Logics
[gL.edu] This article gathers contributions by Alexander Tabelander, developed within the context of the Conceptual clarification about "Information, Knowledge and Philosophy", under the supervisión of J.M. Díaz Nafría.
Overview
This article deals with the concept of logic. To get a brief overview of the evolution of logic since the ancient times a summary of the history is given. Then the general concept of philosophical logic is stated including its divisions in formal and informal logic and classical and non-classical logic. This leads to a very important part of the philosophical logic, the epistemic logic or the logic of knowledge. The following chapter only deals with the epistemic logic. The article is concluded with some problems of this type of logic and a general conclusion.
Introduction
Logic is a very important part of our lives and is used in many different fields like mathematics, philosophy, and more recently artificial intelligence. For a better understanding of the following chapters about logic and logic in the field of epistemology some terms must be introduced. First the term epistemology has to be explained. The word comes from the greek terms “episteme" and "logos". Episteme can be translated as "knowledge", "understanding", or "acquaintance" whereas logos can be translated as "account", "argue", or "reason". Thus, epistemology is often regarded as the study of knowledge. Logic in general is the study of correct reasoning. In many disciplines, including philosophy, mathematics, computer science, and linguistics, logic is a crucial part. Extended logics take the fundamental assumptions of classical logic and apply them to additional areas like metaphysics, ethics, and epistemology.
History of Logic
A short overview of the history of logic is given to boost their understanding and get to know some of the most important contributors to the evolution of the logic trough time. The history of logic can be mainly divided into three parts: the ancient-, medieval-, and modern logic. The basics of our modern logic were laid in ancient times.
Ancient Logic
Big contributions were made in China and India but the most important contributors of the logic in the western hemisphere were the philosophers of Ancient Greece. The development of logic in Ancient Greece can be divided in two parts the logic before and after Aristotle. To name some of the important philosophers before Aristotle and their contribution to logic: The first who used deductive reasoning in a mathematical context were Thales and Pythagoras. The next important philosopher who contributed to logic was Plato. It is not recorded if Plato contributed something to the so-called formal logic, but he contributed something to logic in general by raising three important questions:
1. What is it that can properly be called true or false?
2. What is the nature of the connection between the assumptions of a valid argument and its conclusion?
3. What is the nature of definition?
The next and according to literature most important philosopher, with a huge impact to this day, was Aristotle. It’s said that Aristotle was the first great logician in the history of logic. His work was collected in the “Organon”, which can be translated to “Instrument”. His concept of logic is built around the term “sullogismos“, in english this means “deduction”. He stated a very famous testimony: A deduction is speech in which, certain things having been supposed, something different from those supposed results of necessity because of their being so. Another very important school of logic were the Stoics. The Stoics said that the cosmos functioned rationally. The area of philosophy that investigated reason was logic.[1][2]
Medieval Logic
The next step in the history of logic is the medieval logic. The medieval logic is strongly based on Aristotle’s theories. The theory of Aristotle were the most important logical theory during the middle ages and for a long time it was practically synonymous with logic as a discipline. Modal logic had started to evolve by the second quarter of the fourteenth century, and the notion of the modal syllogism was developed using new distinctions. The first person to just reject Aristotle's thesis in favor of a fresher, more methodical explanation was the philosopher Ockham. But it was the philosopher Buridan who took a great step to systematize all of logic.[1][3]
Modern Logic
Which leads to the last step in the evolution of logic the modern logic. The evolution of modern logic can be roughly be divided into five phases:
- Embryonic period: In this time the idea of a logical calculus was explored and refined, especially by Leibniz, but no schools were established.
- Algebraic period: There were more practitioners like Bool and Schröder and more development around logic. Their goal was to create a calculus that could be used to formalize reasoning about classes, propositions, and probabilities.
- Logicist period: The goal at this time was to unify the logic of all mathematical and scientific speech into one cohesive system that, based on the fundamental tenet that all mathematical facts are logical, forbade the use of any language that was not logical.
Frege, Russell, Whitehead and the early Wittgenstein were the leading logicians of this period. - Metamathematical period: The development of metalogic, the finitist system of Hilbert, the non-finitist system of Löwenheim and Skolem, and the fusion of logic and metalogic in the work of Gödel and Tarski all happened during this period.
