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For any given interpretation, a given formula is either true or false under it. [65] [75] ... If p then q; and if p then r; therefore if p is true then q and r are true
is true only if both A and B are false, or both A and B are true. Whether a symbol means a material biconditional or a logical equivalence , depends on the author’s style. x + 5 = y + 2 ⇔ x + 3 = y {\displaystyle x+5=y+2\Leftrightarrow x+3=y}
Then if is true, that rules out the first disjunct, so we have . In short, P → Q {\displaystyle P\to Q} . [ 3 ] However, if P {\displaystyle P} is false, then this entailment fails, because the first disjunct ¬ P {\displaystyle \neg P} is true, which puts no constraint on the second disjunct Q {\displaystyle Q} .
If the variable that is the final conclusion of a formula takes the value true, then the whole formula takes the value true regardless of the values of the other variables. Consequently if A is true, then Φ, Φ −, Φ + and Φ − →(Φ + →Φ) are all true. So without loss of generality, we may assume that A is false.
The proposition to be proved is P. We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, Q and ¬Q, and appealing to the law of noncontradiction. Since assuming P to be false leads to a contradiction, it is concluded that P is ...
For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris ...
In the most basic sense, there are four possible outcomes for a COVID-19 test, whether it’s molecular PCR or rapid antigen: true positive, true negative, false positive, and false negative.
Therefore φ follows from this formula. It is also easy to show that if the formula is false, then so is φ. Unfortunately, in general there is no such predicate Q'. However, this idea can be understood as a basis for the following proof of the Lemma. Proof. Let φ be a formula of degree k + 1; then we can write it as