Modus tollendo tollens

The modus tollendo tollens (Latin: "the way that, by denying, denies", known as modus tollens, negation of the consequent or law of contraposition)) is a valid argument form and rule of inference in logic propositional. It can be summarized as "If P implies Q, and Q is not true, then P does not it's true".

The modus tollendo tollens is an application of the general truth that if a statement is valid, so is its contraposition. The history of the rule modus tollendo tollens dates back to antiquity, with the Stoics being the first to explicitly state this valid form of argument.

The modus tollendo tollens can be formally set as:

P→ → Q,¬ ¬ Q▪ ▪ ¬ ¬ P{displaystyle {frac {Pto Q,neg Q}{therefore neg P}}}}

where P → Q means "P implies Q", ¬Q means "is not the case Q" ("not Q"), ¬P means "not P". The rule is that whenever P → Q and ¬Q appear by themselves on a line of a logical proof, ¬P can be validly written on a subsequent line.

A simple example of modus tollendo tollens is:

P→ → Q{displaystyle Pto Q} If the water boils, then it will release steam.

¬ ¬ Q{displaystyle neg Q} No. Let go of steam.

▪ ▪ ¬ ¬ P{displaystyle {therefore neg P}} So., No. He's boiling the water.

In this case, P{displaystyle P} It's the boiling water. Q{displaystyle Q} It's "soul steam." Given that ¬ ¬ Q{displaystyle neg Q}I mean, "No. release steam", it can be concluded that ¬ ¬ P{displaystyle neg P}I mean, "the water No. boil."

The modus tollendo tollens is closely related to another valid argument form, the modus ponendo ponens. Both are related to two invalid forms of argument or fallacies: affirmation of the consequent and denial of the antecedent.

Formal notation

The modus tollendo tollens rule can be written in several ways.

Modus tollendo tollens in subsequent notation

P→ → Q,¬ ¬ Q ¬ ¬ P{displaystyle Pto Q,neg Qvdash neg P}

where {displaystyle vdash } is a metalogical symbol that means ¬ ¬ P{displaystyle neg P} is a syntactic consequence of P→ → Q{displaystyle Pto Q} and ¬ ¬ Q{displaystyle neg Q} in some logical system.

Modus tollendo tollens as a statement of truth-functional tautology

This notation is also called the theorem of propositional logic. Is written:

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

where P{displaystyle P} and Q{displaystyle Q} are propositions expressed in some formal system.

Modus tollendo tollens including assumptions

It is written:

Interpreter Interpreter P→ → QInterpreter Interpreter ¬ ¬ QInterpreter Interpreter ¬ ¬ P{displaystyle {frac {Gamma vdash Pto Q~~~Gamma vdash neg} Q{Gamma vdash neg P}}}}}

although since the rule does not change the set of assumptions, this is not strictly necessary.

More complex scripts

Many times, you see more complex rewrites involving modus tollendo, for example, in set theory:

P Q{displaystyle Psubseq Q}

x Q{displaystyle xnotin Q}

▪ ▪ x P{displaystyle therefore xnotin P}

("P is a subset of Q. x is not in Q. Therefore, x is not in P.")

Also in first-order predicate logic:

Русский Русский x:P(x)→ → Q(x){displaystyle forall x:~P(x)to Q(x)}

consuming consuming x:¬ ¬ Q(x){displaystyle exists x:~neg Q(x)}

▪ ▪ consuming consuming x:¬ ¬ P(x){displaystyle therefore exists x:~neg P(x)}

("For all x if x is P then x is Q. There exists some x that is not Q. Therefore, there exists some x that is not P.")

Strictly speaking, these are not instances of tollendo modus, but they can be derived using modus tollendo tollens using some additional measures.

Explanation

The argument has two premises. The first premise is an "if-then" conditional or statement, eg, that if p then q. The second premise is that it is not the case for q ("not q"). From these two premises, it can be logically concluded that it is not the case for p ("not p").

For example:

p1: If the watchdog detects an intruder, the watchdog will bark.

p2: The guard dog didn't bark.

C: Therefore, the watchdog did not detect any intruders.

Assuming that the premises are true (the dog barks if it detects an intruder, and in fact does not bark), it follows that no intruder has been detected. This is a valid argument, since the conclusion cannot be false if the premises are true. (It is conceivable that there was an intruder that the dog did not detect, but that does not invalidate the argument; the first premise is "If the dog detects an intruder"). The important fact is that the dog detects or does not detect an intruder, not whether it exists or not.

Another example:

p1: If I'm the ax killer, then I know how to use an axe.

p2: I don't know how to use an axe.

C: Therefore, I am not the ax murderer.

Relation to modus ponens

Each use of modus tollendo tollens can be converted to a use of modus ponens and a use of transposing the premise that is a material implication. For example:

If p, then q (premises - material involvement)

If not q, then not p. (derivated by transposition)

No. Therefore, no p. (defeated by modus puts)

Similarly, each use of modus ponens can be converted to a use of modus tollendo tollens and transposition.

Justification via truth table

The validity of the modus tollendo tollens can be clearly demonstrated through a truth table.

In the cases of modus tollendo tollens we assume as premises that p → q is true and q is false. There is only one line in the table—the fourth——that satisfies these two conditions. In this, p is false. Therefore, in all cases where p → q is true and q is false, p must also be false.

Formal proof

Via disjunctive syllogism

Step	Proposed	Referral
1	P→ → Q{displaystyle Prightarrow Q}	Premises
2	¬ ¬ Q{displaystyle neg Q}	Premises
3	¬ ¬ P Q{displaystyle neg Plor Q}	Material involvement (1)
4	¬ ¬ P{displaystyle neg P}	Disjunctive siloge (2.3)

Via reductio ad absurdum

Step	Proposed	Referral
1	P→ → ▪ ▪ Q{displaystyle Prightarrow backsim Q}	Premises
2	¬ ¬ Q→ → p{displaystyle neg Qrightarrow p}	Premises
3	P∧ ∧ ♥ ♥ q{displaystyle Pland sim q}	Assumption
4	Q{displaystyle Q}	Modus puts (1,3)
5	p∧ ∧ ¬ ¬ r{displaystyle pland neg r}	Introduction of conjunction (2.4)
6	¬ ¬ P{displaystyle neg P}	Reductio ad absurdum (3.5)