
Starting Point - Gottlob Frege
This is the earliest time I could remember from my consciousness when I found a very interesting book from the MIT’s OCW reading materials - The Logic Book by Merrie Bergmann, James Moor, and Jack Nelson (2008).
The book started with mentioning one of the prominent figure in the field of Logic - Friedrich Ludwig Gottlob Frege (8 November 1848 - 26 July 1925) a German philosopher, logician, and mathematician.
He was a mathematics professor at the University of Jena, and is understood by many to be the father of Analytic Philosophy, concentrating on the philosophy of language, logic, and mathematics. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever.
While Gottlob Frege’s body of work is not extensive, it is incredibly dense and influential. His most important books represent major milestones in the history of logic and philosophy.
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Begriffsschrift (or Concept-Script in English): his first revolutionary work where he unveiled his powerful new system of logic that went far beyond the Aristotelian logic of his day.
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Die Grundlagen der Arithmetik (The Foundations of Arithmetic): a direct and sustained argument against Immanuel Kant’s philosophy of mathematics. Frege dismantled the prevailing Kantian view that arithmetic was “synthetic a priori” (meaning it required a special faculty of intuition) and argued that the principles of arithmetic were “analytic” (derivable from pure logic and definitions alone). Much of The Foundations of Arithmetic is dedicated to refuting Kant’s position and demonstrating the superiority of a purely logical foundation for numbers.
Gottlob Frege and Immanuel Kant are profoundly connected in the history of philosophy, with Frege’s work being a direct response to and a challenge against a core part of Kant’s philosophy. Their relationship is one of intellectual opposition. The heart of their disagreement lies in the nature of mathematical truth.
In his Position, Kant argued that the truths of arithmetic (like 7 + 5 = 12) are synthetic a priori:
- A priori means the knowledge is independent of experience. We don’t need to count objects in the world to know that 7 + 5 = 12.
- Synthetic means the truth is not merely based on definitions. Kant believed that the concept of “12” is not actually contained within the concepts of “7,” “5,” and “plus.” He argued that to arrive at the sum, we need a special faculty he called pure intuition - a kind of mental construction, like counting on our fingers in our mind.
Frege’s Rebuttal, however, stated that the truths of arithmetic are analytic a priori:
- He agreed with Kant that they are a priori
- However, he argued they are analytic, meaning they can be derived from pure logic and definitions alone, without any need for a mysterious “intuition.”
Philosopher in Field of Language
Being one of the founders of analytic philosophy, Frege’s work on logic and language gave rise to the linguistic turn in philosophy. His contributions to the philosophy of language include:
- Function and argument analysis of the proposition;
- Distinction between concept and object (Begriff und Gegenstand);
- Principle of compositionality;
- Context principle; and
- Distinction between the sense and reference (Sinn und Bedeutung) of names and other expressions, sometimes said to involve a mediated reference theory.
As a philosopher of mathematics, Frege attacked the psychologistic appeal to mental explanations of the content of judgment of the meaning of sentences. His original purpose was very far from answering general questions about meaning; instead, he devised his logic to explore the foundations of arithmetic, undertaking to answer questions such as “What is a number?” or “What objects do number-words (‘one’, ‘two’, etc.) refer to?” But in pursuing these matters, he eventually found himself analysing and explaining what meaning is, and thus came to several conclusions that proved highly consequential for the subsequent course of analytic philosophy and the philosophy of language.
Why Would Frege Become Interested in the Philosophy of Language?
A Tool for Mathematics
Frege’s journey into the philosophy of language is a fascinating example of how a problem in one field can force a revolution in another. His interest wasn’t born from a desire to study language for its own sake, but rather as a necessary step to solve a problem in mathematics.
Frege’s ultimate goal was logicism: to prove that all of arithmetic could be derived from the basic principles of logic alone. He wanted to show that mathematical truths weren’t based on some special intuition, as Kant believed, but were simply complex logical truths.
To build these proofs, however, he quickly ran into a major obstacle: ordinary human language. He found German, like any natural language, to be too ambiguous, vague, and context-dependent for the kind of absolute precision required for logical proofs. The grammatical structure of a sentence often hid its true logical form.
His solution was to invent a new language, a “formula language for pure thought,” which he called the Begriffsschrift (“Concept-Script”). This was a perfectly logical and unambiguous formal system. In the process of creating and justifying this system, he was forced to answer fundamental questions about how any language works:
- What is the difference between an object and a concept?
- How do sentences connect to the world to be true or false?
- How can two names, like “the morning star” and “the evening star,” refer to the same object (Venus) yet have different meanings?
Answering these questions, which he had to do to make his logical system work, led him directly to his groundbreaking contributions to the philosophy of language, such as his famous distinction between sense and reference. His work on language was a necessary tool for his work on mathematics.
The Influence of Childhood
While his mathematical goals provided the motivation, his childhood likely provided a crucial part of the mindset.
In childhood, Frege encountered philosophies that would guide his future scientific career. For example, Frege’s father, Carl Alexander Frege, was the co-founder and headmaster of a girls’ high school. his father wrote a textbook on the German language for children aged 9-13, entitled Hülfsbuch zum Unterrichte in der deutschen Sprache für Kinder von 9 bis 13 Jahren (2nd ed., Wismar 1850; 3rd ed., Wismar and Ludwigslust: Hinstorff, 1862) (Help book for teaching German to children from 9 to 13 years old), the first section of which dealt with the structure and logic of language. Growing up in a household where his own father was professionally dedicated to analyzing, structuring, and teaching the rules of grammar would have exposed Frege from a very young age to the idea of language as a formal system.
