Modus ponens and Modus tollens
In logic, modus ponens (Latin: mode that affirms; often abbreviated MP) is a validity, simple argument form. It is a very common form of reasoning, and takes the following form:
- If P, then Q.
- P.
- Therefore, Q.
In logical operator notation:
- P → Q
- P
- ⊢ Q
where ⊢ represents the logical assertion ("Therefore Q is true").
The modus ponens rule may also be written:
- P → Q, P
- Q
The argument form has two premises. 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. From these two premises it can be logically concluded that Q, the consequent of the conditional claim, must be true as well.
Here is 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.
The fact that the argument is valid cannot assure us that any of the statements in the argument are true; the validity of modus ponens tells us that the conclusion must be true if all the premises are true. It is wise to recall that a valid argument within which one or more of the premises are not true is called an unsound argument, whereas if all the premises are true, then the argument is sound. In most logical systems, modus ponens is considered to be valid. However, the instances of its use may be either sound or unsound:
- If the argument is modus ponens and its premises are true, then it is sound.
- The premises are true.
- Therefore, it is a sound argument.
A propositional argument using modus ponens is said to be deductive.
Modus ponens can also be referred to as affirming the antecedent or "Law of Detachment".
In metalogics, modus ponens is the cut rule. The cut-elimination theorem says that the cut is valid (an admissible rule) in some logical calculus (sequent calculus).
For an amusing dialog that problematizes modus ponens, see Lewis Carroll's "What the Tortoise Said to Achilles."
Modus tollens (Latin for "mode that denies") is the formal name for indirect proof or proof by contrapositive (contrapositive inference), often abbreviated to MT. It can also be referred to as denying the consequent, and is a valid form of argument (unlike similarly-named but invalid arguments such as affirming the consequent or denying the antecedent).
Modus tollens has the following argument form:
- If P, then Q.
- Q is false.
- Therefore, P is false.
In logical operator notation:
- P → Q
- ¬Q
- ⊢ ¬P
where ⊢ represents the logical assertion.
Or in set-theoretic form:
- P ⊆ Q
- x ∉ Q
- ∴x∉ P
("P is a subset of Q. x is not in Q. Therefore, x is not in P.")
The argument has two premises. The first premise is the conditional "if-then" statement, namely that P implies Q. The second premise is that Q is false. From these two premises, it can be logically concluded that P must be false. (Why? If P were true, then Q would be true, by premise 1, but it isn't, by premise 2.)
Consider an example:
- If there is fire here, then there is oxygen here.
- There is no oxygen here.
- Therefore, there is no fire here.
Another example:
- If Lizzy was the murderer, then she owns an axe.
- Lizzy does not own an axe.
- Therefore, Lizzy was not the murderer.
Just suppose that the premises are both true. If Lizzy was the murderer, then she really must have owned an axe; and it is a fact that Lizzy does not own an axe. What follows? That she was not the murderer.
It is important to note that when an argument is valid, if the premises are true, the conclusion must follow. Suppose we decide that it is not the case that: if Lizzy was the murderer, then she would have to have owned an axe; Perhaps we have found that she borrowed someone's. This means that the first premise is false. But notice that it does not mean the argument is invalid, since it remains the case that, if the premises are true (and in this case they are not), the conclusion would follow, even though in this particular case the premise is false. An argument can be valid even though it has a false premise. Such an argument can reach a false conclusion.
- If a modus tollens argument has true premises, then it is sound.
- The argument is unsound
- Therefore, its premises are false.
(Of course this particular argument applied to itself would be a paradox)
Modus tollens became somewhat legendary when it was used by Karl Popper in his proposed response to the problem of induction, Falsificationism.
See also
- Affirming the consequent
- Denying the antecedent
- Falsificationism
- Modus ponens - "one man's modus ponens is another man's modus tollens" (Dretske 1995)
- Non sequitur (logic)
External links
See also
- Hypothetical syllogism
- Modus tollens
- Affirming the consequent
- Denying the antecedent
- Inference rule
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