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'second way of understanding logic is as
a formal or mathematical system we have
a precedent of this in the history of
philosophy as well while formal
mathematical logic really becomes its
own kind of subject in maybe say the
19th century in the Medieval World we
had incredibly robust logical analysis
and medieval logicians were really
worried about the structure of argument
in the way that contemporary
mathematical logicians are even if you
go back to Aristotle we're talking about
the form of argument rather than the
content of argument which is that first
distinction you need if you're going to
study formal logic when most
philosophers today talk about logic they
are talking about formal logic so logic
in our second sense and not that first
sense in making you a better thinker so
the study of formal logic really started
to flourish in the 19th and 20th
centuries one of the key figures for
this is gotb frga frer was the inventor
of something called the B Griff shrift
this translates to a concept script from
German his idea was to create a formal
notation which could represent arguments
especially the kinds of arguments that
you would find in mathematics so
logicians would actually translate
arguments into this formal
representation and then using
well-defined rules try to assess whether
or not they were valid or invalid it can
feel tedious to try to spell out all of
these rules but this level of precision
is useful for the sort of work that
logicians want to do formal logic really
originally was supposed to provide a
grounding for all of mathematics but
this actually proved a little bit
difficult due to things like Russell's
Paradox and girdle's incompleteness
Theorem so Russell's Paradox showed an
essential contradiction in fraga's
system it actually showed that fra made
inconsistent assumptions as he was
laying out his entire formal system and
then girdle's incompleteness theorem
shows that there are some truths of
mathematics which cannot be proven but
formal logic is not only useful because
it could provide a grounding for
mathematics it's also useful for our
kind of prise modeling of other
philosophically interesting Concepts and
that's why we have modal Logics temporal
Logics epistemic Logics fuzzy Logics and
so on some philosophers also spend their
time thinking about paradoxes and a
paradox is where we get an absurd
conclusion from plausibly true pries and
reliable inference rules they seem to
show us that there's a problem with the
way we think about reasoning here's a
famous example the Liar's Paradox so
consider the sentence this sentence is
false now let's ask ourselves an OB
question so is that sentence true or
false if you say that it's false then
the sentence is true because after all
it says of itself that it's false and
it's false so it's true that's absurd
right but if it is true then we get the
inverse reasoning since it says of
itself that it's false and we've suppos
that it's true then it actually
describes itself incorrectly and a
sentence that describes itself
incorrectly is false which again is
absurd the liar traps us in a kind of
pattern of circular reasoning we can go
back back and forth concluding that it's
true and then it's false and it's true
and then it's false and if you accept
some common inference rules like modus
ponents you actually get to this
principle called explosion the idea
behind the principle of explosion is
that when you when you've arrived at a
contradiction you can actually prove any
sentence at all and that obviously is
absurd not every sentence is true so
paradoxes tell us something important
but actually figuring out what they tell
us about our language or about our logic
is a matter of debate I didn't even
bother to count how many different kinds
of solutions to the paradoxes there are
on offer but I would guess probably in
the hundreds at least when it comes to
theories when you look at all the
variance and subvariant of the solutions
so to say that there's any kind of
consensus I think would be an
overstatement so logicians study good'

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```markdown
## Logic: The Art of Thinking

### Overview of Logic
Logic is fundamentally the study of reasoning and can be divided into two main categories. The first category focuses on the art of thinking and aims to improve one's thinking skills. A notable historical reference is the 17th-century book by Arnauld and Nicole titled *Logic, or the Art of Thinking*, which serves as a manual for developing rational thought. The authors intended to guide readers in cultivating true beliefs while steering clear of false ones.

### Importance of Critical Thinking
In modern contexts, the principles of logic are often discussed under the umbrella of critical thinking. This includes familiar concepts such as fallacies, which are flawed patterns of reasoning. Understanding these fallacies is crucial because they can lead to incorrect conclusions, even if the premises are true. For instance, the fallacy of guilt by association illustrates how associating a belief with a disreputable person does not inherently disprove that belief.

### Cognitive Biases and Their Impact
In addition to fallacies, cognitive biases play a significant role in our reasoning processes. While some research on cognitive biases has faced scrutiny, many biases are well-documented and can hinder our ability to assess truth accurately. Learning about these biases, much like understanding fallacies, enhances our critical thinking skills and helps us recognize flawed reasoning in our own thoughts.

### Application in Philosophy and Beyond
The study of logic and critical thinking is not only beneficial for philosophy but also for any field of study. By improving our reasoning abilities, we can become better thinkers overall. This foundational knowledge is essential for anyone embarking on a philosophical journey or seeking to enhance their intellectual capabilities.
```
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