History Of Modern Logic
Posted by
Ali Reda
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Posted in
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Posted on
11/15/2014
The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if:
Some cat is feared by every mouse then it follows logically that:
All mice are afraid of at least one cat
The syntax of traditional logic permits exactly four sentence types:
"All As are Bs"
"No As are Bs"
"Some As are Bs"
"Some As are not Bs".
Each type is a quantified sentence containing exactly one quantifier. Since the sentences above each contain two quantifiers ('some' and 'every' in the first sentence and 'all' and 'at least one' in the second sentence), they cannot be adequately represented in TL. The best TL can do is to incorporate the second quantifier from each sentence into the second term, thus rendering the artificial-sounding terms 'feared-by-every-mouse' and 'afraid-of-at-least-one-cat'. This in effect "buries" these quantifiers, which are essential to the inference's validity, within the hyphenated terms. Hence the sentence "Some cat is feared by every mouse" is given the same logical form as the sentence "Some cat is hungry".
Leibniz
Leibniz was the first to dream of the Characteristica Universalis, the perfect language which would provide a direct representation of ideas along with a calculus for the philosophical reasoning.
Bolzano
Bolzano is mainly concerned with three realms:
(1) The realm of language, consisting in words and sentences.
(2) The realm of thought, consisting in subjective ideas and judgments.
(3) The realm of logic, consisting in objective ideas (or ideas in themselves) and propositions in themselves.
According to Bolzano, all propositions are composed out of three (simple or complex) elements: a subject, a predicate and a copula. Instead of the more traditional copulative term 'is', Bolzano prefers 'has'. The reason for this is that 'has', unlike 'is', can connect a concrete term, such as 'Socrates', to an abstract term such as 'baldness'. "Socrates has baldness" is, according to Bolzano, preferable to "Socrates is bald" because the latter form is less basic: 'bald' is itself composed of the elements 'something', 'that', 'has' and 'baldness'. Bolzano also reduces existential propositions to this form: "Socrates exists" would simply become "Socrates has existence".
It is important to understand that an idea does not need to have an object. Bolzano uses object to denote something that is represented by an idea. An idea that has an object represents that object. But an idea that does not have an object represents nothing. (Don't get confused here by terminology: an objectless idea is an idea without a representation.)
Boole
Boole provided a general symbolic method of logical inference. Boole proposed that logical propositions should be expressed by means of algebraic equations. Algebraic manipulation of the symbols in the equations would provide a fail-safe method of logical deduction. By 1 (unity) Boole denoted the "universe of thinkable objects"; literal symbols, such as x, y, z, v, u, etc., were used with the "elective" meaning attaching to adjectives and nouns of natural language. Thus, if x = horned and y = sheep, then the successive acts of election (i.e. choice) represented by x and y, if performed on unity, give the class "horned sheep". Thus (1 – x) would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 – x) (1 – y) would give all things neither horned nor sheep
Frege
The first logical calculus capable of dealing with multi quantifiers was Frege's Begriffsschrift, the ancestor of modern predicate logic, which dealt with quantifiers by means of variable bindings. Frege invented axiomatic predicate logic, Two common quantifiers are the existential ? ("there exists") and universal ? ("for all") quantifiers. The variables could be elements in the universe under discussion, or perhaps relations or functions over that universe. For Frege, an expression refers to an object by way of a sense: thus, two expressions (say, “the morning star” and “the evening star”) may refer to the same object (Venus) but express different senses with different manners of presentation.
Some cat is feared by every mouse then it follows logically that:
All mice are afraid of at least one cat
The syntax of traditional logic permits exactly four sentence types:
"All As are Bs"
"No As are Bs"
"Some As are Bs"
"Some As are not Bs".
Each type is a quantified sentence containing exactly one quantifier. Since the sentences above each contain two quantifiers ('some' and 'every' in the first sentence and 'all' and 'at least one' in the second sentence), they cannot be adequately represented in TL. The best TL can do is to incorporate the second quantifier from each sentence into the second term, thus rendering the artificial-sounding terms 'feared-by-every-mouse' and 'afraid-of-at-least-one-cat'. This in effect "buries" these quantifiers, which are essential to the inference's validity, within the hyphenated terms. Hence the sentence "Some cat is feared by every mouse" is given the same logical form as the sentence "Some cat is hungry".
Leibniz
Leibniz was the first to dream of the Characteristica Universalis, the perfect language which would provide a direct representation of ideas along with a calculus for the philosophical reasoning.
Bolzano
Bolzano is mainly concerned with three realms:
(1) The realm of language, consisting in words and sentences.
(2) The realm of thought, consisting in subjective ideas and judgments.
(3) The realm of logic, consisting in objective ideas (or ideas in themselves) and propositions in themselves.
According to Bolzano, all propositions are composed out of three (simple or complex) elements: a subject, a predicate and a copula. Instead of the more traditional copulative term 'is', Bolzano prefers 'has'. The reason for this is that 'has', unlike 'is', can connect a concrete term, such as 'Socrates', to an abstract term such as 'baldness'. "Socrates has baldness" is, according to Bolzano, preferable to "Socrates is bald" because the latter form is less basic: 'bald' is itself composed of the elements 'something', 'that', 'has' and 'baldness'. Bolzano also reduces existential propositions to this form: "Socrates exists" would simply become "Socrates has existence".
It is important to understand that an idea does not need to have an object. Bolzano uses object to denote something that is represented by an idea. An idea that has an object represents that object. But an idea that does not have an object represents nothing. (Don't get confused here by terminology: an objectless idea is an idea without a representation.)
Boole
Boole provided a general symbolic method of logical inference. Boole proposed that logical propositions should be expressed by means of algebraic equations. Algebraic manipulation of the symbols in the equations would provide a fail-safe method of logical deduction. By 1 (unity) Boole denoted the "universe of thinkable objects"; literal symbols, such as x, y, z, v, u, etc., were used with the "elective" meaning attaching to adjectives and nouns of natural language. Thus, if x = horned and y = sheep, then the successive acts of election (i.e. choice) represented by x and y, if performed on unity, give the class "horned sheep". Thus (1 – x) would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 – x) (1 – y) would give all things neither horned nor sheep
Frege
The first logical calculus capable of dealing with multi quantifiers was Frege's Begriffsschrift, the ancestor of modern predicate logic, which dealt with quantifiers by means of variable bindings. Frege invented axiomatic predicate logic, Two common quantifiers are the existential ? ("there exists") and universal ? ("for all") quantifiers. The variables could be elements in the universe under discussion, or perhaps relations or functions over that universe. For Frege, an expression refers to an object by way of a sense: thus, two expressions (say, “the morning star” and “the evening star”) may refer to the same object (Venus) but express different senses with different manners of presentation.
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