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Logical Investigations
Contains the essays "Thoughts," "Negation," and "Compound Thoughts."
82 pages, Hardcover
First published January 1, 1966
About the author
Gottlob Frege
131 books182 followersFriedrich Ludwig Gottlob Frege (German: [ˈɡɔtloːp ˈfreːɡə]) was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics. While he was mainly ignored by the intellectual world when he published his writings, Giuseppe Peano (1858–1932) and Bertrand Russell (1872–1970) introduced his work to later generations of logicians and philosophers.
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June 18, 2021
I rarely discuss more academic books on here, but I came upon a review of Frege's Logische Untersuchungen that made me think about what I would say if asked why someone should read it. Is his work dry? Is it outdated?
Frege's contributions to the history of ideas surely can be elusive. The foundation for his theory is a Platonic realism about the contents of thought and language that only a mathematician will immediately be willing to embrace. After all, mathematicians already believe in the reality of functions and numbers, so for them it's not so big of a deal. For other readers, however, his metaphysical abstractions will almost certainly appear esoteric at first. More importantly, it's not easy to see why his theory matters.
What we now call his Logische Untersuchungen is a collection of philosophical essays that Frege wrote and published at the very end of his life. The collection mainly consists of three articles, "Der Gedanke", "Die Verneinung", and "Das Gedankengefüge". The articles were published in the journal Beiträge zur Philosophie des deutschen Idealismus, in the years 1918, 1918 and 1923, respectively. In order to get a clearer picture of why his ideas are of interest, I will discuss some of the themes developed in these works.
In "Der Gedanke" Frege develops an understanding of the science of logic according to which its task is to discover the laws of thinking. Talks of "laws" is clearly to be understood as a honorific. One such law is the modus ponens, exemplified by reasoning such as: All men are mortal; Socrates is a man; Therefore, Socrates is mortal. Frege conceives of them as laws of being true. The argument just cited instantiates a structure that guarantees the truth of the conclusion given the truth of the premises.
This leads to one important contribution of Frege to the history of ideas. From this objective state of affairs - the fact that there are laws that govern how truth (or information) is conveyed from sets of sentences to other sentences - follow normative rules of inference, rules that explicate how you should reason. So, by discovering the laws of logic we get a clearer understanding of human rationality.
Since logic is concerned with truth, as opposed to other sciences that strive for truths, Frege goes on to discuss the nature of truth. He observes that we speak of "truth" in regard to different entities, including sentences, ideas ("Vorstellungen"), pictures, and thoughts. A picture is true (well, I guess we would rather say something like "accurate") when we take it to represent something and when there is a satisfying degree of accordance between the representing picture and the thing represented. Naturally, this leads Frege to a discussion of the Correspondence Theory of Truth, the idea that something is true if it is in accordance with the thing it represents.
Frege rejects this famous theory. Approaching the issue, he observes that correspondence is a relation while "truth" is a property word (compare "green" with "is father of"). More importantly, he thinks that a definition of truth in this sense is necessarily circular: "[...] [E]very [...] attempt to define truth collapses [...]. For in a definition certain characteristics would have to be stated. And in application to any particular case the question would always arise whether it were true that the characteristics were present. So one goes round in a circle." I have to admit, I'm not sure about the merit of this criticism.
Frege goes on to analyze the meaning structures expressed by declarative sentences, a theory that had been immensely influential in the philosophy of language as well as in linguistic semantics. According to his analysis, we have to distinguish different components of an utterance of a declarative sentence: 1. the thought expressed (what he calls the "sense" of the sentence), 2. the (prior) mental judgement of the thought being true, and 3. the verbal assertion of that judgment.
As the title of the article suggests, Frege is mainly interested in the first component, the thought. However, in order to sharpen this semantic aspect he distinguishes it from pragmatic concerns you may have about the utterance. Frege is often praised as the father of modern logic and semantics, but he is probably also the first thinker who clearly distinguished between the semantics and pragmatics of sentences in the modern sense.
Before I go on to discuss the nature of thought and the relationship between sentences and thoughts, I should emphasize again the importance of distinguishing the three components. As a first approximation, Frege characterizes the thought as something that can be true or false. He observes that many writers in the philosophical tradition instead took judgements to be the truth-bearers. Talk of judgements takes center stage in the works of authors such as Descartes, Leibniz, or Kant.
