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For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is guaranteed by the truth of P. (Equivalently, it is impossible to have P without Q , or the falsity of Q ensures the falsity of P .) [ 1 ] Similarly, P is sufficient for Q , because P being true always implies that Q is true, but P not being ...
A mixed hypothetical syllogism has two premises: one conditional statement and one statement that either affirms or denies the antecedent or consequent of that conditional statement. For example, If P, then Q. P. ∴ Q. In this example, the first premise is a conditional statement in which "P" is the antecedent and "Q" is the consequent.
Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.
Showing that if the statement holds for an arbitrary number n ≥ b, then the same statement also holds for n + 1. This can be used, for example, to show that 2 n ≥ n + 5 for n ≥ 3 . In this way, one can prove that some statement P ( n ) holds for all n ≥ 1 , or even for all n ≥ −5 .
The example in the previous section used unformalized, natural-language reasoning. Curry's paradox also occurs in some varieties of formal logic.In this context, it shows that if we assume there is a formal sentence (X → Y), where X itself is equivalent to (X → Y), then we can prove Y with a formal proof.
On the other hand, one can affirm with certainty that "if someone does not live in California" (non-Q), then "this person does not live in San Diego" (non-P). This is the contrapositive of the first statement, and it must be true if and only if the original statement is true. Example 2. If an animal is a dog, then it has four legs. My cat has ...
Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples: 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.
The following numerical example illustrates Lindley's paradox. In a certain city 49,581 boys and 48,870 girls have been born over a certain time period. The observed proportion x {\displaystyle x} of male births is thus 49 581 / 98 451 ≈ 0.5036.