display all the ideas for this combination of philosophers
9 ideas
| 17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
| Full Idea: A proof of the consistency of propositional logic was given by Emil Post in 1921. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2.1) |
| 17765 | Propositional language can only relate statements as the same or as different [Walicki] |
| Full Idea: Propositional language is very rudimentary and has limited powers of expression. The only relation between various statements it can handle is that of identity and difference. As are all the same, but Bs can be different from As. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 7 Intro) | |
| A reaction: [second sentence a paraphrase] In predicate logic you could represent two statements as being the same except for one element (an object or predicate or relation or quantifier). |
| 17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
| Full Idea: Boolean connectives are interpreted as functions on the set {1,0}. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 5.1) | |
| A reaction: 1 and 0 are normally taken to be true (T) and false (F). Thus the functions output various combinations of true and false, which are truth tables. |
| 16468 | Non-S5 can talk of contingent or necessary necessities [Stalnaker] |
| Full Idea: One can make sense of necessary versus contingent necessities in a non-S5 modal semantics. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 4.3 n17) | |
| A reaction: In S5 □φ → □□φ, so all necessities are necessary. Does it make any sense to say 'I suppose this might have been necessarily true'? |
| 18823 | To say there could have been people who don't exist, but deny those possible things, rejects Barcan [Stalnaker, by Rumfitt] |
| Full Idea: Stalnaker holds that there could have been people who do not actually exist, but he denies that there are things that could have been those people. That is, he denies the unrestricted validity of the Barcan Formula. | |
| From: report of Robert C. Stalnaker (Counterparts and Identity [1987]) by Ian Rumfitt - The Boundary Stones of Thought 6.2 | |
| A reaction: And quite right too, I should have thought. As they say, Jack Kennedy and Marilyn Monroe might have had a child, but the idea that we should accept some entity which might have been that child but wasn't sounds like nonsense. Except as fiction….. |
| 17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
| Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) |
| 17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
| Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) | |
| A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set. |
| 16449 | In modal set theory, sets only exist in a possible world if that world contains all of its members [Stalnaker] |
| Full Idea: One principle of modal set theory should be uncontroversial: a set exists in a given possible world if and only if all of its members exist at that world. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 2.4) | |
| A reaction: Does this mean there can be no set containing all of my ancestors and future descendants? In no world can we coexist. |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
| Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level. |