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| 13689 | 'Theorems' are formulas provable from no premises at all |
| Full Idea: Formulas provable from no premises at all are often called 'theorems'. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 2.6) |
| 13705 | Truth tables assume truth functionality, and are just pictures of truth functions |
| Full Idea: The method of truth tables assumes truth functionality. Truth tables are just pictures of truth functions. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 6.3) |
| 13706 | Intuitively, deontic accessibility seems not to be reflexive, but to be serial |
| Full Idea: Deontic accessibility seems not to be reflexive (that it ought to be true doesn't make it true). One could argue that it is serial (that there is always a world where something is acceptable). | |||
| From: Theodore Sider (Logic for Philosophy [2010], 6.3.1) |
| 13710 | In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary |
| Full Idea: In D we add to K a new axiom saying that 'what's necessary is possible' (□φ→◊φ), ..and it can then be proved that tautologies are possible and contradictions are not necessary. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 6.4.2) |
| 13711 | System B introduces iterated modalities |
| Full Idea: With system B we begin to be able to say something about iterated modalities. ..S4 then takes a different stand on the iterated modalities, and neither is an extension of the other. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 6.4.4) |
| 13708 | S5 is the strongest system, since it has the most valid formulas, because it is easy to be S5-valid |
| Full Idea: S5 is the strongest system, since it has the most valid formulas. That's because it has the fewest models; it's easy to be S5-valid since there are so few potentially falsifying models. K is the weakest system, for opposite reasons. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 6.3.2) | |||
| A reaction: Interestingly, the orthodox view is that S5 is the correct logic for metaphysics, but it sounds a bit lax. Compare Idea 13707. |
| 13712 | Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and K¬K) |
| Full Idea: Epistemic accessibility should be required to be reflexive (allowing Kφ→φ). S4 allows the 'KK principle', or 'positive introspection' (Kφ→KKφ), and S5 allows 'negative introspection' (¬Kφ→K¬Kφ). | |||
| From: Theodore Sider (Logic for Philosophy [2010], 7.2) |
| 13714 | We can treat modal worlds as different times |
| Full Idea: We can think of the worlds of modal logic as being times, rather than 'possible' worlds. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 7.3.3) |
| 13718 | The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic |
| Full Idea: The Barcan Formula ∀x□Fx→□∀xFx is often regarded as a defect of Simple Quantified Modal Logic, though this most clearly seen in its equivalent form ◊∃xFx→∃x◊Fx. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 9.5.2) | |||
| A reaction: [See Idea 13719 for an explanation why it might be a defect] I translate the first one as 'if xs must be F, then they are always F', and the second one as 'for x to be possibly F, there must exist an x which is possibly F'. Modality needs existence. |
| 13720 | Converse Barcan Formula: □∀αφ→∀α□φ |
| Full Idea: The Converse Barcan Formula reads □∀αφ→∀α□φ (or an equivalent using ◊). | |||
| From: Theodore Sider (Logic for Philosophy [2010], 9.5.2) | |||
| A reaction: I would read that as 'if all the αs happen to be φ, then αs have to be φ'. Put like that, I would have thought that it was obviously false. Sider points out that some new object could turn up which isn't φ. |
| 13723 | System B is needed to prove the Barcan Formula |
| Full Idea: The proof of the Barcan Formula require System B. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 9.7) |
| 13715 | You can employ intuitionist logic without intuitionism about mathematics |
| Full Idea: Not everyone who employs intuitionistic logic is an intuitionist about mathematics. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 7.4.1) | |||
| A reaction: This seems worthy of note, since it may be tempting to reject the logic because of the implausibility of the philosophy of mathematics. I must take intuitionist logic more seriously. |
| 13679 | Maybe logical consequence is more a matter of provability than of truth-preservation |
| Full Idea: Another answer to the question about the nature of logical consequence is a proof-theoretic one, according to which it is more a matter of provability than of truth-preservation. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 1.5) | |||
| A reaction: I don't like this, and prefer the model-theoretic or substitutional accounts. Whether you can prove that something is a logical consequence seems to me entirely separate from whether you can see that it is so. Gödel seems to agree. |
| 13678 | The most popular account of logical consequence is the semantic or model-theoretic one |
