| ;[[Double negative introduction|Double negation introduction]]:
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| : <math>\underline{\varphi \quad \quad}\,\!</math>
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| : <math> \lnot \lnot \varphi\,\!</math>
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| ===Rules for conditionals===
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| ;[[Deduction theorem]] (or ''[[Conditional proof|Conditional Introduction]]''):
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| : <math>\underline{\varphi \vdash \psi}\,\!</math>
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| : <math>\varphi \rightarrow \psi\,\!</math>
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| ;[[Modus ponens]] (or ''Conditional Elimination''):
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| : <math>\varphi \rightarrow \psi\,\!</math>
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| : <math>\underline{\varphi \quad \quad \quad}\,\!</math>
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| : <math>\psi\,\!</math>
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| ;[[Modus tollens]]:
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| : <math>\varphi \rightarrow \psi\,\!</math>
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| : <math>\underline{\lnot \psi \quad \quad \quad}\,\!</math>
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| : <math>\lnot \varphi\,\!</math>
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| ===Rules for conjunctions===
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| ;[[Conjunction introduction|Adjunction]] (or ''Conjunction Introduction''):
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| : <math>\varphi\,\!</math>
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| : <math>\underline{\psi \quad \quad \ \ }\,\!</math>
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| : <math>\varphi \land \psi\,\!</math>
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| ;[[Simplification (logic)|Simplification]] (or ''Conjunction Elimination''):
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| : <math>\underline{\varphi \land \psi}\,\!</math>
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| : <math>\varphi\,\!</math>
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| : <math>\underline{\varphi \land \psi}\,\!</math>
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| : <math>\psi\,\!</math>
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| ===Rules for disjunctions===
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| ;[[Addition (logic)|Addition]] (or ''Disjunction Introduction''):
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| : <math>\underline{\varphi \quad \quad \ \ }\,\!</math>
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| : <math>\varphi \lor \psi\,\!</math>
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| : <math>\underline{\psi \quad \quad \ \ }\,\!</math>
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| : <math>\varphi \lor \psi\,\!</math>
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| ;[[Case analysis]]
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| : <math>\varphi \lor \psi\,\!</math>
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| : <math>\varphi \rightarrow \chi\,\!</math>
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| : <math>\underline{\psi \rightarrow \chi}\,\!</math>
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| : <math>\chi\,\!</math>
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| ;[[Disjunctive syllogism]]:
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| : <math>\varphi \lor \psi\,\!</math>
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| : <math>\underline{\lnot \varphi \quad \quad}\,\!</math>
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| : <math>\psi\,\!</math>
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| : <math>\varphi \lor \psi\,\!</math>
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| : <math>\underline{\lnot \psi \quad \quad}\,\!</math>
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| : <math>\varphi\,\!</math>
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| ===Rules for biconditionals===
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| ;[[Biconditional introduction]]:
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| : <math>\varphi \rightarrow \psi\,\!</math>
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| : <math>\underline{\psi \rightarrow \varphi}\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| ;Biconditional Elimination:
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\varphi \quad \quad}\,\!</math>
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| : <math>\psi\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\psi \quad \quad}\,\!</math>
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| : <math>\varphi\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\lnot \varphi \quad \quad}\,\!</math>
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| : <math>\lnot \psi\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\lnot \psi \quad \quad}\,\!</math>
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| : <math>\lnot \varphi\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\psi \lor \varphi}\,\!</math>
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| : <math>\psi \land \varphi \,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\lnot \psi \lor \lnot \varphi}\,\!</math>
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| : <math>\lnot \psi \land \lnot \varphi \,\!</math>
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| ==Rules of classical [[First-order logic|predicate calculus]]==
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| In the following rules, <math>\varphi(\beta / \alpha)\,\!</math> is exactly like <math>\varphi\,\!</math> except for having the term <math>\beta\,\!</math> everywhere <math>\varphi\,\!</math> has the free variable <math>\alpha\,\!</math>.
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| ;[[Universal generalization|Universal Introduction]] (or ''Universal Generalization''):
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| : <math>\underline{\varphi{(\beta / \alpha)}}\,\!</math>
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| : <math>\forall \alpha\, \varphi\,\!</math>
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| Restriction 1: <math>\beta</math> does not occur in <math>\varphi</math>.
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| <br/> | |
| Restriction 2: <math>\beta</math> is not mentioned in any hypothesis or undischarged assumptions.
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| ;[[Universal instantiation|Universal Elimination]] (or ''Universal Instantiation''):
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| : <math> \forall \alpha\, \varphi\!</math>
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| : <math>\overline{\varphi{(\beta / \alpha)}}\!</math>
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| Restriction: No free occurrence of <math>\alpha\,\!</math> in <math>\varphi\,\!</math> falls within the scope of a quantifier quantifying a variable occurring in <math>\beta\,\!</math>.
