Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

http://i.stack.imgur.com/tFzfO.png

Why is (a) true if (if) is put in there?

share|improve this question

4 Answers 4

Recall that a false statement implies everything. Remember that a statement of the form $$\text{If }P\text{ then }Q$$ is false only in the case when $P$ is true but $Q$ is false.

If $P$ is false then the implication is immediately true, and if $Q$ is true then the implication is immediately true. In the case of (a), both things occur.


See also:

  1. The meaning of implication in logic
  2. not understanding this row of truth table for logical implication
share|improve this answer
    
Hence you can define disjunction as "if not p, then q". –  Doug Spoonwood Dec 17 '12 at 19:51

Note that if we let $$P = 2 + 2 = 5,\; \text{ and}\;\;Q = \text{ triangles have three sides}$$

Then $\;$"$\text{If} \;P \;\text{ then}\;\; Q$"$\;$ is true whenever $\;P\;$ is false, or whenever $Q$ is true (in this case, it happens that both $P$ is false and $Q$ is true).

Equivalently $\;$"$\text{NOT}\; P \; \text{ OR}\;\; Q$"$\;$ is true whenever $\;P\;$ is false, or whenever $\;Q\;$ is true. (In this case, it happens that to be that $P$ is false and $Q$ is true.)

So there is a strong connection between the connectives "OR" = "$\lor$" and "IF (- THEN)"= "$\rightarrow$"; specifically:

$$\lnot P \lor Q \;\iff\; P \rightarrow Q$$

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ $enter image description here

share|improve this answer

In everyday usage, "if $A$ then $B$" suggests a causal or temporal relationship, e.g. $A$ causes $B$. If, over time, $A$ is always false, it would make no sense to say that $A$ causes $B$. There are no causal or temporal relationships in propositional logic, however. In propositional logic $A\rightarrow B$ is simply defined to be $\neg (A\wedge \neg B)$. If $A$ is false, then $\neg (A\wedge \neg B)$ must be true.

share|improve this answer

The logical statement if P then Q is only false is P if true and Q is false.

In this problem P is the statement 2+2=5, while Q is the statement "triangles have three sides". Clearly P is false but Q is true, yet the outcome of the logical statement if P then Q remains true by the line above.

see the section on logical implication

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.