While we have no record of Frege explicitly crediting his father’s work, it’s highly plausible that this upbringing instilled in him a deep-seated sensitivity to the structure of language and an awareness of its potential for both precision and confusion. This early, informal training in thinking analytically about language would have given him a unique foundation for his later, revolutionary work when he realized that ordinary language was inadequate for his logical project.
Bridging to AI
Gottlob Frege’s opposition with Kant and his strong sense of rationality made me think he is a rationalism. In fact, his opposition to Kant stems from a belief that reason is even more powerful than Kant gave it credit for, because Gottlob Frege is a towering figure in the Rationalist tradition, even if he isn’t always labeled as such in introductory texts. His entire philosophical project can be seen as a radical and modernized form of Rationalism.
This triggered my ultimate question: “Is AI related to the Philosophy of Language at all?”
The answer is “Yes, absolutely”. The relationship between AI and the Philosophy of Language is not just a casual connection; it’s a deep, fundamental, and reciprocal one. In many ways, modern AI, especially Large Language Models (LLMs), is a large-scale, practical experiment in the philosophy of language. The core reason is simple: for decades, philosophers have asked theoretical questions about how language works, and now, AI engineers are forced to answer those same questions in a practical way to build systems that can use language.
The Central Problem: What is Meaning?
The biggest overlap is the question of meaning (semantics). Philosophers have long debated how words get their meaning. Do they point to objects in the world? Do they get meaning from a dictionary-like definition in our minds? Or something else?
Modern AI models have a very specific, and philosophically interesting, answer: they learn meaning from statistical relationships between words in vast amounts of text. An AI “knows” that “king” is related to “queen” not because it understands monarchy, but because it has analyzed billions of sentences where those words appear in similar contexts. This strongly resonates with the philosopher Ludwig Wittgenstein’s famous idea that “the meaning of a word is its use in the language.” AI takes this idea to its literal, computational extreme.
This, however, raises a huge philosophical question that AI engineers and philosophers are now grappling with together: Is this statistical “meaning” the same as human understanding? Or is it a sophisticated mimicry without any real comprehension?
The Question of True Understanding
This all leads to the most famous philosophical challenge to AI: John Searle’s “Chinese Room” argument. Searle asks us to imagine a person in a room who doesn’t speak Chinese but has a giant rulebook. Messages in Chinese are passed under the door, and by following the rules in the book, the person can find the correct Chinese symbols to pass back out, creating a convincing conversation.
While the “Chinese Room” argument is most often discussed in the Philosophy of Mind (because its main target is whether a machine can have a mind), its central pivot point - the very concept of “understanding” - is a profoundly epistemological question. Searle is using a problem of mind to make a claim about the nature of knowledge. But we are not going to get deep into Searle’s argument, because what’s important is that we have established the consensus on the deep connection between Philosophy and AI
Philosophy
How to study PhilosophyPeople who are interested in Philosophy but not a Philosophy major doesn’t like the overly complicated style of writing where everything is symbolic or metaphorical and have to dig for the meaning instead of it just being clearly stated. For example, reading Newton’s principia feels struggling for really understanding the work not because the ideas in it are too complicated, but because reaching the ideas feels like going through a maze of unnecessary jargon to reach them, and by the time one has made it through all that, people become too mentally drained to even digest what they’re really saying. Is there anything wrong?
It is an interesting assumption that people without background in academic Philosophy think that one have to read the primary sources with verify little hand holding. Studying both Physics and Philosophy, I have literally never in my life met someone that tries to access Physics by reading the primary academic literature.
Literally nobody interested in Physics starts by trying to read Newton’s principia. If one is interested in Philosophy and doesn’t have any training in it, why not start with Philosophy textbooks just like how we study Physics in college? These university level introductory texts take these complex and often unfriendly primary texts and parse them for us.
Philosophy (‘love of wisdom’ in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, value, mind, and language. It is a rational and critical inquiry that reflects on its methods and assumptions. Philosophical questions can be grouped into several branches. Logic, Metaphysics, Epistemology, and Ethics are sometimes listed as the main branches.
Logic
Axiomatic Systems
Over the centuries, a variety of systems of deductive logic have been developed. One of the oldest is Euclid’s Elements, the axiomatization of plane geometry developed around 300 BCE in classical Greece. All of the truths or theorems of plane geometry can be derived from the five fundamental assumptions or axioms of Euclid’s system. Many have attempted to axiomatize other areas of knowledge, including many of the sciences and many areas of mathematics. For example, Giuseppe Peano successfully axiomatized arithmetic in 1889.
The Five Fundamental Axioms of Euclid’s Elements
- ᾿Ηιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν.
- Καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾿ εὐθείας ἐκβαλεῖν.
- Καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράφεσθαι.
- Καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις εἶναι.
- Καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ, ἐκβαλλομένας τὰς δύο εὐθείας ἐπ᾿ ἄπειρον συμπίπτειν, ἐφ᾿ ἃ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες.
Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner.
The original Ancient Greek text of Euclid’s Elements can be found at:
An axiomatic system consists of two key components:
- Axioms (or Postulates): These are the foundational statements or assumptions that are accepted as true without proof. They are the starting premises of the entire system.
- Rules of Inference: These are the rules of logic that allow us to combine and manipulate the axioms (and any subsequently proven statements) to create new, true statements.
The process of using these rules of inference to derive new statements (theorems) from the initial axioms is deductive reasoning.
Deductive Logic
The systems of deductive logic that we present in this section have their foundations in the work of Gottlob Frege, David Hilbert, Bertrand Russell, and other logicians in the late nineteenth and early twentieth centuries.