I think Frege rightly criticizes their way of using the word "judgment" as often conflating the components distinguished above. They confuse the mental act of judging with the content being judged (what Wilfried Sellars and Robert Brandon calls "the notorious 'ing'/'ed' ambiguity"). Similarly, we need to distinguish the verbal act of asserting from the content being asserted.
In this way Frege introduces the important distinction stressed in speech act theory, that I can express a content verbally with different (illocutionary) force. For instance, other than asserting that Peter left the house, I may demand that Peter left the house, or I may ask whether Peter left the house. Corresponding to the verbal forces, there are types of attitudes I can take towards the thoughts. For instance, I may belief that Peter left the house, or wish that Peter left the house, or wonder whether Peter left the house. In all these cases, the content the same, and the laws of logic concern the contents of sentences.
I spoke of the contents of sentences as being the bearers of truth, rather than the sentence itself. You may think this as unnecessary pedantic. On the contrary, Frege clearly worked out why the distinction is crucial, by talking of thoughts as being expressed by sentences. He observes that one and the same sentence can be used to express different thoughts and that different sentences may express the same thought.
The latter observation is familiar enough, as I guess most people are aware that certain syntactic transformations of sentences won't influence the meaning (as the difference between active and passive modes of the verb). Similarly, the use of so-called indexical expressions require of the heater to interpret them in regard to the conversational context, so that the sentence may say one thing under one circumstance and another thing in a different situation. For instance, "Today it's raining" expresses a different thought than is expressed by uttering the same sentence tomorrow (but the same thought as "Yesterday it was raining" uttered tomorrow).
However, there are two aspects that are much less obvious and for which it's quite likely that Frege was the first author to clearly see the issues. Firstly, two sentences may express the same thought, but present it In forms that appear very different to the heater. For instance, "Angela Merkel is the Chancellor of Germany and she is a woman" has a much more neutral feel to it than "Angela Merkel is the Chancellor of Germany, even though she is a woman". Notice that both sentences are true under the exact same conditions (it's both the case that she is Chancellor and a woman), but the second sentence presents the fact as if there was some sort of a contrast between the component thoughts. "unfortunately" or "luckily" similarly express a certain stance towards the thought expressed. Logic only analyses the meaning of sentences in as far as it concerns truth.
The other aspect discussed by Frege in this context I think is truly fascinating. He points to the fact (if it is a fact) that the way we represent the subject of the sentence (the thing to which we ascribe properties) has bearing on which thought we express with the sentence in a given situation. Frege's gives a rather complicated example about a doctor who said something in a situation and two other people later repeated what he said. However, Frege is much more famous for another example he give in another article, "Über Sinn und Bedeutung".
There he wonders how "The Morning Star is the Evening Star" could be informative. Think about it, if the meaning of names was exhausted by its referent, then the informativeness is surprising, since both names refer to the same star, the Venus, and this is why the identity statement is true. However, if we represent the stars in a certain way ("the brightest star in the morning/evening sky", for instance) it becomes clear that it's possible that someone can find out that one and the same thing may satisfy both descriptions. So, two people uttering the same sentence may express different thoughts, if they represent the subject (or even the predicate) differently.
Frege argues that thoughts form their own domain of reality. As there is the outer world of observable things and the inner world of ideas and decisions, there is the world of thoughts that can be grasped. His argument is quite simple. Thoughts are like mental entities, and unlike the things of the outer world, in that they cannot be observed with our senses. Although certainly problematic, as modern scientific theories postulate many non-observables (atoms, processes of evolution, social structures, what have you), I will grant him that point. More importantly for the rest of the article, Frege argues that thoughts are independent of thinkers, unlike ideas and similar to the things of the outer world.
Frege's argumentative strategy is to reduce the counter-theses - that thoughts are realized in the minds of individuals - to absurdity. What we discuss in the sciences are thoughts. To give his own example, if thoughts where subjective, then we would have to speak of my Pythagorean Theorem and your Pythagorean Theorem, as it would be something that is in your head or in mine. Truth would then be something similar to colors, which are a feature of my mental impressions of things (or so Frege assumes). Scientific dispute would no longer be possible.
I'm again not sure about the merits of these considerations. I feel as if a distinction between individual thoughts (tokens) as representing general thoughts (types) would already solve the issue and would be ontologically much less demanding. Frege also goes on to discuss scepticism about the external world, which takes about one third of the entire paper. The thesis that all external things are in fact ideas is relevant for his argument, since the thesis that thoughts form a realm sui generis depends on the existence of other people (other thinkers), and there wouldn't be if the scepticism was warranted.
