3 Intro to Logic and Reasoning
3.1 History of Logic
As we all know that Artificial Intelligence (‘AI’) is the field of computer science that was developed to enable computers/machines to display behavior that can be characterized as intelligent.
Then you may ask “How should we define intelligence? Why are we humans considered intelligent?”
As I walk you through the history of logic, you will see that our search for the answer to this question has spanned thousands years. Based on this journey of ‘intelligent’ search, we can now shed some light on how and why AI was developed.
As you probably already know humans have long regarded ourselves as intelligent due to our ability to think and reason.
Cogito ergo sum. I think therefore I am - Descartes
Now, let’s begin the journey together to discover one version of the truth of intelligence.
Logic was developed by Aristotle (384-322 BCE). He introduced the formal study of what is now known as ‘formal logic’. His logic was concerned with the form, and not the content of statements or propositions. Basically, Aristotle’s system of logic introduced “hypothetical syllogism”, “temporal model logic” and “inductive logic”. ** add references to definitions
He claims that a proposition is a complex proposition involving 2 terms, a subject and a predicate. Each of them are represented with a noun. The logic form of a proposition is determined by its quantity and it’s quality.
All of Aristotle’s logic revolves around one notion: the deduction (syllogism).
" A deduction is speech in which, certain things having been supposed, something different from those supposed results of necessity because of their being so."
Each of the ‘things supposed’ is a premise of the argument and what ‘results of necessity’ is the conclusion.
Following Greek’s tradition in logic, Cicero (106-43 BCE) introduced the term ‘proposition’ for which we will discuss more in depth later.
Alexander of Aphrodisias (3rd century A.D.) used the term ‘logic’ in the modern sense of distinguishing correct from incorrect reasoning.
In the early twelfth century, Peter Abelard wrote extensive commentaries attempting to articulate issues like opposition, conversion, quantity, and quality, and composed his treatise, ‘the Dialectica’.
Later, William of Sherwood developed mnemonic verse as an aid in mastering syllogisms. Jean Buridan elaborated a theory of consequences which somewhere ??(somewhere) discussed the rules of inference.
Around the same time, Scholastic logician Ramon Llull, used logic to prove the Christian faith. He also remarkably designed machines that would perform logical calculations, and is thus arguably considered as the father of computer programming. This could be said to have started the first idea of logic being converted into a machine along with Charles Babbage’s analytical engine.
The traditional logic period starts with Antoine Arnauld and Pierre Nicole‘s Logic, or the Art of Thinking published in 1662 or the Port Royal Logic in which logic is defined as the ’art of managing one’s reason right in the knowledge of things, both for the instruction of oneself and of others’. It was the most influential work on logic in England until John Stuart Mill’s System of Logic in 1825.
Gottfried Wilhelm Leibniz (1646-1716) having invented calculus, concluded that the whole of logic actually depends on mathematics and thus worked on reducing scientific and philosophical speculation to computation. He also suggested that a universal calculus of reasoning could be devised which would provide an automatic method of solution for all problems which could be expressed in the universal language. The current understanding of the power of Leibniz’s discoveries did not emerge until the 1980s.
During the modern and Contemporary Period (1850 - Present), logicians in the modern period ‘rediscovered’ the Stoic logic of proposition.
People like Augustus De Morgan (1806-1864) proposed some ??(some) theorem in that logic which now bears his name. Considered the founder of symbolic logic and Boolean Algebra, which is the basis of all modern computer arithmetic.
George Boole (1815-1864) gave us Boolean Logic which treats propositions as either true or false. His use of numbers to express the truth values of compound statements have significantly influenced the development of computers and he is regarded as being one of the founders of the field of computer science. Until today, programmers still use his principles to test the truth of program results or user feedback.
John Venn (1834-1923) was a Cambridge logician who published 3 standard texts in logic, The Logic of Chance 1866, symbolic Logic 1881, and The Principles of Empirical Logic 1889. He is remembered for introducing the circular diagrams as a tool to test the validity of syllogisms, known as Venn Diagrams.
Around the same time, John Stuart Mill (1806-1873) made a thorough study about inductive reasoning and introduced methods for checking such arguments now known as ‘Mill’s Methods’. And Charles Sanders Peirce (1839-1914) introduced Pragmatism, at the core of which he argued that ideas should be evaluated solely by their practical effects and not by any intrinsic qualities of reason or logic.
