display all the ideas for this combination of texts
6 ideas
| 10206 | Modal operators are usually treated as quantifiers [Shapiro] |
| Full Idea: It is common now, and throughout the history of philosophy, to interpret modal operators as quantifiers. This is an analysis of modality in terms of ontology. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) |
| 8943 | Three-valued logic says excluded middle and non-contradition are not tautologies [Fisher] |
| Full Idea: In three-valued logic (L3), neither the law of excluded middle (p or not-p), nor the law of non-contradiction (not(p and not-p)) will be tautologies. If p has the value 'indeterminate' then so will not-p. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 07.I) | |
| A reaction: I quite accept that the world is full of indeterminate propositions, and that excluded middle and non-contradiction can sometimes be uncertain, but I am reluctant to accept that what is being offered here should be called 'logic'. |
| 8945 | Fuzzy logic has many truth values, ranging in fractions from 0 to 1 [Fisher] |
| Full Idea: In fuzzy logic objects have properties to a greater or lesser degree, and truth values are given as fractions or decimals, ranging from 0 to 1. Not-p is defined as 1-p, and other formula are defined in terms of maxima and minima for sets. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 07.II) | |
| A reaction: The question seems to be whether this is actually logic, or a recasting of probability theory. Susan Haack attacks it. If logic is the study of how truth is preserved as we move between propositions, then 0 and 1 need a special status. |
| 10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
| Full Idea: One version of the Axiom of Choice says that for every set A of nonempty sets, there is a function whose domain is A and whose value, for every a ∈ A, is a member of a. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 1) |
| 10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
| Full Idea: If the Axiom of Choice says we can choose one member from each of a set of non-empty sets and put the chosen elements together in a set, this licenses the constructor to do an infinite amount of choosing. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3) | |
| A reaction: This is one reason why the Axiom was originally controversial, and still is for many philosophers. |
| 10207 | Anti-realists reject set theory [Shapiro] |
| Full Idea: Anti-realists reject set theory. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
| A reaction: That is, anti-realists about mathematical objects. I would have thought that one could accept an account of sets as (say) fictions, which provided interesting models of mathematics etc. |