IMPORTANTNote that deductive logic is overwhelmingly and fundamentally associated with the Analytic tradition, which is why the materials of this section are drawn from the authors above who are essentially all Analytical philosophers
Although philosophers are primarily referring to deductive logic when they talk about Logic as one of the main branches, the philosophical field of “Logic” is broader than just deduction. It encompasses the systematic study of all forms of correct reasoning. So, while deduction is the main event, the branch also includes:
- Inductive Logic
- Abductive Logic
Unlike axiomatic systems, which are based on a (usually) small number of axioms, the deductive systems to be introduced are based on a small number of reasonably intuitive rules that govern how sentences can be derived from other sentences.
Studying deductive logic by learning to symbolize natural language sentences (such as English sentences) in a formal language, one becomes more aware and more appreciative of the importance of the structure and complexities of natural languages. The specific words that we use have a direct bearing on whether a piece of reasoning is valid or invalid. For example, it is essential to distinguish between “Roberta will pass if she completes all the homework” and “Roberta will pass only if she completes all the homework” if we want to reason well about Roberta’s prospects for passing. Finally, the concepts that we explore here are abstract concepts. Learning to think about abstract concepts and the relations between them is an important skill that is useful in a wide range of theoretical and applied disciplines.
Now let’s explore the core concepts of deductive logic, starting with basic definitions:
Definitions
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An argument is a set of two or more sentences, one of which is designated as the conclusion and the others as the premises (English sentences).
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An argument is logically valid if and only if it is not possible for all the premises to be true and the conclusion false. An argument is logically invalid if and only if it is not logically valid. In this sense, a logically valid argument is truth-preserving (always taking one from truth to another truth, never falsehood)
CorollaryAn argument whose conclusion is logically true is logically valid no matter what premises it has, because it will never lead us from truths to a falsehood
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An argument is logically sound if and only if it is logically valid and all of its premises are true.
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A sentence is logically true if and only if it is not possible for the sentence to be false
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A sentence is logically false if and only if it is not possible for the sentence to be true
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A sentence is logically indeterminate if and only if it is neither logically true nor logically false
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Sentences p and q are Logically equivalent if and only if it is not possible for one of these sentences to be true while the other sentence is false
Corollary- All logically true sentences are logically equivalent
- All logically false sentences are logically equivalent
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A set of sentences is logically consistence if and only if it is possible for all the members of that set to be true
CorollaryAn argument whose premises form logically inconsistent sets is logically valid but never logically sound.
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A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence false
CorollaryEvery logically true sentence is entailed by every set of sentences, including the empty set, because it is not possible for the sentence to be false
Syntax (of Sentential Logic)
The syntax of a language specifies the basic expressions of a language and the rules that determine which combinations of those expressions count as sentences of the language. The syntax of a language does not specify how the sentences of the language are to be interpreted; that is a matter for semantics. The syntax of English, and every other natural language, is enormously complex. Fortunately, the syntax of Sentential Logic is simple, straightforward, and easily learned.
When we use a language to talk about a language, we are using that language as a metalanguage, and the language that we are talking about is the object language. For example, in a German class the instructor uses English to talk about German, and in this instance English is the metalanguage and German is the object language. When a grammar instructor uses English to talk about the rules of English grammar English is both the metalanguage and the object language.
ConventionWe will use the boldface capital letters "", "", "", "" with or without subscripts as in
as metavariables ranging over expressions of object language. These variables are termed “meta” because they are part of the metalanguage.
We now begin introducing formal language of Sentential Logic (SL), starting with the specification of the syntax of SL:
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Sentence Letters: the capital Roman letters ‘A’ through ‘Z’, with or without positive integer subscripts
CAUTION
Although both are denoted by capital letters, expressions (bolded such as ) and sentences (regular like ) are not the same. Expressions can be a sentences but not always.
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Sentential Connectives
- (Negation)
- (Conjunction) - In , and are called conjuncts of the conjunction
- (Disjunction) - In , and are called disjuncts of the conjunction
- (Material Conditionals) - In , is called antecedent and consequent
- (Material Biconditionals)
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Sentence of SL, the definition
- Every sentence letter of SL is a sentence of SL
- If is a sentence of SL, then is a sentence of SL
- If and are sentences of SL, then is a sentence of SL
- If and are sentences of SL, then is a sentence of SL
- If and are sentences of SL, then is a sentence of SL
- If and are sentences of SL, then is a sentence of SL
- Nothing is a sentence of SL unless it can be formed by repeated application of 1 - 6
The followings are not sentences of SL, although they look like very much are
- - no parenthesis as definition 3 above requires
- - no left operand as definition 4 requires
- - is not a sentence by definition
- - is not a symbol of SL
- - neither nor is a sentence of SL
Semantics (of Sentential Logic)
Metaphysics
Overview
Metaphysics is the study of the most general features of reality, such as existence, objects and their properties, wholes and their parts, space and time, events, and causation. Metaphysicians attempt to answer basic questions including “Why is there something rather than nothing?”; “Of what does reality ultimately consist?”; and “Are humans free?”
The Fundamental Question of Metaphysics“Why is there anything at all?” or “Why is there something rather than nothing?” is a question about the reason for basic existence which has been raised or commented on by a range of philosophers and physicists who called it “the fundamental question of metaphysics”
Metaphysics encompasses a wide range of general and abstract topics. It investigates the nature of existence, the features all entities have in common, and their division into categories of being. An influential division is between particulars and universals. Particulars are individual unique entities, like a specific apple. Universals are general features that different particulars have in common, like the color red. Metaphysicians also explore the concepts of space, time, and change, and their connection to causality and the laws of nature. Other topics include how mind and matter are related, whether everything in the world is predetermined, and whether there is free will.
