16 ideas
| 15375 | If terms change their designations in different states, they are functions from states to objects [Fitting] |
| Full Idea: The common feature of every designating term is that designation may change from state to state - thus it can be formalized by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3) | |
| A reaction: Specifying the objects sounds OK, but specifying states sounds rather tough. |
| 15376 | Intensional logic adds a second type of quantification, over intensional objects, or individual concepts [Fitting] |
| Full Idea: To first order modal logic (with quantification over objects) we can add a second kind of quantification, over intensions. An intensional object, or individual concept, will be modelled by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.3) |
| 15378 | Awareness logic adds the restriction of an awareness function to epistemic logic [Fitting] |
| Full Idea: Awareness logic enriched Hintikka's epistemic models with an awareness function, mapping each state to the set of formulas we are aware of at that state. This reflects some bound on the resources we can bring to bear. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) | |
| A reaction: [He cites Fagin and Halpern 1988 for this] |
| 15379 | Justication logics make explicit the reasons for mathematical truth in proofs [Fitting] |
| Full Idea: In justification logics, the logics of knowledge are extended by making reasons explicit. A logic of proof terms was created, with a semantics. In this, mathematical truths are known for explicit reasons, and these provide a measure of complexity. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) |
| 11026 | Classical logic is deliberately extensional, in order to model mathematics [Fitting] |
| Full Idea: Mathematics is typically extensional throughout (we write 3+2=2+3 despite the two terms having different meanings). ..Classical first-order logic is extensional by design since it primarily evolved to model the reasoning of mathematics. | |
| From: Melvin Fitting (Intensional Logic [2007], §1) |
| 11028 | λ-abstraction disambiguates the scope of modal operators [Fitting] |
| Full Idea: λ-abstraction can be used to abstract and disambiguate a predicate. De re is [λx◊P(x)](f) - f has the possible-P property - and de dicto is ◊[λxP(x)](f) - possibly f has the P-property. Also applies to □. | |
| From: Melvin Fitting (Intensional Logic [2007], §3.3) | |
| A reaction: Compare the Barcan formula. Originated with Church in the 1930s, and Carnap 1947, but revived by Stalnaker and Thomason 1968. Because it refers to the predicate, it has a role in intensional versions of logic, especially modal logic. |
| 20660 | At one level maths and nature are very similar, suggesting some deeper origin [Wolfram] |
| Full Idea: At some rather abstract level one can immediately recognise one basic similarity between nature and mathematics ...this suggests that the overall similarity between mathematics and nature must have a deeper origin. | |
| From: Stephen Wolfram (A New Kind of Science [2002], p.772), quoted by Peter Watson - Convergence 17 'Philosophy' | |
| A reaction: Personally I think mathematics has been derived by abstracting from the patterns in nature, and then further extrapolating from those abstractions. So the puzzle in nature is not the correspondence with mathematics, but the patterns. |
| 15377 | Definite descriptions pick out different objects in different possible worlds [Fitting] |
| Full Idea: Definite descriptions pick out different objects in different possible worlds quite naturally. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.4) | |
| A reaction: A definite description can pick out the same object in another possible world, or a very similar one, or an object which has almost nothing in common with the others. |
| 22717 | Self-interest can fairly divide a cake; first person cuts, second person chooses [Poundstone] |
| Full Idea: To fairly divide a cake between two children, the first divides it and the second chooses. …Even division is best, as it anticipates the second child will take the largest piece. Fairness is enforced by the children's self-interests. | |
| From: William Poundstone (Prisoner's Dilemma [1992], 03 'Cake') | |
| A reaction: [compressed] This is introduced as the basic principle of game theory. There is an online video of two cats sharing a dish of milk; each one drinks a bit, then pushes the dish to the other one. I'm sure two children could manage that. |
| 22718 | Formal game theory is about maximising or minimising numbers in tables [Poundstone] |
| Full Idea: At the most abstract level, game theory is about tables with numbers in them - numbers that entities are are efficiently acting to maximise or minimise. | |
| From: William Poundstone (Prisoner's Dilemma [1992], 03 'Curve') | |
| A reaction: A brilliant idea. The question is the extent to which real life conforms to the numberical tables. The assumption that everyone is entirely self-seeking is blatantly false. Numbers like money have diminishing marginal utility. |
| 22719 | The minimax theorem says a perfect game of opposed people always has a rational solution [Poundstone] |
| Full Idea: The minimax theorem says that there is always a rational solution to a precisely defined conflict between two people whose interests are completely opposite. | |
| From: William Poundstone (Prisoner's Dilemma [1992], 03 'Minimax') | |
| A reaction: This is Von Neumann's founding theorem of game theory. It concerns maximising minimums, and minimising maximums. Crucially, I would say that it virtually never occurs that two people have completely opposite interests. There is a common good. |
| 22720 | Two prisoners get the best result by being loyal, not by selfish betrayal [Poundstone] |
| Full Idea: Prisoners A and B can support or betray one another. If both support, they each get 1 year in prison. If one betrays, the betrayer gets 0 and the betrayed gets 3. If they both betray they get 2 each. The common good is to support each other. | |
| From: William Poundstone (Prisoner's Dilemma [1992], 06 'Tucker's') | |
| A reaction: [by Albert Tucker, highly compressed] The classic Prisoner's Dilemma. It is artificial, but demonstrates that selfish behaviour gets a bad result (total of four years imprisonment), but the common good gets only two years. Every child should study this! |
| 22721 | The tragedy in prisoner's dilemma is when two 'nice' players misread each other [Poundstone] |
| Full Idea: The tragedy is when two 'nice' players defect because they misread the other's intentions. The puzzle of the prisoner's dilemma is how such good intentions pave the road to hell. | |
| From: William Poundstone (Prisoner's Dilemma [1992], 11 'Howard's') | |
| A reaction: I really wish these simple ideas were better known. They more or less encapsulate the tragedy of the human race, with its inability to prioritise the common good. |
| 22722 | TIT FOR TAT says cooperate at first, then do what the other player does [Poundstone] |
| Full Idea: The successful TIT FOR TAT strategy (for the iterated prisoner's dilemma) says cooperate on the first round, then do whatever the other player did in the previous round. | |
| From: William Poundstone (Prisoner's Dilemma [1992], 12 'TIT') | |
| A reaction: There are also the tougher TWO TITS FOR A TAT, and the more forgiving TIT FOR TWO TATS. The one-for-one seems to be the main winner, and is commonly seen in animal life (apparently). I recommend this to school teachers. |
| 22723 | Do unto others as you would have them do unto you - or else! [Poundstone] |
| Full Idea: TIT FOR TAT threatens 'Do unto others as you would have them do unto you - or else!'. | |
| From: William Poundstone (Prisoner's Dilemma [1992], 12 'TIT') | |
| A reaction: Essentially human happiness arises if we are all nice, but also stand up firmly for ourselves. 'Doormats' (nice all the time) get exploited. TIT FOR TAT is weak, because it doesn't exploit people who don't respond at all. |
| 20659 | Space and its contents seem to be one stuff - so space is the only existing thing [Wolfram] |
| Full Idea: It seems plausible that both space and its contents should somehow be made of the same stuff - so that in a sense space becomes the only thing in the universe. | |
| From: Stephen Wolfram (A New Kind of Science [2002], p.474), quoted by Peter Watson - Convergence 17 'Philosophy' | |
| A reaction: I presume the concept of a 'field' is what makes this idea possible. |