** what does this [Charles Sanders Peirce] mean for the story?
Gottlob Frege (1848-1925) pronounced that logic is the basis of mathematics and that arithmetic and analysis are a part of logic. He has been called the greatest logician since Aristotle. By developing the predicate calculus (Quantification Theory), he had combined Aristotelian and Stoic’s logics. His work was the foundation and the beginning point for an enormous outpouring of work in formal logic.
In 1903, Bertrand Russell (1872-1970) started his project ‘The Principles of Mathematics’ in which he purposed to prove that ‘all pure mathematics deals exclusively with concepts definable in terms of a very small number of logic principles.’ Russel also continued the development of the predicate calculus and he also found the inconsistency in Frege’s system (because Russell’s Paradox could be derived within Frege’s system).
The next wave of the logic can be called the ‘mathematical school of logic’. This tradition or school, includes the work of Richard Dedekind (1831-1916), Giuseppe Peano (1858-1932), David Hilbert (1862-1943). Ernst Zermelo (1871-1953), and many others since then. Its goal was the axiomatization of particular branches of mathematics, including geometry, arithmetic, analysis, and set theory.
In 1889 Giuseppe Peano published the first version of the logical axiomatization of arithmetic. Five of the nine axioms he came up with are now known as the Peano axioms. One of these axioms was a formalized statement of the principle of mathematical induction.
Ernst Zermelo’s axiomatic set theory was also an attempt to escape Russell’s Paradox. His axioms went well beyond Frege’s axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo-Fraenkel set theory.
Gradually, logic became the branch of mathematics that was to be brought within the axiomatic methodology. Jan Łukasiewicz worked on multi-valued logics. His three-valued propositional calculus, introduced in 1917, was the first explicitly axiomatized non-classical logical calculus. ** what does this mean for logic and AI?
The famous Ludwig Wittgenstein (1819-1951) entered the list of significant logician by being one of the developers of the ‘truth tables’.
This intensive work on mathematical issues culminated in the work of Kurt Gödel (1906-1978), a logician of the caliber of Aristotle and Frege. Using many applications of the rules of logic, Kurt Godel proved his ‘incompleteness theorem’, which proposes that some parts of mathematics are based on ideas that cannot be proved within the system of mathematics.
Gödel was also one of the central figures in the study of computability. Others included Alonzo Church (1903-1995), Alan Turing (1912-1954), and others.
There are plenty of other advances in logic afterwards as well. Logical empiricist Rudolf Carnap (1891-1970) was associated with the famous verfiability principle, according to which a synthetic statement is meaninful only if it is verificable.
Logical positivist A.J. Ayer, on the other hand, wrote in 1936 his ‘Language, Truth, and Logic’ in which he focused on the role of language as the medium through which knowledge is understood and verified.
In 1965, Lotfi A. Zadeh developed ‘fuzzy logic’ which allows imprecise answers to questions in addition to being either clear-cut true or false. This logic now serves as the basis of computer programming designed to mimic human intelligence.
Let’s pause here for a while and take a close look at the progress that made by Alan Turing, Alonzo Church and Kurt Godel during this period.
As we all know that, Alan Turing, the father of computing, created a machine that can accept different instructions for different tasks in 1936 and marked the first step of the AI with his seminal 1950 paper. Turing’s initial investigation of computation stemmed from the programme set out by David Hilbert in 1928. Hibert presented 3 open questions for logic and mathematics. Was mathematics …
complete in a sense that any mathematical assertion could either be proved or disproval
consistent in the sense that false statements could not be derived by a sequence of valid steps and
decidable in the sense that there exists a definite method to decide the truth of falsity of every mathematical assertion.
Within 3 years, Kurt Godel had proved that the axioms of arithmetic are both not complete and consistent. By 1937 both Alonzo Church and Alan Turing had demonstrated that undecidability of particular mathematical assertions. Interestingly, as Godel and Church had depended on demonstrating their results using purely mathematical calculi. Turing chose to take an unusual route of considering mathematical proof as an artifact of human reasoning. He even generalized this notion to a physical machine that he believes it could emulate a human mathematician and in turn there could be a universal machine that can emulate all other computing machines. Then he used this construct to show that certain functions cannot be computed by such a universal machine and in turn, demonstrated the undecidability of assertions associated with such functions.