An important area in metaphysics is ontology. Some theorists identify it with general metaphysics. Ontology investigates concepts like being, becoming, and reality. It studies the categories of being and asks what exists on the most fundamental level. Another subfield of metaphysics is philosophical cosmology. It is interested in the essence of the world as a whole. It asks questions including whether the universe has a beginning and an end and whether it was created by something else
A key topic in metaphysics concerns the question of whether reality only consists of physical things like matter and energy. Alternative suggestions are that mental entities (such as souls and experiences) and abstract entities (such as numbers) exist apart from physical things. Another topic in metaphysics concerns the problem of identity. One question is how much an entity can change while still remaining the same entity. According to one view, entities have essential and accidental features. They can change their accidental features but they cease to be the same entity if they lose an essential feature
The roots of metaphysics lie in antiquity with speculations about the nature and origin of the universe, like those found in the Upanishads in ancient India, Daoism in ancient China, and pre-Socratic philosophy in ancient Greece. During the subsequent medieval period in the West, discussions about the nature of universals were influenced by the philosophies of Plato and Aristotle. The modern period saw the emergence of various comprehensive systems of metaphysics, many of which embraced idealism. In the 20th century, traditional metaphysics in general and idealism in particular faced various criticisms, which prompted new approaches to metaphysical inquiry.
IdealismIdealism in philosophy, also known as philosophical realism or metaphysical idealism, is the set of metaphysical perspectives asserting that, most fundamentally, reality is equivalent to mind, spirit, or consciousness; that reality is entirely a mental construct; or that ideas are the highest type of reality or have the greatest claim to being considered “real”.
As an important area of philosophy, metaphysics discusses things like
- What makes it the case that one event causes another event?
- What are material objects?
- Given that material objects exist, do such things as properties exist?
- What makes it the case that a person may exist at two different times?
What is Metaphysics?
It is practically impossible to get any two philosophers to agree on a single definition of “metaphysics”. What we can do, instead, is to offer 3 different but mutually compatible approaches to characterizing metaphysics:
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Etymological Approach
It seems that during the first century CE, some of Aristotle’s works were being collected and published in Alexandria (Aristotle had died 300 or 400 years earlier, around 322 BCE.). Among these was a collection of Aristotelian writings that was given a name in ancient Greek that is normally translated into modern english as Physics
Aristotle’s Physics is NOT mainly about physicsIn fact, ironically, it is mainly about metaphysics divided into 2 volumes:
- Aristotle: The Physics Vol. I
- Aristotle: The Physics Vol. II
Shortly after the publication of Aristotle’s Physics, another batch of Aristotle’s writings was ready for publication. This second batch of Aristotle’s works contained discussions of such important but disparate philosophical topics as existence, identity, actuality, potentiality, time, change, causation, substance, matter, form, and universals
If we take an etymological approach to characterizing metaphysics, then, we will say that metaphysics is the branch of philosophy concerned with a disparate collection of topics that happen to be associated with one particular collection of writings by Aristotle, namely, the collection that was published after Aristotle’s Physics.
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Big-Picture Approach
The etymological approach unfortunately, doesn’t give us a very satisfying account of what many take to be the most central branch of philosophy. A more conceptual definition is by sensing that various topics within metaphysics do have something essential in common with one another. Attempting to unify those various topics of results in a pronouncement like the following: Metaphysics is the branch of philosophy concerned with fundamental questions about the nature of reality
This big-picture approach, unfortunately, is not without its own problems. One main difficulty is that most branches of inquiry, including biology, economics, and history, and other branches of philosophy are also concerned with reality.
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Definition-by-Example Approach
The problem regarding the fields of biology, economics, and history is easily resolved for stating that they are just not branches of philosophy. The issues with other branches of philosophy, however, requires a subtractive approach, i.e. by saying that there are 4 main branches of philosophy: ethics, epistemology, logics, and metaphysics and characterize metaphysics as what is left over when we subtract ethics, epistemology, and logics from the core area of philosophy. But what exactly is left over when we subtract ethics, epistemology, and logics from the core area of philosophy? Subjectively a good way to answer this question is with specific examples.
Here, then, are some of the topics that metaphysicians deal with
- ontology: roughly, the study of being, including the attempt to come up with a list of all the main categories of things that exist
- the nature of time
- the Mind-Body Problem: roughly, the problem of understanding the relationship between mental phenomena and the physical basis of those phenomena
- the problem of personal identity: roughly, the problem of identifying the conditions under which an earlier person and a later person are one and the same person
- the problem of freedom and determinism: roughly, the problem of specifying what is required in order for a person to be acting freely
- the nature of the laws of nature
- the nature of causation
- the nature of material objects, including questions about the relation between an object and the matter it is made of, and the conditions under which two or more objects compose a further object
Definition of metaphysics by examplesMetaphysics is the branch of philosophy concerned with such topics as ontology, time, the Mind-Body Problem, the problem of personal identity, the problem of freedom and determinism, laws of nature, causation, and material objects
Metaphysical Necessity
As important as its definition, a basic building block of metaphysics is modality, especially the concept that philosophers call metaphysical necessity, which we are going to discuss in this section.
Propositions that are metaphysically necessary must be true. Here are examples:
- Kant is wise or it is not the case that Kant is wise
- 2 + 2 = 4.
- Red is a color.
- All bachelors are unmarried.
Propositions that are not metaphysically necessary are ones that are either metaphysically impossible (have to be false) or metaphysically contingent (can be true and also can be false).
metaphysically impossible examples | metaphysically contingent examples |
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Kant is wise and Kant is not wise | Kant is wise |
2 + 2 = 5 | There are four oranges in the refrigerator |
Red is not a color | Red is the color of some fire engines |
Not all bachelors are unmarried | Asia is the smallest continent |
Other types of necessityMetaphysical necessity is one concept of necessity. There are others. For example, there is an important class of metaphysically contingent propositions that are physically (i.e., lawfully) necessary. For example, existence of a perpetual motion machine would contradict some law of nature (e.g., The Second law of Thermodynamics); hence it is impossible that there be a perpetual motion machine.