I'm sure most readers of philosophy came upon presentations of these arguments, so it's unlikely that they pick up Frege with these interests in mind. Still, his summary of the sceptic's position is decent enough as a refresher. There was one aspect of his argument, though, that I found very noteworthy. The sceptic may point out that we don't see the external world; what we have is only our mental representations of that world. So, when we think about things what we are actually thinking about are the ideas in our mind. If you grant this point, then the sceptic may ask: How would we know that there is something that our ideas represent? How would we know that our mind doesn't create the external world on it's own?
Frege rightly rejects the assumption that our mind addresses ideas. This is an undue reduplication of entities. Ideas need a bearer (the I or Self), but it's not like the I somehow looks at the idea as if through a window into the world. If that were true, then the I would have an idea representing the idea, which obviously would lead to an infinite regress. The I has ideas, but usually the ideas are not what forms the content of our consciousness. What we are conscious of (through our representing ideas) is the external world. This is what Hilary Putnam calls the need for a "second naïveté” in the theory of perception.
"Das Gedankengefüge" is another paper that marked many firsts. The general topic is the fact that compound sentences ("Satzgefüge") may be used to express what Frege calls compound thoughts ("Gedankengefüge"). He defines six such structures and gives words and phrases from natural language that are usually used to form the corresponding compound sentences. The first compound thought he defines as being true if and only if both component thoughts are true, a complex thought canonically expressed by "and" between full sentences and today usually called a conjunction.
It may easily be overlooked that this is one application of what may be Frege's most lasting contribution (other than quantificational logic). In a very general sense, Fregian ontology distinguishes two kinds of entities, things that can exist on their own and things that need to be saturated by other things. The unsaturated entities he conceives of as functions, as things that take an input (the "argument") and return an output (the "value"). Functions play a foundational role in mathematics.
In "Funktion und Begriff" (1891), Frege argued that functions can be much more widely applied than was assumed at his time. It's not only numbers (or other sorts of mathematical entities) that a function may take as input or outputs, but also individuals and truth-values. Surprisingly, he went on to assert that verbs or adjectives express functions that return truth-values. For instance, "wise" expresses a function that returns the value True for all individuals that have the property of being wise. Full sentences are than treated as expressing applications of functions to arguments. For instance, "Socrates is wise" applies the function to the argument Socrates (the referent of the proper name "Socrates") and returns True. Functions that return truth-values he identifies with what is called concepts ("Begriffe") in the philosophical tradition.
Simple thoughts are the result (in the non-temporal sense) of composing individuals and functions. Because of that, he considers the meaning of declarative sentences to be saturated. To return to the topic of compound thoughts, compound sentences contain another component that can be interpreted as expressing functions, the sentential connectives like "and", "or", and "if... then...". These are functions of a different kind, though. Instead of taking individuals as arguments, these functions take truth-values and return truth-values.
This is one instance of Frege's general strategy of taking all semantic composition to be a case of functional application (what Heim and Kratzer call "Frege's Conjecture"). This conjecture has been highly influential in modern generative grammar, since nowadays it's often assumed that to every rule of syntactic composition there is a semantic rule functional application. In this way, the grammar explains our extraordinary ability to produce and understand sentences we never encountered before.
Since it only involves one thought, the negation (of a thought) is not considered a compound thought. However, since it also involves a connective ("It's not the case that" in natural language) the similarity to compound thoughts is obvious. Unlike most presentations in modern-day textbooks, Frege doesn't introduce disjunction, material implication, material equivalence, NAND, and NOR as being on the same level as conjunction and negation. Instead, he uses the latter two to define the others. In effect, he reduces them to these two operations (as authors like Schönfinkel or Sheffer called it).
In language-related terms, he shows that languages that only contain the words "and" and "it's not the case that" and their cognates are just as expressive as languages that additionally contain "if then", "or", "if and only if", or "neither... nor". He shows the same for conditional and negation, and points out that disjunction and negation could have been used as starting point as well. It was only later that Sheffer discovered that in fact there are two truth-functions which allow the reduction all by themselves, NAND (expressed by the Sheffer stroke) and NOR. Well, as is often the case, Peirce showed the same thing even earlier than Frege even published his work.