As we can easily tell, at the heart of Turing’s universal machine is a model of human reasoning and calculation. After putting his idea into practice during the World War II, he came up with a comprehensive paper that provided a philosophical framework for answering the question ‘Can machines think?’ i.e. the invention of the ‘Turing test’ and universality of digital computers. He also goes on to discuss 2 distinct strategies that might be considered possible of achieving a thinking machine:
AI by programming
AI by machine learning
AI using logic, probabilities, learning and background knowledge
As you can see from history, the endeavors of implementing these strategies have been carrying on simulatenously in the last few decades. Here let’s just take a close look at the last proposed approach AI using logic, probabilities, learning and background knowledge.
“It is necessary there to have some other ‘unemotional’ channels of communication. If there are available it is possible to teach a machine by punishments and rewards to obey orders given in some language, e.g. a symbolic language. There orders are to be transmitted through the ‘unemotional’ channels. The use of this language will diminish greatly the number of punishment and reward required.” - Alan Turing
As for how to achieve it, he said:
“Opinions may vary as to the complexity which is suitable in the child machine. One might try to make it as simple as possible, consistent with the general principles. Alternatively, one might have a complete system of logical inference ‘built in’. In the latter case, the store would be largely occupied with definitions and propositions.”
**chain the above paragraph with the one below by relation
Here are some attempts that we have made after Turing’s proposal. After the introduction of resolution-based automatic theorem introduced by Alan Robinson in 1965. The interest of using first-order predicate calculus as a representation for reasoning within AI systems skyrocketed. Also, as introduced in Gordon Plotkin’s thesis, it became possible for us to use resolution theorem proving to investigate a form of machine learning that involves hypothesising logical axioms from observations to background knowledge.
Now let’s go back to our initial question about AI. As we can see from the above journey, AI has been heavily influenced by some of these logical ideas. Undoubtedly, logic has played an crucial role in some central areas of AI research. As much as we are trying to compute our reasoning process, the ultimate goal of a thinking machine is to be able to formalize common sense reasoning, the prescientific reasoning that is used in handling everyday problems.
In order to take on this mission, we just created a whole new set of problems for ourselves to deal with i.e. knowledge representation and reasoning for which we will discuss more in depth in the later chapter. In short, inspired by psychology and neurobiology about how we humans solve problems and represent knowlege of the world.
Talking about the connection of AI, logic and neurophysiology, we have brought up these 2 men, Warren Sturgis McCulloch and Walter Pitts. McCulloch was a psychologist, psychiatrist and philosopher by degree, but he had been working on and thinking about how to apply neurophysiology and logic to model the brain. Upon the time he met Pitts, a homeless young man who had been hanging around the University of Chicago, they realized that they shared a same hero in common: Gottfried Leibniz. As what have mentioned before, Leibniz is the inventor of calculus and he also had attempted to create a computing system that can replicate human thoughts, in which each letter represented a concept and they could be combined and manipulated according to a set of logical rules. As a numerous logicians and philosophers had tried, it is an attempt and a vision that promised to transform the chaotic outside world into the rational and organized system.
In their paper A Logical Calculus of Ideas Immanent in Nervous Activity, they introduced the idea of artificial neural network. This had been inspired by Leibnizian logical calculus and Principia Mathematica, they created a network that uses logical predicate to compute as they were convinced that the brain was just such a machine that uses logic encoded in neural networks to compute.
As you probably have had predicted, these two men started to work on this idea of capturing reasoning with a logical calculus. But this time, they also tried a novel approach which is to combine the knowledge of biological neurons. By stringing the simple neurons into chains and loops, they had shown that it is possible for brain to implement every possible logical operation and outputting anything that could be calculated by one of Turing’s hypothetical machines.
As what they put:“Because of the ‘all-or-none’ character of nervous activity, neural events and the relations among them can be treated by means of propositional logic.”, they divided neurons into 2 groups, *peripheral afferents (or ‘input neurons’) and the rest (’output neurons) and each neuron can be in 2 states, firing or non-firing. They defined every neuron i a predicate which is true when the neuron is firing at the moment t. As for the solution of this network, \(N_i(t) \equiv B\) here B is a conjunction of firings from the previous moment of the peripheral afferents, and i is not an input neurons. It is also worth to mention that this paper has only three references, and all of them are classical works in logic:
Carnap, R. 1938. The Logical Syntax of Language. New York: Harcourt-Brace.