There is also epistemological necessity. For a long time, it was common to hear mathematicians and others say things like, “It is possible that Fermat’s last Theorem is false, there just has been no proof yet.” Prima facie, what were saying was essentially stating that the last Theorem was not epistemological necessity.
Fermat’s Last Theoremhas no integer solutions for and .
Nevertheless, in the mid-1990s, thanks to the mathematical work of Andrew Wiles, Fermat’s last Theorem was proven which made the Theorem metaphysically necessary. This is an example of certain propositions being said to be not epistemological necessity that in fact are metaphysically necessary.
Origin of Metaphysical Necessity
The concept of metaphysical necessity originates from ancient Greek philosophy’s attempts to understand eternal truths and the fundamental nature of being. It was later formalized by Gottfried Wilhelm Leibniz with his theory of possible worlds and given its distinct modern meaning by the 20th-century philosopher Saul Kripke.
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Ancient Greek Roots: Forms and Essences
The earliest seeds of metaphysical necessity can be found in the works of Plato and Aristotle, who sought to identify truths that were unchanging and eternal, as opposed to the fleeting, sensory world.
- Plato’s Theory of Forms: For Plato, the ultimate reality consisted of perfect, eternal “Forms” or “Ideas” (like the Form of Justice or the Form of a Triangle). Truths about these Forms were necessarily true. For example, the properties of a perfect triangle are necessary and unchanging, regardless of any imperfect triangle we might draw in the physical world.
- Aristotle’s Concept of Essence: The most direct ancestor of metaphysical necessity is Aristotle’s concept of essence. The essence of a thing is the set of properties that it cannot lack without ceasing to be what it is. For example, Aristotle would argue it is essential for a human being to be a “rational animal.” This is not just a definition; it’s a claim about the fundamental nature of humanity. Therefore, the statement “a human is a rational animal” is a necessary truth grounded in the very being of a human. This idea—that some properties are essential to an object’s identity—is the core of metaphysical necessity.
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The Framework of Possible Worlds: Leibniz
While the core idea was ancient, the philosopher Gottfried Wilhelm Leibniz in the 17th century provided the formal framework that philosophers still use today. He introduced the idea of possible worlds.
The book by Gottfried Wilhelm Leibniz that introduced the idea of possible worlds is his Théodicée, published in 1710.
Leibniz argued that a truth is metaphysically necessary if it is true not just in our world, but in all possible worlds. For example, the proposition “2 + 2 = 4” is necessary because there is no conceivable world where it is false. Conversely, a proposition like “the sky is blue” is contingent because we can easily imagine a possible world where it is, for example, green. This framework gave philosophers a powerful and intuitive way to analyze and discuss necessity.
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The Modern Revival: Saul Kripke
For a long period, especially after Immanuel Kant linked necessity with a priori knowledge, metaphysical necessity was often conflated with logical necessity or things known purely by reason. The modern, distinct understanding of the concept was established by Saul Kripke in the 1970s.
Kripke’s major contribution was to definitively separate metaphysical necessity from epistemology (how we know things). He did this by arguing for the existence of necessary a posteriori truths - facts that are necessarily true but can only be discovered through empirical investigation. His famous example is “Water is H₂O.”
- It is metaphysically necessary: Once we discovered the chemical structure of water, we realized that its essence is H₂O. Anything with a different chemical makeup would not be water, period. It’s true in all possible worlds.
- It is known a posteriori (not a priori): Humanity did not and could not know this fact simply by reason. It required scientific discovery.
By showing that necessity is a feature of reality itself - not just our language or knowledge - Kripke revived Aristotle’s focus on essences and firmly established metaphysical necessity as a central, independent concept in contemporary philosophy.
In short, its history is a journey from early ideas about what things must be to a rigorous concept of what is true in all possible realities.
Why are we briefly touching the history of metaphysical necessity? Because we are about to step on this historical ground and use it as the basis for investigating another epistemological notion very easily confused with metaphysical necessity - A Priori
Metaphysical Necessity v.s. A Priori
The term a priori is a Latin phrase that literally means “from what comes before.” Its origin traces back to ancient Greek philosophy, was formalized in Latin by medieval thinkers, and was given its modern meaning by Immanuel Kant.
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Ancient Greek Foundations: Aristotle
The conceptual groundwork for the a priori/a posteriori distinction was laid by the Greek philosopher Aristotle. While he didn’t use the Latin terms, in his Posterior Analytics, he distinguished between two kinds of knowledge:
- Knowledge from the cause: This is reasoning from a prior cause to a known effect. For example, knowing that rain will make the ground wet. This is the forerunner of the a priori concept - knowledge that proceeds from established principles or definitions.
- Knowledge from the effect: This is reasoning from an observed effect back to its cause. For example, seeing that the ground is wet and inferring that it must have rained. This is the basis for the a posteriori concept - knowledge gained from experience and observation.
Aristotle’s primary focus was on logic and the structure of scientific demonstration, establishing which arguments were logically prior to others.
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Scholastic and Early Modern Development
The Latin terms “a priori” and “a posteriori” were adopted and popularized by Scholastic philosophers in the Middle Ages, such as Albert of Saxony and, later, Thomas Aquinas. They used the terms to distinguish between two types of arguments, especially in theology:
- Argument a priori: An argument from cause to effect. For example, an argument for God’s existence based on a definition of God as a necessary being (like the ontological argument).
- Argument a posteriori: An argument from effect to cause. For example, arguing for God’s existence by observing the complexity and order of the natural world (the teleological argument).
This usage continued into the early modern period with rationalist philosophers like Gottfried Wilhelm Leibniz.
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Kant’s Modern Definition
The contemporary meaning of a priori was decisively shaped by the 18th-century German philosopher Immanuel Kant. He fundamentally shifted the distinction from being about the order of argument to the justification of knowledge.
