24 ideas
| 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) |
| 11211 | If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt] |
| Full Idea: If a designated conclusion follows from the premisses, but the argument involves two howlers which cancel each other out, then the moral is that the path an argument takes from premisses to conclusion does matter to its logical evaluation. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], II) | |
| A reaction: The drift of this is that our view of logic should be a little closer to the reasoning of ordinary language, and we should rely a little less on purely formal accounts. |
| 11210 | Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt] |
| Full Idea: If 'and' and 'but' really are alike in sense, in what might that likeness consist? Some philosophers of classical logic will reply that they share a sense by virtue of sharing a truth table. | |
| From: Ian Rumfitt ("Yes" and "No" [2000]) | |
| A reaction: This is the standard view which Rumfitt sets out to challenge. |
| 11212 | The sense of a connective comes from primitively obvious rules of inference [Rumfitt] |
| Full Idea: A connective will possess the sense that it has by virtue of its competent users' finding certain rules of inference involving it to be primitively obvious. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], III) | |
| A reaction: Rumfitt cites Peacocke as endorsing this view, which characterises the logical connectives by their rules of usage rather than by their pure semantic value. |
| 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) |
| 11214 | We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt] |
| Full Idea: The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], IV) | |
| A reaction: [compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role). |
| 4061 | The right to life is not a right not to be killed, but not to be killed unjustly [Thomson] |
| Full Idea: Maybe the right to life consists not in the right not to be killed, but in the right not to be killed unjustly. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.131) | |
| A reaction: Sounds tautological. There is no right to life, then, but just the requirement that people behave justly? |
| 4057 | A newly fertilized ovum is no more a person than an acorn is an oak tree [Thomson] |
| Full Idea: A newly fertilized ovum, a newly implanted clump of cells, is no more a person than an acorn is an oak tree. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.125) | |
| A reaction: This relies heavily on the philosopher's concept of a 'person', but it seems right to me. |
| 4695 | Maybe abortion can be justified despite the foetus having full human rights [Thomson, by Foot] |
| Full Idea: Thomson suggests that abortion can be justified without the need to deny that the foetus has the moral rights of a human person. | |
| From: report of Judith (Jarvis) Thomson (A Defense of Abortion [1971]) by Philippa Foot - Killing and Letting Die p.86 | |
| A reaction: Thomson uses a dubious analogy between pregnancy and being hooked up to someone for life-support. Presumably killing an innocent person is occasionally justifiable, but the situation would normally be more abnormal than pregnancy. |
| 4059 | It can't be murder for a mother to perform an abortion on herself to save her own life [Thomson] |
| Full Idea: It cannot seriously be thought to be murder if a mother performs an abortion on herself to save her own life (if, say, she had a serious heart condition). | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.127) | |
| A reaction: An extreme view might condemn such an action, but it can hardly be based on the 'sanctity of life'. |
| 4696 | The foetus is safe in the womb, so abortion initiates its death, with the mother as the agent. [Foot on Thomson] |
| Full Idea: A fetus is not in jeopardy because it is in the womb, so an abortion originates the fatal sequence, and the mother is the agent. Hence Thomson's argument is invalid, and we must return to question of the moral status of the foetus. | |
| From: comment on Judith (Jarvis) Thomson (A Defense of Abortion [1971]) by Philippa Foot - Killing and Letting Die p.86 | |
| A reaction: The problem would be if a 'person' was safe, but only if I continue some sustained effort which is not required of me by normal duties. |
| 4058 | Is someone's right to life diminished if they were conceived by a rape? [Thomson] |
| Full Idea: Can we say that a person has a right to life only if they didn't come into existence through rape, or that the latter have less right to life? | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.126) | |
| A reaction: This would clearly be an inconsistency for some opponents of abortion who allow rape as an exception. |
| 4060 | The right to life does not bestow the right to use someone else's body to support that life [Thomson] |
| Full Idea: Having a right to life does not guarantee having either a right to be given the use of or a right to be allowed continued use of another person's body. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.131) | |
| A reaction: A very nice point. You have a right to your life once you are the sole owner of it. |
| 4062 | No one is morally required to make huge sacrifices to keep someone else alive for nine months [Thomson] |
| Full Idea: No one is morally required to make large sacrifices, of health, and other interests and commitments, for nine months, in order to keep another person alive. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.135) | |
| A reaction: It is a trade-off. It might become a duty if society (or even a husband) urgently needed the baby. |