Linked Questions
27 questions linked to/from Using "implies" to refer to material conditional
89
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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 $q$":
$$\begin{array}{|c|c|c|}
\hline
p&q&p\implies q\\ \hline
T&T&T\\
T&F&F\\
F&T&T\\
F&...
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112
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7
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Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$)
Consider A $\Rightarrow$ B, A $\models$ B, and A $\vdash$ B.
What are some examples contrasting their proper use? For example, give A and B such that A $\models$ B is true but A $\Rightarrow$ B is ...
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102
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5
answers
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True or false? $x^2\ne x\implies x\ne 1$
Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:
$$x^2\ne x\implies x\ne 1$$
I immediately ...
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votes
1
answer
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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 ...
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5
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4
answers
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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 $ ...
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6
votes
4
answers
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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 ...
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8
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6
answers
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“$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$"? ...
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votes
1
answer
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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 ...
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votes
2
answers
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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 ...
- 312
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votes
4
answers
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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 ...
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votes
4
answers
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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 ...
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0
votes
6
answers
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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) ...
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7
votes
3
answers
409
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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"; "...
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5
votes
3
answers
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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 $(x = y)$ ...
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4
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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 ...
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