For Kant, a priori knowledge is knowledge that is absolutely independent of all experience. We don’t need to consult the world to know it’s true; it can be known through reason alone. For example, “7 + 5 = 12” is a priori because its truth isn’t derived from counting objects in the world but from the principles of mathematics.
This was central to his project in the Critique of Pure Reason, where he explored the extent and limits of human reason. He famously investigated how “synthetic a priori” judgments were possible - that is, statements that are not just true by definition (analytic) but also tell us something new about the world (synthetic), yet are known to be universally and necessarily true without experience. This was a direct departure from the Aristotelian and Scholastic focus on logical sequencing, redefining “a priori” as a cornerstone of modern epistemology.
A priori knowledge is epistemological. It is knowledge that is justified without recourse to sensory experience. It is often understood as knowledge gained through reason and logical deduction alone, independently of experience. For example, “All bachelors are unmarried.” We do not need to conduct a survey of bachelors to know this is true. The truth of the statement is contained within the definitions of the words themselves.
Metaphysical necessity refers to a state of affairs that could not be otherwise. A proposition is metaphysically necessary if it is true in every possible world. This is a claim about the fundamental structure of reality.
The key difference, therefore, is that metaphysical necessity is a feature of the world, while a priori is a feature of how we gain knowledge. Metaphysical necessity was characterized only in terms of having to be true, yet a distinctly epistemological concept - knowledge - was brought in to characterize the a priori.
Causation
Unlike many other areas of metaphysics, the current philosophical literature on causation is to a large degree not focused on finding a correct theory. The interesting exploration is more about what causes what in an example - about where the causation lies - than it is about what theory gives the correct verdict about each example
Causes and effects must take place in order to be causes and effects. In addition, causation is a matter of two things always being conjoined, as Hume said:
“We may define a cause to be an object, followed by another, and where all the objects similar to the first are followed by objects similar to the second.”
(Hume, An Inquiry Concerning Human Understanding, p. 79, first published in 1748.)
Spelling out similarity in terms of sharing a property and putting this in terms of events and , we have our first theory for consideration:
Constant Conjunction (by David Hume)causes if and only if there are properties and , such that has and has , and each event of kind is followed by an event of kind .
While David Hume first introduced and extensively analyzed the idea of “constant conjunction”, which is presented above, in A Treatise of Human Nature (1739), he was disappointed with its reception. He later refined and presented his arguments in a more accessible and shorter work, An Enquiry Concerning Human Understanding.
Reading An Enquiry Concerning Human Understanding
It is strongly recommended to read the Introduction section of this book, which unlike other books I have read, offered extraordinary historical background of such treatise. It is this prelude that has taught me that given the substantial amount of religious censorship during Hume’s time, it is very important, in order to fully dismantle the ideas of Hume era’s philosophy, to have a professional understanding of the Latin world including
- The Latin language
- The history (since the Roman Empire)
- The religion’s core (Bible)
Epistemology
Epistemology is the branch of philosophy that studies knowledge. It is also known as theory of knowledge and aims to understand what knowledge is, how it arises, what its limits are, and what value it has. It further examines the nature of truth, belief, justification, and rationality. Some of the questions addressed by epistemologists include “By what method(s) can one acquire knowledge?”; “How is truth established?”; and “Can we prove causal relations?”
One area in epistemology asks how people acquire knowledge. According to empiricists, all knowledge is based on some form of experience. Rationalists reject this view and hold that some forms of knowledge, like innate knowledge, are not acquired through experience
Unlike the fields of Psychology which is also interested in beliefs and related cognitive processes by studying the beliefs people actually have and how people acquire them, epistemology explores how people should acquire beliefs. It determines which beliefs or forms of belief acquisition meet the standards or epistemic goals of knowledge and which ones fail, thereby providing an evaluation of beliefs. In this regard, epistemology is a normative discipline, whereas psychology and cognitive sociology are descriptive disciplines
The word epistemology comes from the ancient Greek terms ἐπιστήμη (episteme, meaning knowledge or understanding) and λόγος (logos, meaning study of or reason), literally, the study of knowledge. Despite its ancient roots, the word itself was only coined in the 19th century to designate this field as a distinct branch of philosophy.
Major Schools of Thought
Skepticism and Fallibilism
Philosophical skepticism questions the human ability to attain knowledge by challenging the foundations upon which knowledge claims rest. Some skeptics limit their criticism to specific domains of knowledge. For example, religious skeptics say that it is impossible to know about the existence of deities or the truth of other religious doctrines. Similarly, moral skeptics challenge the existence of moral knowledge and metaphysical skeptics say that humans cannot know ultimate reality. External world skepticism questions knowledge of external facts, whereas skepticism about other minds doubts knowledge of the mental states of others.
Fallibilism is another response to skepticism. Fallibilists agree with skeptics that absolute certainty is impossible. They reject the assumption that knowledge requires absolute certainty, leading them to the conclusion that fallible knowledge exists. They emphasize the need to keep an open and inquisitive mind, acknowledging that doubt can never be fully excluded, even for well-established knowledge claims like thoroughly tested scientific theories
Empiricism and Rationalism
The debate between empiricism and rationalism centers on the origins of human knowledge. Empiricism emphasizes that sense experience is the primary source of all knowledge. Some empiricists illustrate this view by describing the mind as a blank slate that only develops ideas about the external world through the sense data received from the sensory organs. According to them, the mind can attain various additional insights by comparing impressions, combining them, generalizing to form more abstract ideas, and deducing new conclusions from them. Empiricists say that all these mental operations depend on sensory material and do not function on their own.
