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logic

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 Editor
Sabine Wisbacher
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 French
logique
 German Logik
 
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References

[1]    Cohnitz, Daniel & Estrada-Gonzáles, Luis (2019). An Introduction to the Philosophy of Logic. Cambridge: Cambridge University Press

[2]    Dettmer, H. William (2007). The logical thinking process. Milwaukee, Wis.: ASQ Quality Press

[3]    Mukerji:, Nikil (2017). 10 Gebote des gesunden Menschenverstands. Berlin: Springer Verlag

[4]    Jaquette, Dale (Ed.) (2006). Philosophy of logic. Amsterdam: Elsevier/North Holland

[5]    Spitzer, Manfred (2015). Denken. Stattgart: Schattauer [u. a.]

[6]    Müller, Sandra (2016) “Philosophie der Logik: Klar, ist ja logisch! Aber warum eigentlich?”. Focus Online, 14.12.2016

[7]    The Stanford Encyclopedia of Philosophy [Online].Stanford: Stanford University. https://plato.stanford.edu/ , [Consulted: 15.12.19, 17.12.19, 20.12.19, 22.12.19]

[8]    Logical nihilism [Online]. University pf Notre Dame: Cutis Franks department of philosophy. https://www3.nd.edu/~cfranks/nihilism.pdf, [Consulted: 22.12.19]

[9]    The Internet Encycolpedia of Philosophy [Online]. https://www.iep.utm.edu/ded-ind/ [Consulted: 22.12.19]

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Logic
Sabine Wisbacher (29.12.2019, contribution elaborated within the Seminar "A Journey through Philosophy and Information" facilitated by J.M.Díaz at the Hochschule München)

Abstract

This article discusses the evolution of logic and the questions raised by it. It gives an overview of the development of logic and its application in computer science with special focus on the potential of creating artificial intelligence. The questions dealt with in this context are to what extent logic can be viewed as a strictly mathematical system, whether emotional aspects can be considered as part of a logical structure and if there are other influences that have to be regarded.


An Introduction of the developement of logic

When you think about logic, everyone seems to have an idea of what the term means and what is logical and what isn’t. But when it comes to finding a definition of logic itself, it seems to become more difficult.

Generally speaking, logic is about arguments and the form they may take. It is the science of valid inferences, meaning the science of drawing valid conclusions by deducing or inducing from incoming information or statements. It could be described within the context of information, with the information being used as a rope and knowledge as a net, logic would be what determines the shape and pattern of the knots of that net. Since the human brain represents nothing more than some form of a net, it can be said that our thinking process is ultimately based on a sum of logical structures. Thus one would think that it can be easily reproduced. But there are many things that influence those logical structures, and logic itself is not that distinct.

In the first instance there are two ways to draw a conclusion, the deductive reasoning and the inductive reasoning. The inference of a deductive argument would be valid if the two premises it is based on were true. An example for a deductively valid argument would be: “It is dark outside at night. It is night. So, it is dark outside”. In this type of argumentation, as long as the assumptions on which the conclusion is based on are true, there is no doubt about the veracity of the statement, as long as the structure of the logical connection is assumed to be indisputable (Stanford Encyclopedia of Philosophy, 2019). Inductive reasoning, on the other hand, is based on the arguers assumption that if the premises are all true, it is unlikely that the conclusion will not occur. An inductively strong argument would be: “Every tree I’ve ever seen is green. Therefore, all trees must be green.” As inductive reasoning does not guarantee a valid conclusion it has not always been considered for logic. The theory of logic in its earlier days dealt mainly if not exclusively with deductive arguments (Stanford Encyclopedia of Philosophy, 2019).

The first to formulate a particular form of logic was Aristotle. He introduced the rule of the excluded middle, which is the most basic form of deduction, stating that if there are two contradictory statements either one or the other is true (Müller, 2016).

Φ  (-Φ)

“If it is dark it, is not not dark”

He also introduced his rules of syllogism, which are about deducing a conclusion based on at least two propositions. The proposition would be divided into a minor and a major premise.

(p→q),(q→r) ⊢ (p→ r)

Major Premise: All humans are mortals
Minor Premise: I am human
Conclusion:        I am mortal

After Aristotle, during the new philosophical movements of the Hellenistic period Stoicism was invented by Cyprus, Cleanthes and Chrysippus. Stoicism introduces five indemonstrable argument forms that are also structured in a deductive form (Stanford Encyclopedia of Philosophy, 2019).

(p→q),p ⊢ q
“If it is dark it is night. It is dark. Therefore, it is night.”
modus ponens (en. “imposing mode”)

(p→q),-q  (-q)
 “If it dark it is night. It is not dark. Therefore, it is not night.”
modus tollens (en. “deriving mode”)

-(p \wedge  q),p  (-q) or  (p ∨ q),p (-q)
“It is not dark if it is day. It is dark. Therefore, it is not day”

(p  q),-p   q
“It is not dark if it is day. It is not dark. Therefore, it is day”

 

In extension to those basic statements one can add the expressions of modal logic, deontic logics, temporal logics and doxastic logics. Rather than working with a simple structure of statements, like: “The flower is blue” it is now possible to argue with premises that hold additional information about the modality of the information in the statement, as in “It is possible that the flower is blue”. While with modal logic a necessity or possibility is added to the premises, deontic logic deals with whether something is obligatory, permitted or forbitten. With Temporal Logic, a time in which a premise will be or has been true is added to the argument and doxastic logic deals with arguments that are only based on something that is believed by someone (Stanford Encyclopedia of Philosophy, 2019).

