Linked Questions

96
votes
21answers
15k views

In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?

I am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is: $$\begin{array}{|c|c|c|} \...
43
votes
6answers
9k views

“If everyone in front of you is bald, then you're bald.” Does this logically mean that the first person is bald?

Suppose we have a line of people that starts with person #1 and goes for a (finite or infinite) number of people behind him/her, and this property holds for every person in the line: If everyone ...
4
votes
4answers
2k views

Conditional statements?

My textbook states that for the conditional statement "p implies q", "p is a sufficient condition for q and q is a necessary condition for p." How is this so? One might be lead to believe that p ...
4
votes
4answers
599 views

Are P and Q assumed to be independent?

Suppose P and Q are each true. Classify the following statement as True or False (choose one): If (not P) then (not Q). I put False, the answer was TRUE I assumed that P and Q were independent ...
4
votes
3answers
810 views

Implication and the Deduction theorem

Implication Implication $A\rightarrow B$ is on occasion introduced as shorthand for $B \vee \neg A$. Their truth tables certainly match, and regardless of the valuations of $A$ and $B$, mappings to ...
4
votes
6answers
468 views

How does one know if $A \implies B$ (an implication) is true without knowing if $B$ (the consequent is true) is true?

This might be a weird question but I was trying to distinguish the difference between an implication and modus ponens. I think the distinction is clear in my head (but I have something missing), modus ...
1
vote
6answers
585 views

When is implication true?

If we have $p\implies q$, then the only case the logical value of this implication is false is when $p$ is true, but $q$ false. So suppose I have a broken soda machine - it will never give me any can ...
2
votes
2answers
1k views

Why will an implication be true when the hypothesis is false? [duplicate]

When $p$ and $q$ are propositions such that "if $p$ then $q$" (which is an implication) is false only in the condition when $p$ (which is hypothesis) is true but $q$ (which is conclusion) is false. ...
1
vote
2answers
293 views

A counterexample for equivalency of $P \implies Q$ and $\neg Q \implies \neg P$?

Suppose P is when it rains and Q is when a garden's soil is dry. Now, I cover the garden with a cover. It rains. So I say a true statement: If it rains the garden's soil is not (won't be) wet. And its ...
0
votes
3answers
344 views

Does it make sense to say that a number is larger than every member of the empty set? [duplicate]

Can any number be said to be larger than every member of the empty set? I have been asked to compare a positive integer to the empty set, but I'm not sure how to compare...
0
votes
2answers
255 views

Defining Material Conditional

Defining the conditional statement "$\phi \implies \psi$" so as to capture what is meant by implication requires that we ignore the issue of causation. From Keith Devlin's Introduction to ...
0
votes
2answers
254 views

Why is $a \implies b$ is true when $a$ is false [duplicate]

I understand that: $True \implies True$, is true. $True \implies False$, is False. But why is it that $False \implies True$, is True. and $False \implies False$, is True. If $a$ is false I don'...
-4
votes
2answers
105 views

Reference request: How to understand $p\to q$? [duplicate]

I have studied calculus, linear algebra, some naive set theory, some analysis. These were not satisfactory at all. I want to find a way to understand the mathematical system in an axiomatic way. The ...
1
vote
1answer
94 views

Deductive Logic : If A -> B it can be deduced neg(A) -> neg(B)

I am having a hard time proving the following $$ (a \to b) \vdash (\lnot a \to \lnot b) $$ I followed the book advice and first proved that $ (a \to b) \vdash (\lnot \lnot a \to \lnot \lnot b) $ ...
-1
votes
2answers
148 views

Should we change the truth table for the material conditional?

Having studied logic, I still cannot understand the conditional. At first, it was because (as with most things I learn) it was a problem with my understanding. I now believe it is because there is an ...

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