15413 | With four tense operators, all complex tenses reduce to fourteen basic cases |
Full Idea: Fand P as 'will' and 'was', G as 'always going to be', H as 'always has been', all tenses reduce to 14 cases: the past series, each implying the next, FH,H,PH,HP,P,GP, and the future series PG,G,FG,GF,F,HF, plus GH=HG implying all, FP=PF which all imply. | |
From: John P. Burgess (Philosophical Logic [2009], 2.8) | |
A reaction: I have tried to translate the fourteen into English, but am not quite confident enough to publish them here. I leave it as an exercise for the reader. |
15415 | The temporal Barcan formulas fix what exists, which seems absurd |
Full Idea: In temporal logic, if the converse Barcan formula holds then nothing goes out of existence, and the direct Barcan formula holds if nothing ever comes into existence. These results highlight the intuitive absurdity of the Barcan formulas. | |
From: John P. Burgess (Philosophical Logic [2009], 2.9) | |
A reaction: This is my reaction to the modal cases as well - the absurdity of thinking that no actually nonexistent thing might possibly have existed, or that the actual existents might not have existed. Williamson seems to be the biggest friend of the formulas. |
15430 | Is classical logic a part of intuitionist logic, or vice versa? |
Full Idea: From one point of view intuitionistic logic is a part of classical logic, missing one axiom, from another classical logic is a part of intuitionistic logic, missing two connectives, intuitionistic v and → | |
From: John P. Burgess (Philosophical Logic [2009], 6.4) |
15431 | It is still unsettled whether standard intuitionist logic is complete |
Full Idea: The question of the completeness of the full intuitionistic logic for its intended interpretation is not yet fully resolved. | |
From: John P. Burgess (Philosophical Logic [2009], 6.9) |
15429 | Relevance logic's → is perhaps expressible by 'if A, then B, for that reason' |
Full Idea: The relevantist logician's → is perhaps expressible by 'if A, then B, for that reason'. | |
From: John P. Burgess (Philosophical Logic [2009], 5.8) |
15404 | Technical people see logic as any formal system that can be studied, not a study of argument validity |
Full Idea: Among the more technically oriented a 'logic' no longer means a theory about which forms of argument are valid, but rather means any formalism, regardless of its applications, that resembles original logic enough to be studied by similar methods. | |
From: John P. Burgess (Philosophical Logic [2009], Pref) | |
A reaction: There doesn't seem to be any great intellectual obligation to be 'technical'. As far as pure logic is concerned, I am very drawn to the computer approach, since I take that to be the original dream of Aristotle and Leibniz - impersonal precision. |
15405 | Classical logic neglects the non-mathematical, such as temporality or modality |
Full Idea: There are topics of great philosophical interest that classical logic neglects because they are not important to mathematics. …These include distinctions of past, present and future, or of necessary, actual and possible. | |
From: John P. Burgess (Philosophical Logic [2009], 1.1) |
15427 | The Cut Rule expresses the classical idea that entailment is transitive |
Full Idea: The Cut rule (from A|-B and B|-C, infer A|-C) directly expresses the classical doctrine that entailment is transitive. | |
From: John P. Burgess (Philosophical Logic [2009], 5.3) |
15421 | Classical logic neglects counterfactuals, temporality and modality, because maths doesn't use them |
Full Idea: Classical logic neglects counterfactual conditionals for the same reason it neglects temporal and modal distinctions, namely, that they play no serious role in mathematics. | |
From: John P. Burgess (Philosophical Logic [2009], 4.1) | |
A reaction: Science obviously needs counterfactuals, and metaphysics needs modality. Maybe so-called 'classical' logic will be renamed 'basic mathematical logic'. Philosophy will become a lot clearer when that happens. |
15403 | Philosophical logic is a branch of logic, and is now centred in computer science |
Full Idea: Philosophical logic is a branch of logic, a technical subject. …Its centre of gravity today lies in theoretical computer science. | |
From: John P. Burgess (Philosophical Logic [2009], Pref) | |
A reaction: He firmly distinguishes it from 'philosophy of logic', but doesn't spell it out. I take it that philosophical logic concerns metaprinciples which compare logical systems, and suggest new lines of research. Philosophy of logic seems more like metaphysics. |
15407 | Formalising arguments favours lots of connectives; proving things favours having very few |
Full Idea: When formalising arguments it is convenient to have as many connectives as possible available.; but when proving results about formulas it is convenient to have as few as possible. | |
From: John P. Burgess (Philosophical Logic [2009], 1.4) | |
A reaction: Illuminating. The fact that you can whittle classical logic down to two (or even fewer!) connectives warms the heart of technicians, but makes connection to real life much more difficult. Hence a bunch of extras get added. |
