2

I got a question for implication, P implies Q is true when P is true, Q is true at the same time. P implies Q is false when P is true, but Q is false.

My question is if P is true, but sometimes Q is true and sometimes is false, then, can I say the implication is false or true.

For example: P is " It is a Christmas day" ; Q is "it is Wednesday"

Does P implies Q true or false? Since Christmas can be any day from Monday to Sunday. Sometimes Christmas is Monday, sometimes Christmas is Wednesday.

Which row of the following table it is belonged to?

the true table

Vincent
  • 23
  • 1
    It is false. It must always be the case that when $P$ is true then $Q$ is true. Otherwise, the case where $Q$ is false and $P$ is true would provide a counter-example. – John Douma Sep 08 '19 at 21:17
  • @JohnDouma yes. But the question is how to denote it in the truth table. – Mihail Sep 08 '19 at 21:18
  • 1
    The second row shows there is one case where $P\implies Q$ is false. – John Douma Sep 08 '19 at 21:22
  • 1
    You are mixing two different usage of "if..., then...". "If (today) is a Christmas day, then it is Wednesday" is not the same as "for every day (if it is a Christmas day, then it is Wednesday)". – Mauro ALLEGRANZA Sep 09 '19 at 14:27
  • @ John Douma Thank you for your answer. But may I know whether there is any book stating this idea. Because I need to quote it in my work. – Vincent May 10 '20 at 10:04

4 Answers4

1

A statement is true under a particular interpretation, i.e. under one row in the truth table. Truth is only defined relative to interpretations (which, in the case of propositional logic, are the assignment functions/valuation functions, where each row in the truth table stands for one assignment). Whenever you are talking about truth, you are talking about one row in the truth table.

Think of interpretations as situations. You already observed that sometimes the atomic statement $Q$ is true, and sometimes $Q$ is false -- its truth value changes depending on the situation. It's just the same for the implication $P \to Q$, whose truth value is determined by the changing truth values of $P$ and $Q$. When it's Christmas and it's Wednesday, this is a situation depicted by first row in the table, and the implication is true in that situation. When it's Christmas day and it's Monday, this corresponds to the second row in the truth table, and the implication is false in that situation. And of course there are also situations where it's not Christmas at all and the implication is true no matter what weekday it is, which is captured by the last two rows of the truth table. So sometimes $P \to Q$ is true and sometimes it's false, depending on which interpretation (row in the truth table) you are talking about.

A different question is whether a statement is valid (tautological), i.e. always true, independently of a particular interpretation. A statement is valid iff it is true under all possible interpretations -- that is, iff it has $1$ in all rows of the truth table. An implication is valid if whenever $P$ is true, $Q$ is true as well. So in order for the implication $P \to Q$ to be valid, we'd need to have that whenever it's Christmas, it's Wednesday. This is not the case with your example: There are situations where $P$ is true but $Q$ is false -- namely those situations where it's Christmas but not Wednesday --, and therefore, the implication $P \to Q$ is not valid.

So the answer to your question is: Sometimes $P \to Q$ is true and sometimes it's false, just like $P$ and $Q$ are sometimes true and sometimes false, depending on which interpretation (~= situation = assignment = row in the truth table) you are talking about. $P \to Q$ is not valid, since there are interpretations where it is false.

Natalie Clarius
  • 10,530
1

I would say that if it is possible for the truth-value of Q to vary with context, then propositional logic probably doesn't apply to the problem.

PMar
  • 36
0

Yeah, it seems counter-intuitive until you realize what "P implies Q" actually means. It does NOT mean that P causes Q, or that Q causes P, or that, historically, Q has followed P in time. It means ONLY that, at the moment in question (usually the present), we do NOT have both P being true and Q being false.

$P\implies Q\space\space \equiv \space\space \neg [P \land \neg Q]$

$P$ and $Q$ may be two, completely unrelated, logical propositions.

This is often given as The Definition of logical implication in introductory textbooks, but it and each line of the truth table can be derived from other widely accepted rules of logical inference. For formal proofs, see my brief blog posting, If Pigs Could Fly.

Dan Christensen
  • 14,529
0

According to the definition, a proposition P is defined as a sentence which is either True or False, but not both. And also it doesn't depend on the time. The truth value of a proposition is true if it is a true statement, and false if it is a false statement. For example, $1+3=5$ is a proposition because one can determine its value clearly. What a nice day ! is not a proposition because one can't decide about its value. The point is even, for example, $x+3=5$ is not a proposition, however, one can determine its value when $x=2$, or when $x\neq2$.

Maryam Ajorlou
  • 449
  • This doesn't answer the question of whether the implication above is true or false. – Natalie Clarius Sep 10 '19 at 10:17
  • I think my answer is natural. According to the definition of the proposition in propositional logic, one can not assign a truth value table to those sentences which their truth value is not clear, or they are sometimes true or sometimes false. – Maryam Ajorlou Sep 10 '19 at 10:33