Modus ponens

In logic, modus ponendo ponens (Latin: mode that affirms by affirming;^[1] often abbreviated to MP or modus ponens) is a valid, simple argument form sometimes referred to as affirming the antecedent or the law of detachment. Logic is the study of the principles of valid demonstration and Inference. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in Logic refers generally to a property of In Logic, the argument form or test form of an Argument results from replacing the different words or sentences that make up the argument with letters It is closely related to another valid form of argument, modus tollens or "denying the consequent". In Classical logic, modus tollens (or modus tollendo tollens) ( Latin for "the way that denies by denying" has the following Argument form In Classical logic, modus tollens (or modus tollendo tollens) ( Latin for "the way that denies by denying" has the following Argument form

Modus ponens is a very common rule of inference, and takes the following form:

If P, then Q. In Logic, a rule of inference (also called a transformation rule) is a function from sets of formulae to formulae

Therefore, Q. ^[2]

1 Formal notation
2 Explanation
3 Justification via truth table
4 See also
5 References
6 External links

Formal notation

The modus ponens rule may be written in logical operator notation:

$P \to Q, P \vdash Q$

where $\vdash$ represents the logical assertion (that Q is true). Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence In Proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction.

Or in set-theoretic form:

$P\subseteq Q$

$x\in P$

$\therefore x\in Q$

("P is a subset of Q. x is an element of P. Therefore, x is an element of Q. ")

It can also be written as:

$\qquad\frac{P \rightarrow Q, P}{Q}$

Explanation

The argument form has two premises. The first premise is the "if–then" or conditional claim, namely that P implies Q. The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain Conditionals in Logic The second premise is that P, the antecedent of the conditional claim, is true. An antecedent is the first half of a Hypothetical Proposition. From these two premises it can be logically concluded that Q, the consequent of the conditional claim, must be true as well. A consequent is the second half of a hypothetical Proposition. In Artificial Intelligence, modus ponens is often called forward reasoning. In Classical logic, modus ponendo ponens ( Latin: mode that affirms by affirming; often abbreviated to MP or modus ponens) is a

An example of an argument that fits the form modus ponens:

If today is Tuesday, then I will go to work.

Today is Tuesday.

Therefore, I will go to work.

This argument is valid, but this has no bearing on whether any of the statements in the argument are true; the validity of modus ponens means that the conclusion must be true if all the premises are true. The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in Logic refers generally to a property of The meaning of the word truth extends from Honesty, Good faith, and Sincerity in general to agreement with Fact or Reality An argument can be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premises are true, then the argument is sound. In Logic, an argument is a Set of one or more Declarative sentences (or "propositions") known as the Premises along In Mathematical logic, a Logical system has the soundness property If and only if its Inference rules prove only formulas that are In Mathematical logic, a Logical system has the soundness property If and only if its Inference rules prove only formulas that are A propositional argument using modus ponens is said to be deductive. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" Deductive reasoning is Reasoning which uses deductive Arguments to move from given statements ( Premises to Conclusions which must be true if the

In metalogics, modus ponens is the cut rule. Metalogic is the study of the Metatheory of Logic. While logic is the study of the manner in which logical systems can be used to decide the correctness The cut-elimination theorem says that the cut is valid (an admissible rule) in some logical calculus (sequent calculus). The cut-elimination theorem is the central result establishing the significance of the Sequent calculus. In Logic, a Rule of inference is admissible in a Formal system if the set of theorems of the system is closed under the rule In Proof theory and Mathematical logic, the sequent calculus is a widely known Proof calculus for First-order logic (and Propositional logic

Justification via truth table

The validity of modus ponens can be clearly demonstrated by use of a truth table. A truth table is a Mathematical table used in Logic — specifically in connection with Boolean algebra, Boolean functions and Propositional

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

In instances of modus ponens we assume as premises that p → q is true and p is true. 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.

References

^ Stone, Jon R. In Logic, a hypothetical syllogism has two uses In Propositional logic it expresses a rule of inference while in the History of logic, it is a short-hand In Classical logic, modus tollens (or modus tollendo tollens) ( Latin for "the way that denies by denying" has the following Argument form Modus tollendo ponens (literally mode which by denying affirms) or MTP, is a valid, simple Argument form that is today known as In Classical logic, modus tollens (or modus tollendo tollens) ( Latin for "the way that denies by denying" has the following Argument form Affirming the consequent, sometimes also called Converse error, is a Formal fallacy, committed by reasoning in the form: If P Denying the antecedent, sometimes also called Inverse error, is a Formal fallacy, committed by reasoning in the form: If P, then A disjunctive syllogism, historically known as Modus tollendo ponens, is a classically valid, simple Argument form: P or Q In Logic, a rule of inference (also called a transformation rule) is a function from sets of formulae to formulae (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London, UK: Routledge: 60. .
^ Jago, Mark (2007). Formal Logic. Humanities-Ebooks LLP. ISBN 978-1-84760-041-7.

External links

Modus ponens at Wolfram MathWorld

Dictionary

-noun

(philosophy, logic) A valid form of argument in which the antecedent of a conditional proposition is affirmed, thereby entailing the affirmation of the consequent. Modus ponens has the form:

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