At first sight, this result may not exactly be exciting. However, think about it. People like to compare languages, saying things like "Hungarian is richer than English" or "Turkish is a poetic language". Usually, these statements are rather ill-defined. For much simpler languages, languages of propositional logic, Frege was able to explain what expressiveness actually amounts to and how you could qualify the issues involved. So, he came to a fully principles conclusion.
Rating: 4/5
Frege's contributions to the history of ideas surely can be elusive. The foundation for his theory is a Platonic realism about the contents of thought and language that only a mathematician will immediately be willing to embrace. After all, mathematicians already believe in the reality of functions and numbers, so for them it's not so big of a deal. For other readers, however, his metaphysical abstractions will almost certainly appear esoteric at first. More importantly, it's not easy to see why his theory matters.
What we now call his Logische Untersuchungen is a collection of philosophical essays that Frege wrote and published at the very end of his life. The collection mainly consists of three articles, "Der Gedanke", "Die Verneinung", and "Das Gedankengefüge". The articles were published in the journal Beiträge zur Philosophie des deutschen Idealismus, in the years 1918, 1918 and 1923, respectively. In order to get a clearer picture of why his ideas are of interest, I will discuss some of the themes developed in these works.
In "Der Gedanke" Frege develops an understanding of the science of logic according to which its task is to discover the laws of thinking. Talks of "laws" is clearly to be understood as a honorific. One such law is the modus ponens, exemplified by reasoning such as: All men are mortal; Socrates is a man; Therefore, Socrates is mortal. Frege conceives of them as laws of being true. The argument just cited instantiates a structure that guarantees the truth of the conclusion given the truth of the premises.
This leads to one important contribution of Frege to the history of ideas. From this objective state of affairs - the fact that there are laws that govern how truth (or information) is conveyed from sets of sentences to other sentences - follow normative rules of inference, rules that explicate how you should reason. So, by discovering the laws of logic we get a clearer understanding of human rationality.
Since logic is concerned with truth, as opposed to other sciences that strive for truths, Frege goes on to discuss the nature of truth. He observes that we speak of "truth" in regard to different entities, including sentences, ideas ("Vorstellungen"), pictures, and thoughts. A picture is true (well, I guess we would rather say something like "accurate") when we take it to represent something and when there is a satisfying degree of accordance between the representing picture and the thing represented. Naturally, this leads Frege to a discussion of the Correspondence Theory of Truth, the idea that something is true if it is in accordance with the thing it represents.
Frege rejects this famous theory. Approaching the issue, he observes that correspondence is a relation while "truth" is a property word (compare "green" with "is father of"). More importantly, he thinks that a definition of truth in this sense is necessarily circular: "[...] [E]very [...] attempt to define truth collapses [...]. For in a definition certain characteristics would have to be stated. And in application to any particular case the question would always arise whether it were true that the characteristics were present. So one goes round in a circle." I have to admit, I'm not sure about the merit of this criticism.
Frege goes on to analyze the meaning structures expressed by declarative sentences, a theory that had been immensely influential in the philosophy of language as well as in linguistic semantics. According to his analysis, we have to distinguish different components of an utterance of a declarative sentence: 1. the thought expressed (what he calls the "sense" of the sentence), 2. the (prior) mental judgement of the thought being true, and 3. the verbal assertion of that judgment.
As the title of the article suggests, Frege is mainly interested in the first component, the thought. However, in order to sharpen this semantic aspect he distinguishes it from pragmatic concerns you may have about the utterance. Frege is often praised as the father of modern logic and semantics, but he is probably also the first thinker who clearly distinguished between the semantics and pragmatics of sentences in the modern sense.
Before I go on to discuss the nature of thought and the relationship between sentences and thoughts, I should emphasize again the importance of distinguishing the three components. As a first approximation, Frege characterizes the thought as something that can be true or false. He observes that many writers in the philosophical tradition instead took judgements to be the truth-bearers. Talk of judgements takes center stage in the works of authors such as Descartes, Leibniz, or Kant.
I think Frege rightly criticizes their way of using the word "judgment" as often conflating the components distinguished above. They confuse the mental act of judging with the content being judged (what Wilfried Sellars and Robert Brandon calls "the notorious 'ing'/'ed' ambiguity"). Similarly, we need to distinguish the verbal act of asserting from the content being asserted.
