Discuss subjects related to logic and mathematics.
Averroes
Posts: 73
Joined: Sun Sep 10, 2017 9:01 am

So we have the following deductive sequents.
$$1. \sim A \vdash A \supset B \hspace{5em} (PMI)$$
$$2.\ \ \ A \vdash B \supset A \hspace{5em} (PMI)$$

Intuitively, by reference to the truth table, you can understand that they are true. For the first one, when $\sim A$ is true, $A$ is false and $A \supset B$ is true. For the second, when $A$ is true, $B \supset A$ is true. But this is semantic entailment, i.e there is no interpretation where the premises are true and the conclusion false. In SD, we need to prove these using inference rules.

We can prove these as follows.

$1. \sim A \vdash A \supset B$

$1. \sim A \hspace{10em} Assumption$
$\ \ \ 2. \ A \hspace{10em} Assumption$
$\ \ \ \ \ \ 3. \sim B \hspace{8em} Assumption$
$\ \ \ \ \ \ 4.\ A \hspace{9em} 1\ Reiteration$
$\ \ \ \ \ \ 5.\sim A \hspace{8em} 2 \ Reiteration$
$\ \ \ 6.\ B \hspace{10em} 3-5 \sim E$
$7. \ A \supset B \hspace{10em} 2,6 \supset I$

Compare with (the content is the same as above but different display):

Code: Select all

1. ~A                     Assunmption
2.A                   Assumption
3.~B              Assumption
4.A                1 Reiteration
5.~A                2 Reiteration
6.B                3-5 ~E
7. A ⊃ B               2,6 ⊃I

______________

$2.\ A \vdash B \supset A$

$1.\ A \hspace{10em} Assumption$
$\ \ \ 2.\ B \hspace{9em} Assumption$
$\ \ \ 3.\ A \hspace{9em} 1,\ Reiteration$
$3.\ B \supset A \hspace{10em} 1,2 ⊃I$

Compare with (the content is the same thing as above):

Code: Select all

1. A                Assumption
2. B             Assumption
3. A              1 Reiteration
3.B ⊃ A            1,2 ⊃I

__________________

It is called LaTex notation. For example, the first deduction above looks like this in the post editor:

Code: Select all

$1. \sim A \hspace{10em} Assumption$
$\ \ \ 2. \ A \hspace{10em} Assumption$
$\ \ \ \ \ \ 3. \sim B \hspace{8em} Assumption$
$\ \ \ \ \ \ 4.\ A \hspace{9em} 1\ Reiteration$
$\ \ \ \ \ \ 5.\sim A \hspace{8em} 2 \ Reiteration$
$\ \ \ 6.\ B \hspace{10em} 3-5 \sim E$
$7. \ A \supset B \hspace{10em} 2,6 \supset I$
I have added functionality to the program of the forum such that when a poster type some specific codes within two dollar signs, an effect is produced when displayed. You can also display a lot of cool math stuffs, like matrices. Let us display a magic square.

$$\begin{vmatrix} 12 & 1 & 14 & 7 \\ 13 & 8 & 11 & 2 \\ 3 & 10 & 5 & 16 \\ 6 &15 & 4 & 9 \end{vmatrix}$$

Code: Select all

$$\begin{vmatrix} 12 & 1 & 14 & 7 \\ 13 & 8 & 11 & 2 \\ 3 & 10 & 5 & 16 \\ 6 &15 & 4 & 9 \end{vmatrix}$$
Later, you will be studying predicate logic. Such sentences as $\forall x \exists y (Lxy)$. It is so easy to write that now. It looks like this in the post editor:

Code: Select all

$\forall x \exists y (Lxy)$

The syntax of the language is shown here: https://www.abisource.com/~msevior/math-reference.html

One just has to put this syntax between a single pair of dollar signs or a double pair of dollar signs. Integrals, summations, greek alphabets, and all mathematical symbols can be displayed nicely with this language and functionality. What do you think, isn't this cool?

Averroes
Posts: 73
Joined: Sun Sep 10, 2017 9:01 am