| Full Idea: On the question of the nature of genuine logical consequence, ...the most popular answer is the semantic, or model-theoretic one. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 1.5) | |||
| A reaction: Reading the literature, one might be tempted to think that this is the only account that anyone takes seriously. Substitutional semantics seems an interesting alternative. |
| 13682 | Maybe logical consequence is impossibility of the premises being true and the consequent false |
| Full Idea: The 'modal' account of logical consequence is that it is not possible for the premises to be true and the consequent false (under some suitable notion of possibility). | |||
| From: Theodore Sider (Logic for Philosophy [2010], 1.5) | |||
| A reaction: Sider gives a nice summary of five views of logical consequence, to which Shapiro adds substitutional semantics. |
| 13680 | Maybe logical consequence is a primitive notion |
| Full Idea: There is a 'primitivist' account, according to which logical consequence is a primitive notion. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 1.5) | |||
| A reaction: While sympathetic to substitutional views (Idea 13674), the suggestion here pushes me towards thinking that truth must be at the root of it. The trouble, though, is that a falsehood can be a good logical consequence of other falsehoods. |
| 13722 | A 'theorem' is an axiom, or the last line of a legitimate proof |
| Full Idea: A 'theorem' is defined as the last line of a proof in which each line is either an axiom or follows from earlier lines by a rule. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 9.7) | |||
| A reaction: In other words, theorems are the axioms and their implications. |
| 13696 | When a variable is 'free' of the quantifier, the result seems incapable of truth or falsity |
| Full Idea: When a variable is not combined with a quantifier (and so is 'free'), the result is, intuitively, semantically incomplete, and incapable of truth or falsity. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 4.2) |
| 13700 | A 'total' function must always produce an output for a given domain |
| Full Idea: Calling a function a 'total' function 'over D' means that the function must have a well-defined output (which is a member of D) whenever it is given as inputs any n members of D. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 5.2) |
| 13703 | λ can treat 'is cold and hungry' as a single predicate |
| Full Idea: We might prefer λx(Fx∧Gx)(a) as the symbolization of 'John is cold and hungry', since it treats 'is cold and hungry' as a single predicate. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 5.5) |
| 13688 | Good axioms should be indisputable logical truths |
| Full Idea: Since they are the foundations on which a proof rests, the axioms in a good axiomatic system ought to represent indisputable logical truths. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 2.6) |
| 13687 | No assumptions in axiomatic proofs, so no conditional proof or reductio |
| Full Idea: Axiomatic systems do not allow reasoning with assumptions, and therefore do not allow conditional proof or reductio ad absurdum. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 2.6) | |||
| A reaction: Since these are two of the most basic techniques of proof which I have learned (in Lemmon), I shall avoid axiomatic proof systems at all costs, despites their foundational and Ockhamist appeal. |
| 13690 | Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms |
| Full Idea: The style of proof called 'induction on formula construction' (or 'on the number of connectives', or 'on the length of the formula') rest on the fact that all formulas are built up from atomic formulas according to strict rules. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 2.7) | |||
| A reaction: Hence the proof deconstructs the formula, and takes it back to a set of atomic formulas have already been established. |
| 13691 | Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' |
| Full Idea: A proof by induction starts with a 'base case', usually that an atomic formula has some property. It then assumes an 'inductive hypothesis', that the property is true up to a certain case. The 'inductive step' then says it will be true for the next case. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 2.7) | |||
| A reaction: [compressed] |
| 13685 | Natural deduction helpfully allows reasoning with assumptions |
| Full Idea: The method of natural deduction is popular in introductory textbooks since it allows reasoning with assumptions. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 2.5) | |||
| A reaction: Reasoning with assumptions is generally easier, rather than being narrowly confined to a few tricky axioms, You gradually show that an inference holds whatever the assumption was, and so end up with the same result. |
| 13686 | We can build proofs just from conclusions, rather than from plain formulae |