In this way Frege introduces the important distinction stressed in speech act theory, that I can express a content verbally with different (illocutionary) force. For instance, other than asserting that Peter left the house, I may demand that Peter left the house, or I may ask whether Peter left the house. Corresponding to the verbal forces, there are types of attitudes I can take towards the thoughts. For instance, I may belief that Peter left the house, or wish that Peter left the house, or wonder whether Peter left the house. In all these cases, the content the same, and the laws of logic concern the contents of sentences.
I spoke of the contents of sentences as being the bearers of truth, rather than the sentence itself. You may think this as unnecessary pedantic. On the contrary, Frege clearly worked out why the distinction is crucial, by talking of thoughts as being expressed by sentences. He observes that one and the same sentence can be used to express different thoughts and that different sentences may express the same thought.
The latter observation is familiar enough, as I guess most people are aware that certain syntactic transformations of sentences won't influence the meaning (as the difference between active and passive modes of the verb). Similarly, the use of so-called indexical expressions require of the heater to interpret them in regard to the conversational context, so that the sentence may say one thing under one circumstance and another thing in a different situation. For instance, "Today it's raining" expresses a different thought than is expressed by uttering the same sentence tomorrow (but the same thought as "Yesterday it was raining" uttered tomorrow).
However, there are two aspects that are much less obvious and for which it's quite likely that Frege was the first author to clearly see the issues. Firstly, two sentences may express the same thought, but present it In forms that appear very different to the heater. For instance, "Angela Merkel is the Chancellor of Germany and she is a woman" has a much more neutral feel to it than "Angela Merkel is the Chancellor of Germany, even though she is a woman". Notice that both sentences are true under the exact same conditions (it's both the case that she is Chancellor and a woman), but the second sentence presents the fact as if there was some sort of a contrast between the component thoughts. "unfortunately" or "luckily" similarly express a certain stance towards the thought expressed. Logic only analyses the meaning of sentences in as far as it concerns truth.
The other aspect discussed by Frege in this context I think is truly fascinating. He points to the fact (if it is a fact) that the way we represent the subject of the sentence (the thing to which we ascribe properties) has bearing on which thought we express with the sentence in a given situation. Frege's gives a rather complicated example about a doctor who said something in a situation and two other people later repeated what he said. However, Frege is much more famous for another example he give in another article, "Über Sinn und Bedeutung".
There he wonders how "The Morning Star is the Evening Star" could be informative. Think about it, if the meaning of names was exhausted by its referent, then the informativeness is surprising, since both names refer to the same star, the Venus, and this is why the identity statement is true. However, if we represent the stars in a certain way ("the brightest star in the morning/evening sky", for instance) it becomes clear that it's possible that someone can find out that one and the same thing may satisfy both descriptions. So, two people uttering the same sentence may express different thoughts, if they represent the subject (or even the predicate) differently.
Frege argues that thoughts form their own domain of reality. As there is the outer world of observable things and the inner world of ideas and decisions, there is the world of thoughts that can be grasped. His argument is quite simple. Thoughts are like mental entities, and unlike the things of the outer world, in that they cannot be observed with our senses. Although certainly problematic, as modern scientific theories postulate many non-observables (atoms, processes of evolution, social structures, what have you), I will grant him that point. More importantly for the rest of the article, Frege argues that thoughts are independent of thinkers, unlike ideas and similar to the things of the outer world.
Frege's argumentative strategy is to reduce the counter-theses - that thoughts are realized in the minds of individuals - to absurdity. What we discuss in the sciences are thoughts. To give his own example, if thoughts where subjective, then we would have to speak of my Pythagorean Theorem and your Pythagorean Theorem, as it would be something that is in your head or in mine. Truth would then be something similar to colors, which are a feature of my mental impressions of things (or so Frege assumes). Scientific dispute would no longer be possible.
I'm again not sure about the merits of these considerations. I feel as if a distinction between individual thoughts (tokens) as representing general thoughts (types) would already solve the issue and would be ontologically much less demanding. Frege also goes on to discuss scepticism about the external world, which takes about one third of the entire paper. The thesis that all external things are in fact ideas is relevant for his argument, since the thesis that thoughts form a realm sui generis depends on the existence of other people (other thinkers), and there wouldn't be if the scepticism was warranted.
I'm sure most readers of philosophy came upon presentations of these arguments, so it's unlikely that they pick up Frege with these interests in mind. Still, his summary of the sceptic's position is decent enough as a refresher. There was one aspect of his argument, though, that I found very noteworthy. The sceptic may point out that we don't see the external world; what we have is only our mental representations of that world. So, when we think about things what we are actually thinking about are the ideas in our mind. If you grant this point, then the sceptic may ask: How would we know that there is something that our ideas represent? How would we know that our mind doesn't create the external world on it's own?