Even though rationalists usually accept sense experience as one source of knowledge, they argue that certain forms of knowledge are directly accessed through reason without sense experience, like knowledge of mathematical and logical truths
Analytic Philosophy v.s. Continental Philosophy
In the late 19th and early 20th centuries there was one of the most fascinating and puzzling divisions in modern thought. The divergence is the result of a philosophical fork in the road that occurred as philosophers tried to grapple with the legacy of Immanuel Kant. Both traditions, in their own way, are a response to the questions he raised.
The Analytic Path: A Revolution in Logic
One path was largely set by the revolution in logic initiated by Gottlob Frege and championed by figures like Bertrand Russell, G.E. Moore, and Ludwig Wittgenstein in the United Kingdom. They believed that many of the great philosophical problems were actually the result of confusions caused by the imprecision of ordinary language. Their goal was to achieve the clarity and certainty of mathematics and science in philosophy.
The core methodology was logical analysis. By using the powerful new formal logic (which was a massive leap beyond Aristotle’s system), they could break down complex philosophical statements into their simplest components and evaluate them with rigorous precision. The ideal was to solve, or in some cases dissolve, philosophical problems
This path led to the Anglophone Philosophy or Analytic Philosophy whose approach resonated deeply with the strong empiricist tradition in British philosophy (going back to John Locke and David Hume). Furthermore, the rise of Nazism in the 1930s led to the migration of many leading members of the Vienna Circle - the epicenter of this new “scientific” philosophy - to the United States and the UK (hence the name “Anglophone”). They established themselves in universities, trained the next generation of philosophers, and cemented the dominance of the analytic style.
Analytic philosophers were most interested in Kant’s epistemology and his analysis of the structure of judgments. They saw his work as a brilliant but ultimately flawed attempt to understand the foundations of knowledge, and they sought to “correct” it using the more powerful tools of modern logic. The debate between Frege and Kant over the nature of arithmetic is a perfect example of this.
The Continental Path: History, Experience, and Interpretation
Meanwhile, in continental Europe, particularly in Germany and France, philosophy took a very different turn. Instead of rejecting the grand philosophical tradition, figures like Edmund Husserl, Martin Heidegger, Jean-Paul Sartre, and Jacques Derrida sought to engage with it and reinterpret it. They were less concerned with scientific precision and more interested in understanding the nature of human existence, consciousness, history, and power. They felt that reducing philosophy to logical analysis missed the richness and complexity of lived experience.
The methodology are more varied, but they often involve phenomenology (the study of conscious experience), hermeneutics (the theory of interpretation), and a deep engagement with the history of philosophy itself. They believe that we cannot understand a philosophical concept without understanding its historical development.
This path gave birth to Continental Philosophy and was a natural continuation of the German Idealist tradition of Kant and Hegel. The university systems in France and Germany had long emphasized the study of the history of philosophy as a central part of a philosophical education. Moreover, the immense social and political traumas of the 20th century - two world wars, the Holocaust, the Cold War - made the Continental themes of anxiety, freedom, power, and historical meaning feel deeply and urgently relevant.
Continental philosophers were more engaged with Kant’s metaphysics, his ethics, and his ideas about the limits of reason. They were fascinated by the “noumenal world” (reality-in-itself) that Kant said was unknowable, and they explored themes of freedom, subjectivity, and the human condition that were central to his moral philosophy.
The Converging Two Paths
Despite the two distinct philosophical streams that flowed on opposite sides of the English Channel, a significant and positive shift in the philosophical landscape over the last few decades, however, has made the division become less rigid today. What I mean by that is the old, almost hostile, separation between the two traditions is breaking down. The stereotypes of the hyper-logical, science-obsessed Analytic philosopher and the obscure, literary Continental philosopher are becoming increasingly outdated. This convergence is happening in several important ways:
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The Rediscovery of Shared History: For a long time, Analytic philosophers often ignored the history of philosophy, treating problems as timeless logical puzzles. That has changed dramatically. There is now a huge movement within Analytic philosophy to engage seriously with historical figures, including those central to the Continental tradition.
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Overlapping Subject Matter: The thematic walls have started to crumble. Philosophers from both traditions are finding themselves asking the same questions, even if they approach them with different methods.
- From Analytic to Continental: Analytic philosophers are now deeply engaged in topics that were once considered the exclusive territory of Continental thought. The philosophy of mind, for instance, now seriously discusses consciousness, embodiment, and subjective experience - topics central to phenomenology (a core Continental method).
- From Continental to Analytic: Conversely, thinkers in the Continental tradition are increasingly engaging with cognitive science, logic, and even analytic philosophy of language to sharpen their arguments. The idea that a Continental philosopher would be ignorant of science and logic is no longer a given.
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The Influence of “Bridge Figures”: Certain 20th-century philosophers have proven to be so profound that their work is indispensable to both traditions, creating a shared intellectual space. The most important bridge figure is the later Ludwig Wittgenstein. His early work was a cornerstone of Analytic philosophy, but his later work, with its focus on “language-games,” “forms of life,” and the idea that meaning is use, has deep resonances with Continental themes of culture, context, and the limits of formal systems. We can find his influence everywhere, from Analytic philosophy of language to Continental social theory.
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Globalization and Institutional Change: Finally, the practical, geographical barriers have fallen. The internet, international conferences, and the ease of translation mean that philosophers are no longer isolated in their respective “anglophone” or “European” bubbles. It is now common for a philosophy department in the U.S. or U.K. to have specialists in both traditions, leading to a cross-pollination of ideas among students and faculty.
But what could possibly bridge such a deep intellectual and cultural divide? The convergence wasn’t the result of a single event or a formal peace treaty, but rather a gradual and organic process driven by a recognition that both traditions, in their most dogmatic forms, had reached certain limitations.
