20 ideas
| 13838 | A decent modern definition should always imply a semantics [Hacking] |
| Full Idea: Today we expect that anything worth calling a definition should imply a semantics. | |
| From: Ian Hacking (What is Logic? [1979], §10) | |
| A reaction: He compares this with Gentzen 1935, who was attempting purely syntactic definitions of the logical connectives. |
| 13833 | 'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction [Hacking] |
| Full Idea: 'Dilution' (or 'Thinning') provides an essential contrast between deductive and inductive reasoning; for the introduction of new premises may spoil an inductive inference. | |
| From: Ian Hacking (What is Logic? [1979], §06.2) | |
| A reaction: That is, inductive logic (if there is such a thing) is clearly non-monotonic, whereas classical inductive logic is monotonic. |
| 13834 | Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking] |
| Full Idea: If A |- B and B |- C, then A |- C. This generalises to: If Γ|-A,Θ and Γ,A |- Θ, then Γ |- Θ. Gentzen called this 'cut'. It is the transitivity of a deduction. | |
| From: Ian Hacking (What is Logic? [1979], §06.3) | |
| A reaction: I read the generalisation as 'If A can be either a premise or a conclusion, you can bypass it'. The first version is just transitivity (which by-passes the middle step). |
| 13835 | Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with [Hacking] |
| Full Idea: Only the cut rule can have a conclusion that is less complex than its premises. Hence when cut is not used, a derivation is quite literally constructive, building up from components. Any theorem obtained by cut can be obtained without it. | |
| From: Ian Hacking (What is Logic? [1979], §08) |
| 13845 | The various logics are abstractions made from terms like 'if...then' in English [Hacking] |
| Full Idea: I don't believe English is by nature classical or intuitionistic etc. These are abstractions made by logicians. Logicians attend to numerous different objects that might be served by 'If...then', like material conditional, strict or relevant implication. | |
| From: Ian Hacking (What is Logic? [1979], §15) | |
| A reaction: The idea that they are 'abstractions' is close to my heart. Abstractions from what? Surely 'if...then' has a standard character when employed in normal conversation? |
| 13840 | First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking] |
| Full Idea: First-order logic is the strongest complete compact theory with a Löwenheim-Skolem theorem. | |
| From: Ian Hacking (What is Logic? [1979], §13) |
| 13844 | A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking] |
| Full Idea: Henkin proved that there is no first-order treatment of branching quantifiers, which do not seem to involve any idea that is fundamentally different from ordinary quantification. | |
| From: Ian Hacking (What is Logic? [1979], §13) | |
| A reaction: See Hacking for an example of branching quantifiers. Hacking is impressed by this as a real limitation of the first-order logic which he generally favours. |
| 13842 | Second-order completeness seems to need intensional entities and possible worlds [Hacking] |
| Full Idea: Second-order logic has no chance of a completeness theorem unless one ventures into intensional entities and possible worlds. | |
| From: Ian Hacking (What is Logic? [1979], §13) |
| 13837 | With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically [Hacking] |
| Full Idea: My doctrine is that the peculiarity of the logical constants resides precisely in that given a certain pure notion of truth and consequence, all the desirable semantic properties of the constants are determined by their syntactic properties. | |
| From: Ian Hacking (What is Logic? [1979], §09) | |
| A reaction: He opposes this to Peacocke 1976, who claims that the logical connectives are essentially semantic in character, concerned with the preservation of truth. |
| 13839 | Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers [Hacking] |
| Full Idea: For some purposes the variables of first-order logic can be regarded as prepositions and place-holders that could in principle be dispensed with, say by a system of arrows indicating what places fall in the scope of which quantifier. | |
| From: Ian Hacking (What is Logic? [1979], §11) | |
| A reaction: I tend to think of variables as either pronouns, or as definite descriptions, or as temporary names, but not as prepositions. Must address this new idea... |
| 13843 | If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking] |
| Full Idea: A Löwenheim-Skolem theorem holds for anything which, on my delineation, is a logic. | |
| From: Ian Hacking (What is Logic? [1979], §13) | |
| A reaction: I take this to be an unusually conservative view. Shapiro is the chap who can give you an alternative view of these things, or Boolos. |
| 19405 | Substances are in harmony, because they each express the one reality in themselves [Leibniz] |
| Full Idea: Every substance expresses the whole sequence of the universe in accordance with its own viewpoint or relationship to the rest, so that all are in perfect correspondence with one another. | |
