Modus putting ponens

The modus ponendo ponens (Latin: "the way that, when affirming, affirms"¹, also called modus ponens, elimination of implication, separation rule, affirmation of the antecedent, usually abbreviated MP) is a form of valid argument (deductive reasoning) and one of the rules of inference in propositional logic. It can be summarized as & #34;if P implies Q; y if P is true; then Q is also true." The history of modus ponendo ponens dates back to antiquity.

The modus ponendo ponens can be formally stated as:

P→ → Q,P▪ ▪ Q{displaystyle {frac {Pto Q,;P}{therefore Q}}

where the rule is when "P → Q" and "P" appear by themselves on the same line of a logical test, Q may be validly written on a subsequent line. Note that the premise of P and the implication "dissolve", their only trace being the symbol Q which is kept for later use, e.g., in a more complex deduction.

An example of a modus ponendo ponens is:

If it's raining, I'll wait for you in the theater.

It's raining.

So.I'll wait for you in the theater.

Although the modus ponendo ponens is one of the most used concepts in logic, it should not be confused with a logical law. Rather, it is one of the accepted mechanisms for constructing deductive proofs that includes the "definition rule" and the "substitution rule". Modus ponendo ponens allows you to remove a conditional statement from a logical proof or argument (the antecedents) and therefore not carry these antecedents forward in a long, constant string of symbols. For this reason, the modus ponendon ponens is sometimes called the separation rule. Enderton, for example, observed that "the modus ponendon ponens can produce shorter formulas from longer ones", and Russell noted that "the process of inference cannot be reduced to symbols. Its only record is the occurrence of ⊦ Q [the consequent]...an inference modus ponendo ponens is not so much the casting of a true premise, but rather the dissolution of an implication".

The modus ponendo ponens is closely related to another valid argument form, the modus tollendo tollens. Both are related to two invalid forms of argument or fallacies: affirmation of the consequent and denial of the antecedent. Additionally, the constructive dilemma is the disjunctive version of the modus ponendo ponens. The hypothetical syllogism is closely related to the modus ponendo ponens and is sometimes thought of as the "double modus ponens."

Formal notation

The rule of the modus ponendo ponens can be written in the following notation:

P→ → Q,P Q{displaystyle Pto Q,;P;vdash ;;Q}

where ⊢ is a metalogical symbol meaning that Q is a syntactic consequence of P → Q and P in some logical system;

or as the statement of a truth-functional tautology or theorem of propositional logic:

((P→ → Q)∧ ∧ P)→ → Q{displaystyle (Pto Q)land P)to Q}

where P, and Q are propositions expressed in some formal system.

Explanation

The argument form has two premises (hypothesis). The first premise is the "if-then" or conditional claim, namely: that P implies Q. The second premise is that P, the antecedent of the conditional claim, is true. Accepting the premises necessarily implies that Q, the consequent or apodosis of the conditional claim, must also be true. In artificial intelligence, modus ponens is usually called forward chaining.

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

If it's Tuesday today, then John will go to work.

Today is Tuesday.

Therefore, John will go to work.

This argument is valid, but this tells us nothing about whether the premises required by the argument are true. For modus ponens to be a solid argument as well as valid, the premises must be true. A valid but weak argument may or may not be false. The example argument is only strong on Tuesdays and when, in fact, it is known that John actually goes to work on Tuesdays.

In single-completion sequential calculus, the modus ponens is the cutting rule. The break elimination theorem for a calculus says that every proof involving Break can be transformed (usually by a constructive method) into a proof without break, and hence break is admissible.

The Curry-Howard correspondence between tests and programs relates the modus ponens to the function application: if f is a function of type P → Q and x is of type P, so f x is of type Q.

Relationship with the Modus Tollens

Any Modus ponens rule can be tested by a Modus Tollens and transposition rule. The proof is the following.

1. P → Q

2. P / Q

3.~Q → ~P 1 Transposition

4.~~ P 2 Double denial

5.~~~ Q 3.4 Modus Tollens

6. Q 5 Double denial

Justification by truth table

The validity of modus ponens in classical two-valued logic can be clearly demonstrated using a truth table.

In cases of modus ponens it is assumed as a premise that p → q is true and p is true. Only one line of the truth table—the first—satisfies these two conditions (p and p → q). Along these lines, q is also true. Therefore, whenever p → q is true and p is true, q must also be true..

Via tollendo ponens

Step	Proposed	Referral
1	P→ → Q{displaystyle Prightarrow Q}	Premises
2	P{displaystyle P}	Premises
3	¬ ¬ P Q{displaystyle neg Plor Q}	Material involvement (1)
4	Q{displaystyle Q}	Modus tollendo puts (2,3)