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I was looking at an exercise where it asked the following:

$$\begin{array}{ccc} p&q&p\rightarrow q \\ T&T&T \\ &\ldots \end{array}$$

So, for the third column, I just put $T$ which was correct but I didn't understand what $\rightarrow$ meant. I have seen $\implies$ but I haven't the arrow. Are they the same thing?

Thanks a bunch!

Jeel Shah
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3 Answers3

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The $\rightarrow$ symbol is a connective. It's a symbol which connects two propositions in the context of propositional logic (and its extensions, first-order logic, and so on).

The truth table of $\rightarrow$ is defined to be that $p\rightarrow q$ is false if and only if $p$ is true and $q$ is false.

Indeed this is the same meaning of $\implies$, but the difference is that $p\implies q$ is a statement about propositions, whereas $p\rightarrow q$ is a proposition. In some contexts, though, people don't make this distinction between material implication (the connective) and logical implication (the $\implies$ arrow). But they are not the same thing in every context of propositional logic.

Asaf Karagila
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  • I'll add here that A. N. Prior's textbook Formal Logic has parts of it which read like the following: "Rule: Detachment ($\alpha$, D$\alpha$D$\beta$$\gamma$ $\rightarrow$ $\gamma$) and (In all cases the sole rule beside substitution is E-detachment: $\alpha$, E$\alpha$$\beta$ $\rightarrow$ $\beta$. And in my opinion Prior's symbolism comes as clearer here than writing {E$\alpha$$\beta$, $\alpha$} $\vdash$ $\beta$, since the "$\rightarrow$" symbol suggests that one transitions from the left-hand side to the right hand side. – Doug Spoonwood Nov 10 '14 at 19:26
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    could you please elaborate on when exactly they are not the same thing? and as a non english native i have to ask what is the difference between statement and proposition. – peter Nov 24 '20 at 12:31
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    @peter: A proposition is a formal object in propositional logic (e.g. an atomic proposition is a variable, and if $p$ and $q$ are propositions, then $p\land q, p\lor q,\lnot p$ and $p\to q$ are all propositions), whereas a statement is simply something we as people make when we discuss mathematical objects. In other words, $p\to q$ is a proposition, whereas $p\implies q$ is simply a shorthand for "Every assignment which evaluates $p$ as true will also evaluate $q$ as true". The completeness theorem tells us that $p\to q$ is true for every assignment if and only if $p\implies q$. – Asaf Karagila Nov 24 '20 at 12:35
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It is a material conditional, or otherwise known as $p$ implies $q$, or if $p$, then $q$

The truth table for that is as follows

p  q  p implies q
T  T  T 
T  F  F
F  T  T
F  F  T

$\rightarrow$ can also be written as $\implies$.

In computer science, $p \implies q$ can be rewritten as (not p) or q, or !p||q

scrblnrd3
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1

Given $p$, then we have $q$.

or $p$ implies $q$.

The two arrows mean the same thing.

ireallydonknow
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