Modal Logic:

It is necessary that…

It is possible that…


◊A=-□-A
“It is possible that A” means the same as “It is not necessarily not A”

Deontic Logic:

O

P

F

It is obligatory that...

It is permitted that…

It is forbitten that…

 

PA = -O-A
“It is permitted that A” means the same as “It is not obligatory not A”

FA = O -A
“It is forbitten that A” means the same as “It is obligatory that it is not A”

Temporal Logic:

G

F

H

P

It will always be the case that…

It will be the case that…

It has always been the case that…

It was the case that…

 

FA = -G-A
“It will always be the case that A” is the same as “It won’t be the case that it is not A”

HA = -P-A
“It has always been the case that A” means the same as “It wasn’t the case that there wasn’t A”

Doxastic Logic:

xB

It is believed that…

 

xB{b_1,b_2,...b_n}
“It is believed that B”, part of which is that “It is believed that b_1,2...n”

Now with this we are already able to discuss more complex statements within the frame of deductive reasoning but we are still in a system of formal logic, meaning we are still using very specific syntax and clearly defined proof procedures. To be able to understand and analyze actual arguments, one would have to deal with informal logic. Informal logic finally moves away from pure logica docens (en. “logic in theory”) to logica utens (en. “logic in use”) as it is not so much about finding theorems that connect conclusion and premise but rather about finding patterns in actual argumentations (Stanford Encyclopedia of Philosophy, 2019).

Informal logic adds many new variables to the logic structure. A very interesting addition that is made by informal logic is the consideration of emotions in an argument (Stanford Encyclopedia of Philosophy, 2019). This is the first influence on logic that cannot be simply explained or represented mathematically and which raises further questions about the veracity of the entire logical structure. Logic, when influenced by emotions, depends on the personality of the arguer, on his perspective on the topic and thus on the sum of his experience made prior to the argument.

A very similar approach is followed by the theory of fuzzy logic, taking into account that our decisions are often accompanied by uncertainty. Not only can the initial situation be too complex to describe precisely and the conclusion depend on what we want, but there is also the possibility of future events influencing the present argument which cannot be predicted. Thus we talk about logical pluralism, the theory on the existence of more than one possible true logic.

Logic nihilism even states that there is no logic at all. Although it was developed from the more recent discussions about logical pluralism, it is neither part of it nor part of logical singularism (Universitiy of Notre Dame - Curtis Franks department of philosophy, 2019).

So the essential questions seem to be at first whether logic is revisable and secondly whether logic is independent from experience or not. For the first question we would have to say yes, since logic has changed and evolved substantially over the decades, however, there are also some basic elements of logic that we still consider to be true. To answer the second question, we must first understand what it is that makes the laws of logic true, the metaphysics of logic. This brings us to the question of reality, for which there are many different answers (Cohnitz & Estrada-Gonzàles, 2019).

Ultimately, the different approaches to the topic of logic and the different theories of logic itself as well as the different possible underling theories of reality lead to the fact that it becomes impossible to find a defined foundation for the assertions of logic. The many factors that are involved in real arguments of logic also make it increasingly complex to comprehensively understand and prove conclusions.

________________________________________________________

[1]    Cohnitz, Daniel & Estrada-Gonzáles, Luis (2019). An Introduction to the Philosophy of Logic. Cambridge: Cambridge University Press

[2]    Dettmer, H. William (2007). The logical thinking process. Milwaukee, Wis.: ASQ Quality Press

[3]    Mukerji:, Nikil (2017). 10 Gebote des gesunden Menschenverstands. Berlin: Springer Verlag

[4]    Jaquette, Dale (Ed.) (2006). Philosophy of logic. Amsterdam: Elsevier/North Holland

[5]    Spitzer, Manfred (2015). Denken. Stattgart: Schattauer [u. a.]

[6]    Müller, Sandra (2016) “Philosophie der Logik: Klar, ist ja logisch! Aber warum eigentlich?”. Focus Online, 14.12.2016

[7]    The Stanford Encyclopedia of Philosophy [Online].Stanford: Stanford University. https://plato.stanford.edu/ , [Consulted: 15.12.19, 17.12.19, 20.12.19, 22.12.19]

[8]    Logical nihilism [Online]. University pf Notre Dame: Cutis Franks department of philosophy. https://www3.nd.edu/~cfranks/nihilism.pdf, [Consulted: 22.12.19]

[9]    The Internet Encycolpedia of Philosophy [Online]. https://www.iep.utm.edu/ded-ind/ [Consulted: 22.12.19]


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