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| ;Existential Introduction (or ''Existential Generalization''):
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| : <math>\underline{\varphi(\beta / \alpha)}\,\!</math>
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| : <math>\exists \alpha\, \varphi\,\!</math> | |
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| Restriction: No free occurrence of <math>\alpha\,\!</math> in <math>\varphi\,\!</math> falls within the scope of a quantifier quantifying a variable occurring in <math>\beta\,\!</math>.
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|
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| ;Existential Elimination (or ''Existential Instantiation''):
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| : <math>\exists \alpha\, \varphi\,\!</math>
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| : <math>\underline{\varphi(\beta / \alpha) \vdash \psi}\,\!</math>
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| : <math>\psi\,\!</math>
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| Restriction 1: No free occurrence of <math>\alpha\,\!</math> in <math>\varphi\,\!</math> falls within the scope of a quantifier quantifying a variable occurring in <math>\beta\,\!</math>.
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| <br/>
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| Restriction 2: There is no occurrence, free or bound, of <math>\beta\,\!</math> in <math>\psi\,\!</math>.
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| ==Table: Rules of Inference - a short summary==
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| The rules above can be summed up in the following table.<ref>Kenneth H. Rosen: ''Discrete Mathematics and its Applications'',Fifth Edition, p. 58.</ref> The "[[Tautology (logic)|Tautology]]" column shows how to interpret the notation of a given rule.
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| {| class="wikitable"
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| |-
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| ! Rule of inference
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| ! Tautology
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| ! Name
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| |-
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| |<math>\begin{align}
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| p \\
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| \therefore \overline{p \vee q} \\
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| \end{align}</math>
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| | <math>p \rightarrow (p \vee q)</math>
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| | Addition
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| |-
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| |<math>\begin{align}
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| p \wedge q \\
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| \therefore \overline{p \quad \quad \quad} \\
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| \end{align}</math>
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| | <math>(p \wedge q) \rightarrow p</math>
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| | Simplification
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| |-
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| |<math>\begin{align}
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| p\\
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| q\\
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| \therefore \overline{p \wedge q} \\
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| \end{align}</math>
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| | <math>((p) \wedge (q)) \rightarrow (p \wedge q)</math>
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| | Conjunction
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| |-
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| |<math>\begin{align}
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| p\\
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| p \rightarrow q\\
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| \therefore \overline{q \quad \quad \quad} \\
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| \end{align}</math>
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| | <math>((p \wedge (p \rightarrow q)) \rightarrow q</math>
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| | [[Modus ponens]]
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| |-
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| |<math>\begin{align}
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| \neg q\\
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| p \rightarrow q\\
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| \therefore \overline{\neg p \quad \quad \quad} \\
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| \end{align}</math>
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| | <math>((\neg q \wedge (p \rightarrow q)) \rightarrow \neg p</math>
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| | [[Modus tollens]]
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| |-
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| |<math>\begin{align}
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| p \rightarrow q\\
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| q \rightarrow r\\
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| \therefore \overline{p \rightarrow r} \\
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| \end{align}</math>
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| | <math>((p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow (p \rightarrow r)</math>
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| | Hypothetical syllogism
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| |-
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| |<math>\begin{align}
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| p \vee q \\
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| \neg p \\
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| \therefore \overline{q \quad \quad \quad} \\
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| \end{align}</math>
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| | <math>((p \vee q) \wedge \neg p) \rightarrow q</math>
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| | Disjunctive syllogism
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| |-
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| |<math>\begin{align}
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| p \vee q \\
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| \neg p \vee r \\
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| \therefore \overline{q \vee r} \\
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| \end{align}</math>
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| | <math>((p \vee q) \wedge (\neg p \vee r)) \rightarrow (q \vee r)</math>
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| | Resolution
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| |}
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| All rules use the basic logic operators. A complete table of "logic operators" is shown by a [[truth table]], giving definitions of all the possible (16) truth functions of 2 [[Boolean algebra (logic)|boolean variables]] (''p'', ''q''):
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| {| class="wikitable" style="margin:1em auto 1em auto; text-align:center;"
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| ! ''p'' || ''q''
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| ! 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7
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| !| 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15
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| |-
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| ! T || T
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| | || F || F || F || F || F || F || F || F || || T || T || T || T || T || T || T || T
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| |-
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| ! T || F
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| | || F || F || F || F || T || T || T || T || || F || F || F || F || T || T || T || T
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| |-
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| ! F || T
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| | || F || F || T || T || F || F || T || T || || F || F || T || T || F || F || T || T
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| |-
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| ! F || F
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| | || F || T || F || T || F || T || F || T || || F || T || F || T || F || T || F || T
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| |}
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| where T = true and F = false, and, the columns are the logical operators: '''0''', false, [[Contradiction]]; '''1''', NOR, [[Logical NOR]]; '''2''', [[Converse nonimplication]]; '''3''', '''¬p''', [[Negation]]; '''4''', [[Material nonimplication]]; '''5''', '''¬q''', Negation; '''6''', XOR, [[Exclusive disjunction]]; '''7''', NAND, [[Logical NAND]]; '''8''', AND, [[Logical conjunction]]; '''9''', XNOR, [[If and only if]], [[Logical biconditional]]; '''10''', '''q''', [[Projection function]]; '''11''', if/then, [[Logical implication]]; '''12''', '''p''', Projection function; '''13''', then/if, [[Converse implication]]; '''14''', OR, [[Logical disjunction]]; '''15''', true, [[Tautology (logic)|Tautology]].
