Friday, November 20, 2015

Some Thoughts about Logic...

Ramon Llull (c. 1232 - c.1315),
besides many other things also a philosopher and logician

OK, let's try to find an answer to the question why are we putting so much emphasis on logic in that course..

What's the general objective of logic?

The main objective of a logic (...and there is not a unique logic but many) is to express by means of a formal language the knowledge about certain phenomena or a certain portion of the real world.  To provide a complete description of the real world is clearly impossible, and maybe also useless. However, typically one is interested only in a portion of the real world. Thus, logic serves us to create a model of the real world. For this purpose, we adopt a language. The phrases of this language are used to describe objects of the real world, their properties, and facts that hold. This language can be informal (natural language) or a formal (mathematical logic). 

So, we are creating a formal (mathematical) model of the real world that serves our purpose. The mathematical model constitutes an abstraction of a portion of the real world. It represents the real world in the shape of mathematical objects, such that sets, relations, functions, etc.

But, you might say, we don't need logic to create a model. Especially in everyday life, we don't need a mathematical model to survive.

However, in science in order to show that a certain argumentation is correct, we provide mathematical models that describe in an abstract and concise manner specific aspects of the real world. Based on these mathematical models, logic allows formally correct reasoning.

Logic allows us reasoning

For this purpose, a precise set of deterministic rules, called inference rules, provide the basic reasoning steps which are considered to be correct by everybody. A formally correct reasoning allows to show that a certain knowledge is a logical consequence of a given set of facts. Furthermore, correct reasoning chains are constructed by concatenating applications of simple inference rules, that allow to transform the initial knowledge into the conclusion one wants to derive.

The connection to the real world

The connection between the real world, the formal model and the formal language used to describe the model is the interpretation. Considering model-theoretic semantics, a formal interpretation is the mapping of our logical language to a (simpler) mathematical model (as e.g. set theory). The formal interpretation allows us to define a notion of truth. The main objective of the formal interpretation is that it allows to define when a formula is true in a model.

Finally, the goal of logic is the formalization of what is true or false on a particular world.

...and we are not the first to find out about that :)

According to medieval philosopher Ramon Llull (who also was a poet, a theologian, a mystic, a mathematician, and a martyr):
Logic is „the art and the science to distinguish between truth or lie with the help of reason, to accept truth and to reject lie.

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