| Full Idea: We can construct proofs not out of well-formed formulae ('wffs'), but out of sequents, which are some premises followed by their logical consequence. We explicitly keep track of the assumptions upon which the conclusion depends. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 2.5.1) | |||
| A reaction: He says the method of sequents was invented by Gerhard Gentzen (the great nazi logician) in 1935. The typical starting sequents are the introduction and elimination rules. E.J. Lemmon's book, used in this database, is an example. |
| 13697 | Valuations in PC assign truth values to formulas relative to variable assignments |
| Full Idea: A valuation function in predicate logic will assign truth values to formulas relative to variable assignments. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 4.2) | |||
| A reaction: Sider observes that this is a 'double' relativisation (due to Tarski), since propositional logic truth was already relative to an interpretation. Now we are relative to variable assignments as well. |
| 13684 | The semantical notion of a logical truth is validity, being true in all interpretations |
| Full Idea: The semantical notion of a logical truth is that of a valid formula, which is true in all interpretations. In propositional logic they are 'tautologies'. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 2.3) | |||
| A reaction: This implies that there is a proof-theoretic account of logical truth as well. Intuitively a logical truth is a sequent which holds no matter which subject matter it refers to, so the semantic view sounds OK. |
| 13704 | It is hard to say which are the logical truths in modal logic, especially for iterated modal operators |
| Full Idea: It isn't clear which formulas of modal propositional logic are logical truths, ...especially for sentences that contain iterations of modal operators. Is □P→□□P a logical truth? It's hard to say. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 6.3) | |||
| A reaction: The result, of course, is that there are numerous 'systems' for modal logic, so that you can choose the one that gives you the logical truths you want. His example is valid in S4 and S5, but not in the others. |
| 13724 | In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation |
| Full Idea: In model theory one normally defines some notion of truth in a model, and then uses it to define validity as truth in all models, and semantic consequence as the preservation of truth in models. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 10.1) |
| 13698 | In a complete logic you can avoid axiomatic proofs, by using models to show consequences |
| Full Idea: You can establish facts of the form Γ|-φ while avoiding the agonies of axiomatic proofs by reasoning directly about models to conclusions about semantic consequence, and then citing completeness. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 4.5) | |||
| A reaction: You cite completeness by saying that anything which you have shown to be a semantic consequence must therefore be provable (in some way). |
| 13699 | Compactness surprisingly says that no contradictions can emerge when the set goes infinite |
| Full Idea: Compactness is intuitively surprising, ..because one might have thought there could be some contradiction latent within some infinite set, preventing it from being satisfiable, only discovered when you consider the whole set. But this can't happen. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 4.5) |
| 13701 | A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically |
| Full Idea: A single second-order sentence has second-order semantic consequences which are all and only the truths of arithmetic, but this is cold comfort because of incompleteness; no axiomatic system draws out the consequences of this axiom. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 5.4.3) |
| 13692 | A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation |
| Full Idea: For a 'precisification' we take a trivalent interpretation and preserve the T and F values, and then assign all the third values in some way to either T or F. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 3.4.5) | |||
| A reaction: [my informal summary of Sider's formal definition] |
| 13695 | Supervaluational logic is classical, except when it adds the 'Definitely' operator |
| Full Idea: Supervaluation preserves classical logic (even though supervaluations are three-valued), except when we add the Δ operator (meaning 'definitely' or 'determinately'). | |||
| From: Theodore Sider (Logic for Philosophy [2010], 3.4.5) |
| 13693 | A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification |
| Full Idea: In a 'supervaluation' we take a trivalent interpretation, and assign to each wff T (or F) if it is T (or F) in every precisification, leaving the third truth-value in any other cases. The wffs are then 'supertrue' or 'superfalse' in the interpretation. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 3.4.5) | |||