15424 | Asserting a disjunction from one disjunct seems odd, but can be sensible, and needed in maths |
Full Idea: Gricean implicature theory might suggest that a disjunction is never assertable when a disjunct is (though actually the disjunction might be 'pertinent') - but the procedure is indispensable in mathematical practice. | |
From: John P. Burgess (Philosophical Logic [2009], 5.2) | |
A reaction: He gives an example of a proof in maths which needs it, and an unusual conversational occasion where it makes sense. |
15409 | All occurrences of variables in atomic formulas are free |
Full Idea: All occurrences of variables in atomic formulas are free. | |
From: John P. Burgess (Philosophical Logic [2009], 1.7) |
15414 | The denotation of a definite description is flexible, rather than rigid |
Full Idea: By contrast to rigidly designating proper names, …the denotation of definite descriptions is (in general) not rigid but flexible. | |
From: John P. Burgess (Philosophical Logic [2009], 2.9) | |
A reaction: This modern way of putting it greatly clarifies why Russell was interested in the type of reference involved in definite descriptions. Obviously some descriptions (such as 'the only person who could ever have…') might be rigid. |
15406 | 'Induction' and 'recursion' on complexity prove by connecting a formula to its atomic components |
Full Idea: There are atomic formulas, and formulas built from the connectives, and that is all. We show that all formulas have some property, first for the atomics, then the others. This proof is 'induction on complexity'; we also use 'recursion on complexity'. | |
From: John P. Burgess (Philosophical Logic [2009], 1.4) | |
A reaction: That is: 'induction on complexity' builds a proof from atomics, via connectives; 'recursion on complexity' breaks down to the atomics, also via the connectives. You prove something by showing it is rooted in simple truths. |
15425 | The sequent calculus makes it possible to have proof without transitivity of entailment |
Full Idea: It might be wondered how one could have any kind of proof procedure at all if transitivity of entailment is disallowed, but the sequent calculus can get around the difficulty. | |
From: John P. Burgess (Philosophical Logic [2009], 5.3) | |
A reaction: He gives examples where transitivity of entailment (so that you can build endless chains of deductions) might fail. This is the point of the 'cut free' version of sequent calculus, since the cut rule allows transitivity. |
15426 | We can build one expanding sequence, instead of a chain of deductions |
Full Idea: Instead of demonstrations which are either axioms, or follow from axioms by rules, we can have one ever-growing sequence of formulas of the form 'Axioms |- ______', where the blank is filled by Axioms, then Lemmas, then Theorems, then Corollaries. | |
From: John P. Burgess (Philosophical Logic [2009], 5.3) |
15408 | 'Tautologies' are valid formulas of classical sentential logic - or substitution instances in other logics |
Full Idea: The valid formulas of classical sentential logic are called 'tautologically valid', or simply 'tautologies'; with other logics 'tautologies' are formulas that are substitution instances of valid formulas of classical sentential logic. | |
From: John P. Burgess (Philosophical Logic [2009], 1.5) |
15418 | Validity (for truth) and demonstrability (for proof) have correlates in satisfiability and consistency |
Full Idea: Validity (truth by virtue of logical form alone) and demonstrability (provability by virtue of logical form alone) have correlative notions of logical possibility, 'satisfiability' and 'consistency', which come apart in some logics. | |
From: John P. Burgess (Philosophical Logic [2009], 3.3) |
15412 | Models leave out meaning, and just focus on truth values |
Full Idea: Models generally deliberately leave out meaning, retaining only what is important for the determination of truth values. | |
From: John P. Burgess (Philosophical Logic [2009], 2.2) | |
A reaction: This is the key point to hang on to, if you are to avoid confusing mathematical models with models of things in the real world. |
15411 | We only need to study mathematical models, since all other models are isomorphic to these |
Full Idea: In practice there is no need to consider any but mathematical models, models whose universes consist of mathematical objects, since every model is isomorphic to one of these. | |
From: John P. Burgess (Philosophical Logic [2009], 1.8) | |
A reaction: The crucial link is the technique of Gödel Numbering, which can translate any verbal formula into numerical form. He adds that, because of the Löwenheim-Skolem theorem only subsets of the natural numbers need be considered. |
15416 | We aim to get the technical notion of truth in all models matching intuitive truth in all instances |
Full Idea: The aim in setting up a model theory is that the technical notion of truth in all models should agree with the intuitive notion of truth in all instances. A model is supposed to represent everything about an instance that matters for its truth. | |
From: John P. Burgess (Philosophical Logic [2009], 3.2) |
15428 | The Liar seems like a truth-value 'gap', but dialethists see it as a 'glut' |
