Linked Questions

89 votes
16 answers

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

Provided we have this truth table where "$p\implies q$" means "if $p$ then $q$": $$\begin{array}{|c|c|c|} \hline p&q&p\implies q\\ \hline T&T&T\\ T&F&F\\ F&T&T\\ F&...
user701510's user avatar
  • 1,033
112 votes
7 answers

Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$)

Consider A $\Rightarrow$ B, A $\models$ B, and A $\vdash$ B. What are some examples contrasting their proper use? For example, give A and B such that A $\models$ B is true but A $\Rightarrow$ B is ...
user287424's user avatar
  • 1,239
102 votes
5 answers

True or false? $x^2\ne x\implies x\ne 1$

Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following: $$x^2\ne x\implies x\ne 1$$ I immediately ...
Chris's user avatar
  • 1,041
40 votes
1 answer

What's the difference between material implication and logical implication?

When I read the definitions of material and logical implications, they seem to me pretty much equivalent. Could someone give me an example illustrating the difference? (BTW, I have no problem with ...
kjo's user avatar
  • 14k
5 votes
4 answers

What's the difference between $\equiv$ and $\leftrightarrow$ in a formal proof?

Some texts and online sources seem to imply that they're interchangeable, while others say that there is at least a small nuanced difference between the two. For example, I've read that $ \equiv $ ...
Mirrana's user avatar
  • 8,891
6 votes
4 answers

Implication Logic Truth Table Explained

The question that has bothered me for a while has been answered and closed here (Implication in mathematics - How can A imply B when A is False?) and probably many other posts. Although all the ...
Javed Iqbal's user avatar
8 votes
6 answers

“$B$ does not follow from $A$” seems different from $\lnot(A\to B)$

I am a beginner in logic. I have a basic question about implication. If "$B$ follows from $A$", we can write $A\rightarrow B$; but what about "$B$ does not follow from $A$"? ...
Samuel's user avatar
  • 549
14 votes
1 answer

Distinction between equality, logical equivalence and biconditionality.

This is my first post here so I'm not sure about the proper etiquette, but I have quite a few questions all pertaining to one underlying concept, and I hope it's okay that I've grouped them all into ...
bbbbbbbgggf's user avatar
7 votes
2 answers

Are mathematical definitions logical equivalences or material equivalences?

Lets consider representing logical equivalence using the symbol $\equiv$ and material equivalence with the symbol $\longleftrightarrow$. I know that the formulas $P$ and $Q$ are logically equivalent ...
fire-bee's user avatar
  • 312
5 votes
4 answers

What an implication means when stating the solutions of an equation?

I am having trouble understanding what the following statement means: $$x^2=4 \implies x=2 \hspace{1em} \text{or} \hspace{1em} x=-2.$$ How can we relate this statement to a material conditional? I am ...
Anton's user avatar
  • 247
4 votes
4 answers

Translating "therefore" in propositional logic and using contradictions to prove an argument.

So I have the statements: A Football player is either running at the oval or sitting in the library. If he is running, then he is not reading a book. The player is reading a book. Therefore, he is ...
tinkertailor's user avatar
0 votes
6 answers

Understanding the p implies q statement

The p implies q statement is often described in various ways including: (1) if p then q (i.e. whenever p is true, q is true) (2) ...
Sandeep's user avatar
  • 269
7 votes
3 answers

Is $\curvearrowright$ a valid symbol for "implies that"?

I learned in high school from my favorite math teacher (who also has a PhD in mathematics) that the $\curvearrowright$ symbol means "implies that" (in German "daraus folgt"; "...
user3187119's user avatar
5 votes
3 answers

What is the name of the symbol $=$ in English?

What is the name of the symbol $=$ in English? Wikipedia says people use "equals sign" than "equality". Is it right? Is it to avoid ambiguity between the equality formula $(x = y)$ ...
Paalon's user avatar
  • 89
2 votes
4 answers

Propositional Logic Question Related to Understanding "→" and Tautologies

I should preface this with that I have never studied logic before. When answering my question, please assume that I know nothing about formal logic. Just now, I was reading a different question and ...
Gnosis's user avatar
  • 23

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