26 ideas
| 8092 | Logic was merely a branch of rhetoric until the scientific 17th century [Devlin] |
| Full Idea: Until the rise of what we call the scientific method in the seventeenth century, logic was regarded largely as one aspect of rhetoric - a study of how one person't argument could convince another. | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch.11) | |
| A reaction: This may well give the main reason why the Greeks invented logic in the first place. Aristotle wrote a book on rhetoric, and that was where the money was. Leibniz is clearly a key figure in the change of attitude. |
| 8081 | 'No councillors are bankers' and 'All bankers are athletes' implies 'Some athletes are not councillors' [Devlin] |
| Full Idea: Most people find it hard to find any conclusion that fits the following premises: 'No councillors are bankers', and 'All bankers are athletes'. There is a valid conclusion ('Some athletes are not councillors') but it takes quite an effort to find it. | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 2) | |
| A reaction: A nice illustration of the fact that syllogistic logic is by no means automatic and straightforward. There is a mechanical procedure, but a lot of intuition and common sense is also needed. |
| 8085 | Modern propositional inference replaces Aristotle's 19 syllogisms with modus ponens [Devlin] |
| Full Idea: Where Aristotle had 19 different inference rules (his valid syllogisms), modern propositional logic carries out deductions using just one rule of inference: modus ponens. | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 4) | |
| A reaction: At first glance it sounds as if Aristotle's guidelines might be more useful than the modern one, since he tells you something definite and what implies what, where modus ponens just seems to define the word 'implies'. |
| 8086 | Predicate logic retains the axioms of propositional logic [Devlin] |
| Full Idea: Since predicate logic merely extends propositional logic, all the axioms of propositional logic are axioms of predicate logic. | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 4) | |
| A reaction: See Idea 7798 for the axioms. |
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: [He credits Sierpinski (1918) with this discovery] |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
| 8091 | Situation theory is logic that takes account of context [Devlin] |
| Full Idea: In many respects, situation theory is an extension of classical logic that takes account of context. | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 8) | |
| A reaction: John Barwise is cited as the parent of this movement. Many examples show that logical form is very hard to pin down, because word-meaning depends on context (e.g. 'several crumbs' differs from 'several mountains'). |
| 8087 | Golden ages: 1900-1960 for pure logic, and 1950-1985 for applied logic [Devlin] |
| Full Idea: The period from 1900 to about 1960 could be described as the golden age of 'pure' logic, and 1950 to 1985 the golden age of 'applied' logic (e.g. applied to everyday reasoning, and to theories of language). | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 4) | |
| A reaction: Why do we always find that we have just missed the Golden Age? However this supports the uneasy feeling that the golden age for all advances in human knowledge is just coming to an end. Biology, including the brain, is the last frontier. |
| 8089 | Montague's intensional logic incorporated the notion of meaning [Devlin] |
| Full Idea: Montague's intensional logic was the first really successful attempt to develop a mathematical framework that incorporates the notion of meaning. | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 8) | |
| A reaction: Previous logics, led by Tarski, had flourished by sharply dividing meaning from syntax, and concentrating on the latter. |
| 8082 | Where a conditional is purely formal, an implication implies a link between premise and conclusion [Devlin] |
| Full Idea: Implication involves some form of link or causality between the antecedent and the consequent of an if-then; normally it says that the conclusion is a consequence of the premise (where conditionals are just defined by 'true' and 'false'). | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 2) | |
| A reaction: This distinction is a key one when discussing 'If-then' sentences. Some are merely formal conditionals, but others make real claims about where you can get to from where you are. |
| 8072 | Sentences of apparent identical form can have different contextual meanings [Devlin] |
| Full Idea: "Safety goggles must be worn in the building" is clear enough, but "dogs must always be carried on the escalator" doesn't require us to head off in search of a dog. | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 1) | |
| A reaction: A nice illustration of how the requirements of logical form will often take us beyond the strict and literal meaning of a sentence, into context, tone, allusion and subjective aspects. |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| Full Idea: A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key. |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| Full Idea: In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic. |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| Full Idea: The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory. |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| Full Idea: Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| Full Idea: Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| Full Idea: 'Recursion theory' is the subject of what can and cannot be solved by computing machines | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Ch.9) | |
| A reaction: This because 'recursion' will grind out a result step-by-step, as long as the steps will 'halt' eventually. |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| Full Idea: In 1936 Church showed that Principia Mathematica is undecidable if it is ω-consistent, and a year later Rosser showed that Peano Arithmetic is undecidable, and any consistent extension of it. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int IV) |
| 8075 | Space and time are atomic in the arrow, and divisible in the tortoise [Devlin] |
| Full Idea: The arrow paradox starts with the assumption that space and time are atomic; the tortoise starts with the opposite assumption that space and time are infinitely divisible. | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 2) | |
| A reaction: Aquinas similarly covers all options (the cosmos has a beginning, or no beginning). The nature of movement in a space which involves quantum leaps remains metaphysically puzzling. Where is a particle at half of the Planck time? |
| 8088 | People still say the Hopi have no time concepts, despite Whorf's later denial [Devlin] |
| Full Idea: The Hopi time myth does not appear to have been stopped for a moment by the fact that Whorf himself subsequently wrote that the Hopi language does indeed have words for past, present, and future | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 5) | |
| A reaction: Arguments for relativism based on the Hopi seem now to be thoroughly discredited. Sensible people never believed them in the first place. |
| 8073 | How do we parse 'time flies like an arrow' and 'fruit flies like an apple'? [Devlin] |
| Full Idea: How do people identify subject and verb in the sentences "time flies like an arrow" and "fruit flies like an apple"? | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 1) | |
| A reaction: A nice illustration of the fact that even if we have an innate syntax mechanism, it won't work without some semantics, and some experience of the environmental context of utterances. |
| 8076 | The distinction between sentences and abstract propositions is crucial in logic [Devlin] |
| Full Idea: The distinction between sentences and the abstract propositions that they express is one of the key ideas of logic. A logical argument consists of propositions, assembled together in a systematic fashion. | |
| From: Keith Devlin (Goodbye Descartes [1997], Ch. 2) | |
| A reaction: He may claim that arguments consist of abstract propositions, but they always get expressed in sentences. However, the whole idea of logical form implies the existence of propositions - there is something which a messy sentence 'really' says. |
| 1590 | The just man does not harm his enemies, but benefits everyone [Plato] |
| Full Idea: First, Socrates, you told me justice is harming your enemies and helping your friends. But later it seemed that the just man, since everything he does is for someone's benefit, never harms anyone. | |
| From: Plato (Clitophon [c.372 BCE], 410b) | |
| A reaction: Socrates certainly didn't subscribe to the first view, which is the traditional consensus in Greek culture. In general Socrates agreed with the views later promoted by Jesus. |