Frege rightly rejects the assumption that our mind addresses ideas. This is an undue reduplication of entities. Ideas need a bearer (the I or Self), but it's not like the I somehow looks at the idea as if through a window into the world. If that were true, then the I would have an idea representing the idea, which obviously would lead to an infinite regress. The I has ideas, but usually the ideas are not what forms the content of our consciousness. What we are conscious of (through our representing ideas) is the external world. This is what Hilary Putnam calls the need for a "second naïveté” in the theory of perception.
"Das Gedankengefüge" is another paper that marked many firsts. The general topic is the fact that compound sentences ("Satzgefüge") may be used to express what Frege calls compound thoughts ("Gedankengefüge"). He defines six such structures and gives words and phrases from natural language that are usually used to form the corresponding compound sentences. The first compound thought he defines as being true if and only if both component thoughts are true, a complex thought canonically expressed by "and" between full sentences and today usually called a conjunction.
It may easily be overlooked that this is one application of what may be Frege's most lasting contribution (other than quantificational logic). In a very general sense, Fregian ontology distinguishes two kinds of entities, things that can exist on their own and things that need to be saturated by other things. The unsaturated entities he conceives of as functions, as things that take an input (the "argument") and return an output (the "value"). Functions play a foundational role in mathematics.
In "Funktion und Begriff" (1891), Frege argued that functions can be much more widely applied than was assumed at his time. It's not only numbers (or other sorts of mathematical entities) that a function may take as input or outputs, but also individuals and truth-values. Surprisingly, he went on to assert that verbs or adjectives express functions that return truth-values. For instance, "wise" expresses a function that returns the value True for all individuals that have the property of being wise. Full sentences are than treated as expressing applications of functions to arguments. For instance, "Socrates is wise" applies the function to the argument Socrates (the referent of the proper name "Socrates") and returns True. Functions that return truth-values he identifies with what is called concepts ("Begriffe") in the philosophical tradition.
Simple thoughts are the result (in the non-temporal sense) of composing individuals and functions. Because of that, he considers the meaning of declarative sentences to be saturated. To return to the topic of compound thoughts, compound sentences contain another component that can be interpreted as expressing functions, the sentential connectives like "and", "or", and "if... then...". These are functions of a different kind, though. Instead of taking individuals as arguments, these functions take truth-values and return truth-values.
This is one instance of Frege's general strategy of taking all semantic composition to be a case of functional application (what Heim and Kratzer call "Frege's Conjecture"). This conjecture has been highly influential in modern generative grammar, since nowadays it's often assumed that to every rule of syntactic composition there is a semantic rule functional application. In this way, the grammar explains our extraordinary ability to produce and understand sentences we never encountered before.
Since it only involves one thought, the negation (of a thought) is not considered a compound thought. However, since it also involves a connective ("It's not the case that" in natural language) the similarity to compound thoughts is obvious. Unlike most presentations in modern-day textbooks, Frege doesn't introduce disjunction, material implication, material equivalence, NAND, and NOR as being on the same level as conjunction and negation. Instead, he uses the latter two to define the others. In effect, he reduces them to these two operations (as authors like Schönfinkel or Sheffer called it).
In language-related terms, he shows that languages that only contain the words "and" and "it's not the case that" and their cognates are just as expressive as languages that additionally contain "if then", "or", "if and only if", or "neither... nor". He shows the same for conditional and negation, and points out that disjunction and negation could have been used as starting point as well. It was only later that Sheffer discovered that in fact there are two truth-functions which allow the reduction all by themselves, NAND (expressed by the Sheffer stroke) and NOR. Well, as is often the case, Peirce showed the same thing even earlier than Frege even published his work.
At first sight, this result may not exactly be exciting. However, think about it. People like to compare languages, saying things like "Hungarian is richer than English" or "Turkish is a poetic language". Usually, these statements are rather ill-defined. For much simpler languages, languages of propositional logic, Frege was able to explain what expressiveness actually amounts to and how you could qualify the issues involved. So, he came to a fully principles conclusion.
Rating: 4/5
January 26, 2016
Boring as hell and outdated. Got nothing out of this, I'm sorry.
August 9, 2021
昨天看了the thought这篇，到现在还在头疼...
Displaying 1 - 3 of 3 reviews