- Period after World War II: Model theory, proof theory, computability theory, and set theory are the four interrelated but distinct fields of study that emerged from the study of mathematical logic.
[1][4]
Philosophy of Logic
Logic is a wide field of studies with many categories and subdivisions. It is sometimes very difficult to distinguish between them. Logic is also a crucial part in many disciplines like philosophy, mathematics, computer science and linguistics. This article only deals with the philosophy of logic and some categories of this type of logic including some important concepts and philosophers. Logic is at least closely connected to the study of sound reasoning philosophically.
An epistemic, mental activity is reasoning. Thus, logic and epistemology are at the very least strongly related.
The philosophy of logic can be manly divided in the formal and informal logic and the formal logic can be divided again in classic and non-classic logic.
First the difference between the formal and informal logic is stated. The formal logic uses a formal language to state arguments whereas the informal logic uses a natural language to state arguments. One thing they have in common is that both types try to assess an argument's validity. This leads to some advantages and disadvantages of both forms.
Formal Logic
Advantages:
• more precise
• paradigmatic form
Disadvantages:
• is constrained in terms of the criteria that are utilized to produce precise criteria for evaluation
• theories must be translated into a formal language
• formal languages are restrictive
Informal Logic
Advantages:
• takes many criteria into account, therefore, is more flexible
• builds logic suited to this purpose
Disadvantages:
• less percise
• problems of natural language like ambiguities and vague expressions
To conclude if precision is necessary like in sciences the formal logic is needed. It is more precise and the validity of its premises guarantees the validity of its conclusion. The informal logic on the other hand extends the formal logic because of the use of a natural language and therefore has many cases where it can be used.[5][6]
The next step is the division of the formal logic into the so-called classical logic and the non-classical ones. The non-classical logic can be divided again into the extended logic and the deviant logic.
Classical Logic
Propositional logic and first-order logic are the two main types of classical logic. It is employed in many different domains and is typically regarded as the archetypal form of logic by philosophers. It describes the function these notions play in drawing correct conclusions, and is focused on a small set of fundamental logical concepts. These fundamental concepts include propositional connectives like "and", "or", and "if-then", as well as quantifiers that represent concepts like "all", "some", and others.[7]
Non Classical
Extended logics and deviant logics are two more divisions of non-classical logic. Extended logics build upon classical logic, whereas deviant logics challenge some of its fundamental premises.[8][9]
Extended Logic
The basic concepts and terminology of classical logic are accepted by extended logics. The validity of classical logic's theorems in them serves as evidence of this. However, they go beyond classical logic by introducing new symbols and theorems. One important part of the extended logic is the modal logic and so the epistemic logic or the logic of knowledge.Modal logic, when used narrowly, is the study of reasoning including the words "necessarily" and "potentially." The phrase "modal logic," however, is used more widely to refer to a group of logics that share a common set of rules and a wide range of symbol types.There are numerous of them, such as logics for believing, tense and other temporal expressions, deontic (moral) phrases like "it is required that" and "it is permitted that," among others. In the formal examination of philosophical reasoning, where terms from the modal family are both frequent and perplexing, a comprehension of modal logic is very helpful. Important uses for modal logic may be found in computer science.One important part of modal logic is the modal approach of epistemic logic In the second part of this article this type of logic and its core statements are explained.[8]
Deviant Logic
Deviant logics are types of logic because they aim to do the same thing as classical logic, namely, to explain which inferences are reasonable. By providing a distinct account, they diverge from classical logic. The law of excluded middle, a legitimate type of inference in classical logic, is rejected by deviant logic, for instance.[8][9]
Epistemic Logic