Hilbert, D. and W. Ackermann. 1927. Grundzüge der Theoretischen Logik. Berlin: Springer.
Russell, B. and A. N. Whitehead. 1925. Principa Mathematics. Cambridge University Press.
Here is also a longer version of the story of Pitts and Mccolloch that I won’t cover here. In short, there are plenty of other brilliant minds like Jerome Lettvin, Norbert Weiner, von Neumann have contributed to the invention of cybernetics. You can think of the story of Pitts and McCulloch is also the story of cybernetics which was born out of the influences of ideas from a variety of domains and fields, and in a way a neural network symbolizes this interaction.
If you are interested in knowing more about the cybernetics and the story of McColloch and Pitts Walter, don’t forget to check this article out: The Man Who Tried to Redeem the World with Logic.
After the age of Pitts and McColloch, there were 2 major trens underlying the research in Artificial Intelligence around 1960s. The first trend produced thre program that uses symbolic reasoning/deducive logical systems. People like Herbert A. Simon, Allen Newell and Cliff Shaw have created some working programs based on this principles. The reason the symbolic systems are somewhat appealing is that they seemed to be able to provide the control and extensibility that neural network could not. Due to the function that symbolic systems have achieved i.e. proving theorems and playing chess, we would conclude that
symbolic thinking is considered a rare and desirable aspect of intelligence in humans, but it comes rather natural to computers which have much more trouble with reproducing ’low-level’intelligent behavior that comes very natural to human, such as recognizing the animal in the picture and picking up objects.
This is the famous Moravec’s Paradox discovered by Hans Moravec in 1980s. You can simplify the statement as “Robots find the difficult things easy and the easy things difficult.”
All those discoveries helped us to begin doubting about our initial belief “we think therefore we are”. Brought us back to our initial profound philosophical question:“what makes us human intellient?”
Again, we stumble upon the crux of the issue - most of the intellectual and scientific discipline of modernity are ultimately premised upon philosophical assumptions. As much as we hate to admit the possible limit of AI, we may need to revisit this question with a new framework or mental model. As philosophier Hubert Dreyfus suggested, drawing from the work of Martin Heidegger and Maurice Merleau-Ponty the brain does not create internal representations of objects in the world. The brain simply learns how to see the world directly. Or as Rodney Brooks said:“the best model for the world is the world itself.” In constrast to the classic AI ‘computational theory of mind’, the embodied robot will continuously refer to its sensors rather than to an internal world model/representation just as we humans do.
Though it seems true that the ‘thinking machine’ cannot be built with logic solely, logic in AI currently is still a large and rapidly growing field. One of the most influential figure in logical AI is John McCarthy. McCarthy was also one of the founders of AI, and consistently advocated a research methodology that uses logical techniques to formalize the reasoning problems that AI needs to solve. You may wonder, what motivates them to continue integrating logic with AI. The answer is that a logical formalization helps us to understand the reasoning problem itself. However, as we currently know, the key to solve a problem may not need an thorough understanding of what the reasoning problems are. It is in fact quite controversial in the context of AI, an alternative methodology would be for such a machine to learn or evolve the desired behaviors.
3.1.1 The End of Logic?
The intellectual project to capture thinking in terms of logic was at first thought to solve the problem of reasoning only to find out later that what makes human intelligent isn’t just reasoning, but reasoning under uncertain conditions as well as learning not just from rules but from the world itself.
So it is not the end of logic, for it powers so much of modern computing, but like any grand theory that takes on something as complex as human level thinking it has found it’s practical place as a tool forever in our intellectual and computational toolbelt.
I would like to leave you with this last provoking statement from Longuet-Higgins:“We need AI not to build machines, but to understand humans.”
**this is really amazing and thorough. I like the way the second half reads better than the first because it’s less of a reading of the facts. It would be great if you could relate the discovering together and their relation to AI. As a next step, I would leave this how it is with my corrections and then write a second one that is half as long and a kid could read. Great job.