For example, both traditions found that their own tools weren’t enough. The Analytic project to solve all problems with logic couldn’t adequately address complex human experiences like consciousness or political justice. At the same time, some Continental philosophy was criticized for becoming too detached from scientific and logical standards. This created a mutual need to look outside their own toolkits.
Such limits of purity initiated the shared return to the very foundations of philosophy, especially to the monumental figures Immanuel Kant and Aristotle. Analytic philosophers later started to realize that one cannot fully understand a philosophical problem without understanding its history. This led them back to the classic texts, not as adversaries, but as sources of profound insight. And it was here, in the shared reading of common ancestors, that they found themselves standing on the same ground as their Continental counterparts.
The old Analytic approach was exemplified by Gottlob Frege, who read Kant primarily to refute his theory of mathematics. The focus was narrow and adversarial. The new Analytic approach involves a much deeper engagement. Analytic epistemologists and ethicists began to see the power in Kant’s ideas about the mind’s active role in structuring experience and his arguments for universal moral duties. Suddenly, they were grappling with the same Kantian problems of subjectivity, reason, and moral law that had animated Continental phenomenology and critical theory for decades. They were, in a sense, finally having the same conversation.
Besides engagement with Kant, the same is true for Aristotle. For decades, Analytic ethics was dominated by debates between utilitarianism (judging actions by their consequences) and deontology (judging actions by rules, heavily influenced by Kant). Beginning in the mid-20th century, there was a massive revival of Virtue Ethics, a direct return to Aristotle. Analytic philosophers started seriously discussing what it means to live a “flourishing life” (eudaimonia) and the importance of character and virtue. In doing so, they found themselves exploring the very same Aristotelian concepts of being, purpose (telos), and human nature that had been central to the work of Continental thinkers like Martin Heidegger.
This rediscovery of a shared heritage was the most crucial factor. It revealed that the two traditions were not alien species but estranged siblings who had simply inherited and emphasized different aspects of the same family legacy. Ultimately, the convergence was driven by the realization that the fundamental questions of philosophy - What is real? What can we know? How should we live? - are too big for any single method or tradition to claim ownership over. And the What can we know? How should we live? - are too big for any single method or tradition to claim ownership over. And the best way to approach them is often to return to the great thinkers like Aristotle and Kant who first framed them for us all.
Unifying AI’s Determinism & Human’s Libertarianism
Given the conflict - a world of AI operating on predictable, causal, and deterministic principles clashing with our deep-seated belief in our own free, undetermined, libertarian will - is a modern manifestation of one of the oldest problems in philosophy, what lessons can we learn from this historical unification and apply it to the future conflicts such as dealing with the AI’s Determinism and human’s Libertarianism? The historical rapprochement we’ve been exploring offers several key strategies for addressing it. The reconciliation between Analytic and Continental philosophy offers not just hope, but a practical roadmap for resolving the apparent contradiction between AI’s determinism and human libertarianism.
The most crucial lesson from that unification, based on what we’ve seen so far, is: When two seemingly irreconcilable positions emerge, the path forward is often found not by pushing further into one’s own camp, but by digging deeper into the shared historical foundations from which both positions grew.
The most productive ground was found when both traditions returned to figures like Kant and Aristotle, who had already framed the essential questions. The determinism vs. libertarianism debate is the quintessential Kantian problem.
Immanuel Kant faced this exact dilemma in the 18th century. On one hand, the scientific worldview of Newton described a universe of unshakeable cause and effect (determinism). On the other hand, our experience of morality and reason demands that we believe we are free (libertarianism). His solution was not to pick one over the other, but to argue that they operate in two different domains:
- The Phenomenal Realm: This is the world of science, of cause and effect, of space and time. It is the world as it appears to us through the senses and is structured by our minds. In this realm, everything, including the workings of our brains and the circuits of an AI, is subject to the laws of nature and is therefore deterministic. An AI is a purely phenomenal entity.
- The Noumenal Realm: This is the world as it is “in itself,” outside of our experience of space, time, and causality. Kant argued that we, as rational beings, are also members of this realm. It is from this noumenal self that our free will and moral duty originate. Our freedom is not a magical ability to break the laws of physics in the phenomenal world; it is the capacity to act according to a law we give ourselves through reason.
The Kantian framework offers a stunningly relevant way to resolve the conflict. We can fully accept that an AI is a deterministic system. We can even accept that our own bodies and brains are deterministic systems within the phenomenal realm of science. However, we can simultaneously maintain that human freedom and moral responsibility belong to a different logical category - that of rational agents acting for reasons. An AI, no matter how complex, operates on causes. A human, in their capacity as a moral agent, operates on reasons. The conflict dissolves when we realize the two concepts are not describing the same thing. The AI’s determinism and our libertarianism do not contradict each other because they are answers to questions asked from two different, but equally valid, standpoints. The challenge, though, is not to prove one side right, but to understand the coherent, unified system of thought that allows both to be true.
Defining the Core Concepts with RigorA side-lession on analytical philosophy’s perspective is that we should precisely define before hitting the start button
The early Analytic philosophers, especially Gottlob Frege, insisted that many philosophical disputes were the result of using ambiguous language. They believed that by clarifying our terms, we could dissolve the problems. This is urgently needed in the AI debate.
What do we actually mean by “choice”? When an AI like AlphaGo makes a move that no human predicted, is that a “choice”? It is a deterministic outcome of its programming and training, yet it appears creative and unconstrained. What do we mean by “consciousness” or “understanding”? As the Chinese Room argument shows, a system can be functionally perfect (deterministic) without possessing what we feel is genuine understanding (the seat of our libertarian freedom). By rigorously defining what we value in “human freedom” - is it unpredictability? Is it moral responsibility? Is it creativity? - we may find that some aspects are not threatened by determinism at all, while others need to be protected.