A branch of epistemology called epistemic logic is concerned with logical approaches to knowing, belief, and concepts that are connected to these. Although any logic having an epistemic interpretation can be referred to as an epistemic logic, modal logics are now the most common form of epistemic logic. To show the concept of the epistemic logic some examples are stated. The examples are shown in the formal operators that are used. When talking about epistemic logic there are two principles. One is where a subject knows something, and one is where a subject beliefs something. Here are some examples to clarify how the basics of epistemic logic work: When “a” knows “P”, then “P” is true. And when “a” knows “P” but also “Q”, then “a” knows “P” and “Q”. Both of these statements are correct. But the following is not correct and wouldn’t work: I do not know “P”. So, I know that “P” is not. But in the world of epistemic logic one usually doesn’t write sentences but letters, like some people are more familiar with from mathematics. For “knowledge” and “belief” the letters “K” and “B” are used. So, for the first example ““a” knows “P”” one would write KaP If “a beliefs P” then that would be BaP. To write the complete first example “When “a” knows “P”, then “P” is true.” one again takes the KaP and to write “then “P” is true” one writes “→P” so the complete sentence of the first example can be condensed into KaP→P. If a knows that Q is a logical consequence of P, then it is not possible for P to be true but Q to be not true. That means when a knows that P is true, all logical consequences of P must also be true. So if a knows P, then a also knows Q. To make this more understandable an easy mathematical example is shown next: Let’s say a is a person named Stella, P is the definition of a prime number, a number only exactly dividable by itself and 1, and Q is the knowledge that 7 is a prime number. When Stella (a) knows the definition of a prime number (P), Stella would also know that 7 is a prime number (Q). So there is also no possibility for Stella (a) to know the definition (P) but not know that 7 is a prime number (Q). Unfortunately, this logic can also be prone to mistakes especially when we are talking about beliefs and not knowledge. Let’s take Spiderman as an example. Stella beliefs that Spiderman can shoot spider webs. BStellaWebs(Spiderman) Stella also beliefs that Peter Parker can not shoot spider webs. BStella¬Webs(Peter Parker) Therefore, Stella beliefs that Peter Parker is not Spiderman. BStella(Spiderman)≠(Peter Parker) Which would be false. This is also known as the epistemic fallacy or the Masked-Man fallacy. The fallacy happens because of an immediate identity between Stella’s knowledge of an object with the object itself, which isn’t allowed under the law of “The identity of indiscernibles”. So unfortunately, this method of reasoning is not fool proof.[10]
Conclusion
To conclude, logic is a very important part of our daily lives and is indispensable in many areas like philosophy, mathematics, computer science, linguistics and so on. The beginning dates back to the ancient times, but the most was done in more recent times. Every logic has its own unique problems, therefore it is important that before a certain type of logic is used it was assessed before that this type of logic is valid in this field. Therefore, logic is not infallible and has to be used with care.
References
[1] Spade P. Vincent and Hintikka, . Jaakko J. (2022). history of logic.Encyclopedia Britannica.Retrieved from: https://www.britannica.com/topic/history-of-logic.[18.12.2022]
[2] Smith, Robin.( 2022 ). Aristotle’s Logic.The Stanford Encyclopedia of Philosophy.Retrieved from: https://plato.stanford.edu/archives/win2022/entries/aristotle-logic/ .[19.12.2022]
[3] Lagerlund Henrik.( 2022 ).Medieval Theories of the Syllogism.The Stanford Encyclopedia of Philosophy.Retrieved from:https://plato.stanford.edu/archives/sum2022/entries/medieval-syllogism/.[20.12.2022]
[4] Józef Maria Bocheński.(2017).A History of Formal Logic.Chelsea Publishing Company
[5] Groarke Leo.( 2022 ).Informal Logic. The Stanford Encyclopedia of Philosophy.Retrieved from: https://plato.stanford.edu/archives/win2022/entries/logic-informal/.[20.12.2022]
[6] Craig Edward (1996). Routledge Encyclopedia of Philosophy: Genealogy to Iqbal. Routledge.
[7] Shapiro, Stewart and Teresa Kouri Kissel.( 2022 ).Classical Logic The Stanford Encyclopedia of Philosophy. Retrieved from: https://plato.stanford.edu/archives/win2022/entries/logic-classical/.[20.12.2022]
[8] Audi Robert (ed.) (1999). The Cambridge Dictionary of Philosophy. New York City: Cambridge University Press.
[9] Hintikka J. J. (2022). philosophy of logic. Encyclopedia Britannica.Retrieved from: https://www.britannica.com/topic/philosophy-of-logic.[20.12.2022]
[10] Rendsvig Rasmus and John Symons.(2021). Epistemic Logic. The Stanford Encyclopedia of Philosophy.Retrieved from:https://plato.stanford.edu/archives/sum2021/entries/logic-epistemic/.[20.12.2022]