| From: Gottfried Leibniz (Identity in Substances and True Propositions [1686], p.98) | |
| A reaction: Thus 'expression' (something like mapping what is exterior) is the mechanism through which God achieves harmony in the universe. Instants of time are said to be successive moments of perfect harmony. |
| 6230 | If the soul were a tabula rasa, with no innate ideas, there could be no moral goodness or justice [Cudworth] |
| Full Idea: The soul is not a mere rasa tabula, a naked and passive thing, with no innate furniture of its own, nor any thing in it, but what was impressed upon it without; for then there could not possibly be any such thing as moral good and evil, just and unjust. | |
| From: Ralph Cudworth (On Eternal and Immutable Morality [1688], Bk IV Ch 6.4) | |
| A reaction: He goes on to quote Hobbes saying there is no good in objects themselves. I don't see why we must have an innate moral capacity, provided that we have a capacity to make judgements. |
| 6228 | Senses cannot judge one another, so what judges senses cannot be a sense, but must be superior [Cudworth] |
| Full Idea: The sight cannot judge of sounds, nor the hearing of light and colours; wherefore that which judges of all the senses and their several objects, cannot be itself any sense, but something of a superior nature. | |
| From: Ralph Cudworth (On Eternal and Immutable Morality [1688], Ch.II.VI.1) | |
| A reaction: How nice to find a seventeenth century English writer rebelling against empiricism! |
| 6229 | Sense is fixed in the material form, and so can't grasp abstract universals [Cudworth] |
| Full Idea: Sense which lies flat and grovelling in the individuals, and is stupidly fixed in the material form, is not able to rise up or ascend to an abstract universal notion. | |
| From: Ralph Cudworth (On Eternal and Immutable Morality [1688], Ch.III.III.2) | |
| A reaction: This still strikes me as being one of the biggest problems with reductive physicalism, that a lump of meat in your head can grasp abstractions (whatever they are) and universal concepts. Personally I am a physicalist, but it is weird. |
| 6227 | Keeping promises and contracts is an obligation of natural justice [Cudworth] |
| Full Idea: To keep faith and perform covenants is that which natural justice obligeth to absolutely. | |
| From: Ralph Cudworth (On Eternal and Immutable Morality [1688], Ch.II.4) | |
| A reaction: A nice example of an absolute moral intuition, but one which can clearly be challenged. Covenants (contracts) wouldn't work unless everyone showed intense commitment to keeping them, even beyond the grave, and we all benefit from good contracts. |
| 6225 | Obligation to obey all positive laws is older than all laws [Cudworth] |
| Full Idea: Obligation to obey all positive laws is older than all laws. | |
| From: Ralph Cudworth (On Eternal and Immutable Morality [1688], Ch.II.3) | |
| A reaction: Clearly villains can pass wicked laws, so there can't be an obligation to obey all laws (even if they are 'positive', which seems to beg the question). Nevertheless this is a good reason why laws cannot be the grounding of morality. |
| 6224 | An omnipotent will cannot make two things equal or alike if they aren't [Cudworth] |
| Full Idea: Omnipotent will cannot make things like or equal one to another, without the natures of likeness and equality. | |
| From: Ralph Cudworth (On Eternal and Immutable Morality [1688], Ch.II.I) | |
| A reaction: This is one of the many classic 'paradoxes of omnipotence'. The best strategy is to define omnipotence as 'being able to do everything which it is possible to do'. Anything beyond that is inviting paradoxical disaster. |
| 6223 | If the will and pleasure of God controls justice, then anything wicked or unjust would become good if God commanded it [Cudworth] |
| Full Idea: If the arbitrary will and pleasure of God is the first and only rule of good and justice, it follows that nothing can be so grossly wicked or unjust but if it were commanded by this omnipotent Deity, it must forthwith become holy, just and righteous. | |
| From: Ralph Cudworth (On Eternal and Immutable Morality [1688], Ch.I.I.5) | |
| A reaction: This is the strong (Platonic) answer to the Euthyphro Question (Idea 336). One answer is that God would not command in such a way - but why not? We may say that God and goodness merge into one, but we are interested in ultimate authority. |
| 6226 | The requirement that God must be obeyed must precede any authority of God's commands [Cudworth] |
| Full Idea: If it were not morally good and just in its own nature before any positive command of God, that God should be obeyed by his creatures, the bare will of God himself could not beget any obligation upon anyone. | |
| From: Ralph Cudworth (On Eternal and Immutable Morality [1688], Ch.II.3) | |
| A reaction: This strikes me as a self-evident truth, and a big problem for anyone who wants to make God the source of morality. You don't have to accept anyone's authority just because they are powerful or clever (though they do bestow a certain natural authority!). |