| A reaction: [my non-symbolic summary] Sider says the Ts and Fs in the precisifications are assigned 'in any way you like', so supervaluation is a purely formal idea, not a technique for eliminating vagueness. |
| 13694 | We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings |
| Full Idea: We can introduce 'sharpenings', to make vague terms precise without disturbing their semantics. Then truth (or falsity) becomes true(false)-in-all-sharpenings. You are only 'rich' if you are rich-on-all-sharpenings of the word. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 3.4.5) | |||
| A reaction: Not very helpful. Lots of people might be considered rich in many contexts, but very few people would be considered rich in all contexts. You are still left with some vague middle ground. |
| 13683 | A relation is a feature of multiple objects taken together |
| Full Idea: A relation is just a feature of multiple objects taken together. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 1.8) | |||
| A reaction: Appealingly simple, especially for a logician, who can then just list the relevant objects as members of a set, and the job is done. But if everyone to the left of me is also taller than me, this won't quite do. |
| 13702 | The identity of indiscernibles is necessarily true, if being a member of some set counts as a property |
| Full Idea: The identity of indiscernibles (∀x∀y(∀X(Xx↔Xy)→x=y) is necessarily true, provided that we construe 'property' very broadly, so that 'being a member of such-and-such set' counts as a property. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 5.4.3) | |||
| A reaction: Sider's example is that if the two objects are the same they must both have the property of being a member of the same singleton set, which they couldn't have if they were different. |
| 13721 | 'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist |
| Full Idea: 'Strong necessity' requires the truth of 'necessarily φ' is all possible worlds. 'Weak necessity' merely requires that 'necessarily φ' be true in all worlds in which objects referred to within φ exist. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 9.6.3) | |||
| A reaction: This seems to be a highly desirably distinction, given the problem of Idea 13719. It is weakly necessary that humans can't fly unaided, assuming we are referring the current feeble wingless species. That hardly seems to be strongly necessary. |
| 13707 | Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible |
| Full Idea: Some argue that metaphysical accessibility is intransitive. The individuals involved mustn't be too different from the actual world. A world in which I am a frog isn't metaphysically possible. Perhaps the logic is modal system B or T. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 6.3.1) | |||
| A reaction: This sounds rather plausible and attractive to me. We don't want to say that I am necessarily the way I actually am, though, so we need criteria. Essence! |
| 13709 | Logical truths must be necessary if anything is |
| Full Idea: On any sense of necessity, surely logical truths must be necessary. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 6.4) |
| 13716 | 'If B hadn't shot L someone else would have' if false; 'If B didn't shoot L, someone else did' is true |
| Full Idea: To show the semantic difference between counterfactuals and indicative conditionals, 'If Booth hadn't shot Lincoln someone else would have' is false, but 'If Booth didn't shoot Lincoln then someone else did' is true. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 8) | |||
| A reaction: He notes that indicative conditionals also differ in semantics from material and strict conditionals. The first example allows a world where Lincoln was not shot, but the second assumes our own world, where he was. Contextual domains? |
| 13717 | Transworld identity is not a problem in de dicto sentences, which needn't identify an individual |
| Full Idea: There is no problem of transworld identification with de dicto modal sentence, for their evaluation does not require taking an individual from one possible world and reidentifying it in another. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 9.2) | |||
| A reaction: If 'de dicto' is about the sentence and 'de re' is about the object (Idea 5732), how do you evaluate the sentence without at least some notion of the object to which it refers. Nec the Prime Minister chairs the cabinet. Could a poached egg do the job? |
| 13719 | Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost |
| Full Idea: A problem with the Barcan Formula is it might be possible for there to exist a ghost, even though there in fact exists nothing that could be a ghost. There could have existed some 'extra' thing which could be a ghost. | |||
| From: Theodore Sider (Logic for Philosophy [2010], 9.5.2) | |||
| A reaction: Thus when we make modal claims, do they only refer to what actually exists, or is specified in our initial domain? Can a claim enlarge the domain? Are domains 'variable'? Simple claims about what might have existed seem to be a problem. |