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| Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples:
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| * The column-14 operator (OR), shows ''Addition rule'': when ''p''=T (the hypothesis selects the first two lines of the table), we see (at column-14) that ''p''∨''q''=T.
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| *: We can see also that, with the same premise, another conclusions are valid: columns 12, 14 and 15 are T.
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| * The column-8 operator (AND), shows ''Simplification rule'': when ''p''∧''q''=T (first line of the table), we see that ''p''=T.
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| *: With this premise, we also conclude that ''q''=T, ''p''∨''q''=T, etc. as showed by columns 9-15.
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| * The column-11 operator (IF/THEN), shows ''Modus ponens rule'': when ''p''→''q''=T and ''p''=T only one line of the truth table (the first) satisfies these two conditions. On this line, ''q'' is also true. Therefore, whenever p → q is true and p is true, q must also be true.
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| Machines and well-trained people use this [[Lookup table|look at table approach]] to do basic inferences, and to check if other inferences (for the same premises) can be obtained.
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| ===Example 1===
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| Let us consider the following assumptions: "If it rains today, then we will not go on a canoe today. If we do not go on a canoe trip today, then we will go on a canoe trip tomorrow. Therefore (Mathematical symbol for "therefore" is <math>\therefore</math>), if it rains today, we will go on a canoe trip tomorrow.
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| To make use of the rules of inference in the above table we let <math>p</math> be the proposition "If it rains today", <math>q</math> be " We will not go on a canoe today" and let <math>r</math> be "We will go on a canoe trip tomorrow". Then this argument is of the form:
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| <math>\begin{align} | |
| p \rightarrow q\\
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| q \rightarrow r\\
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| \therefore \overline{p \rightarrow r} \\
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| \end{align}</math>
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| ===Example 2===
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| Let us consider a more complex set of assumptions: "It is not sunny today and it is colder than yesterday". "We will go swimming only if it is sunny", "If we do not go swimming, then we will have a barbecue", and "If we will have a barbecue, then we will be home by sunset" lead to the conclusion "We will be home before sunset."
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| Proof by rules of inference: Let <math>p</math> be the proposition "It is sunny this today", <math>q</math> the proposition "It is colder than yesterday", <math>r</math> the proposition "We will go swimming", <math>s</math> the proposition "We will have a barbecue", and <math>t</math> the proposition "We will be home by sunset". Then the hypotheses become <math>\neg p \wedge q, r \rightarrow p, \neg r \rightarrow s</math> and <math>s \rightarrow t</math>. Using our intuition we conjecture that the conclusion might be <math>t</math>. Using the Rules of Inference table we can proof the conjecture easily:
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| {| class="wikitable"
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| ! Step
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| ! Reason
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| | 1.<math>\neg p \wedge q</math>
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| | Hypothesis
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| |-
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| | 2. <math>\neg p</math>
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| | Simplification using Step 1
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| |-
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| | 3. <math>r \rightarrow p</math>
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| | Hypothesis
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| |-
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| | 4. <math>\neg r</math>
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| | Modus tollens using Step 2 and 3
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| |-
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| | 5. <math>\neg r \rightarrow s</math>
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| | Hypothesis
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| |-
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| | 6. <math>s</math>
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| | Modus ponens using Step 4 and 5
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| |-
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| | 7. <math>s \rightarrow t</math>
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| | Hypothesis
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| |-
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| | 8. <math>t</math>
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| | Modus ponens using Step 6 and 7
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| |}
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| ==References== | |
| <references/>
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| {{Logic}}
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| {{DEFAULTSORT:List Of Rules Of Inference}}
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| [[Category:Rules of inference|*]]
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| [[Category:Mathematics-related lists|Rules of inference]]
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| [[Category:Philosophy-related lists|Rules of inference]]
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| [[de:Schlussregel]]
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| [[it:Elenco di regole di inferenza]]
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| [[he:חוקי היקש]]
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