Full Idea: It is a common view that the liar sentence ('This very sentence is not true') is an instance of a truth-value gap (neither true nor false), but some dialethists cite it as an example of a truth-value glut (both true and false). | |
From: John P. Burgess (Philosophical Logic [2009], 5.7) | |
A reaction: The defence of the glut view must be that it is true, then it is false, then it is true... Could it manage both at once? |
10185 | Set theory is the standard background for modern mathematics |
Full Idea: In present-day mathematics, it is set theory that serves as the background theory in which other branches of mathematics are developed. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: [He cites Bourbaki as an authority for this] See Benacerraf for a famous difficulty here, when you actually try to derive an ontology from the mathematicians' working practices. |
10184 | Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure |
Full Idea: On the structuralist interpretation, theorems of analysis concerning the real numbers R are about all complete ordered fields. So R, which appears to be the name of a specific structure, is taken to be a variable ranging over structures. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: Since I am beginning to think that nearly all linguistic expressions should be understood as variables, I find this very appealing, even if Burgess hates it. Terms slide and drift, and are vague, between variable and determinate reference. |
10189 | There is no one relation for the real number 2, as relations differ in different models |
Full Idea: One might meet the 'Van Inwagen Problem' by saying that the intrinsic properties of the object playing the role of 2 will differ from one model to another, so that no statement about the intrinsic properties of 'the' real numbers will make sense. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5) | |
A reaction: There seems to be a potential confusion among opponents of structuralism between relations at the level of actual mathematical operations, and generalisations about relations, which are captured in the word 'patterns'. Call them 'meta-relations'? |
10186 | If set theory is used to define 'structure', we can't define set theory structurally |
Full Idea: It is to set theory that one turns for the very definition of 'structure', ...and this creates a problem of circularity if we try to impose a structuralist interpretation on set theory. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: This seems like a nice difficulty, especially if, like Shapiro, you wade in and try to give a formal account of structures and patterns. Resnik is more circumspect and vague. |
10187 | Abstract algebra concerns relations between models, not common features of all the models |
Full Idea: Abstract algebra, such as group theory, is not concerned with the features common to all models of the axioms, but rather with the relationships among different models of those axioms (especially homomorphic relation functions). | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: It doesn't seem to follow that structuralism can't be about the relations (or patterns) found when abstracting away and overviewing all the models. One can study family relations, or one can study kinship in general. |
10188 | How can mathematical relations be either internal, or external, or intrinsic? |
Full Idea: The 'Van Inwagen Problem' for structuralism is of explaining how a mathematical relation (such as set membership, or the ratios of an ellipse) can fit into one of the three scholastics types of relations: are they internal, external, or intrinsic? | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5) | |
A reaction: The difficulty is that mathematical objects seem to need intrinsic properties to get any of these three versions off the ground (which was Russell's complaint against structures). |
15420 | De re modality seems to apply to objects a concept intended for sentences |
Full Idea: There is a problem over 'de re' modality (as contrasted with 'de dicto'), as in ∃x□x. What is meant by '"it is analytic that Px" is satisfied by a', given that analyticity is a notion that in the first instance applies to complete sentences? | |
From: John P. Burgess (Philosophical Logic [2009], 3.9) | |
A reaction: This is Burgess's summary of one of Quine's original objections. The issue may be a distinction between whether the sentence is analytic, and what makes it analytic. The necessity of bachelors being unmarried makes that sentence analytic. |
15419 | General consensus is S5 for logical modality of validity, and S4 for proof |
Full Idea: To the extent that there is any conventional wisdom about the question, it is that S5 is correct for alethic logical modality, and S4 correct for apodictic logical modality. | |
From: John P. Burgess (Philosophical Logic [2009], 3.8) | |
A reaction: In classical logic these coincide, so presumably one should use the minimum system to do the job, which is S4 (?). |
15417 | Logical necessity has two sides - validity and demonstrability - which coincide in classical logic |
Full Idea: Logical necessity is a genus with two species. For classical logic the truth-related notion of validity and the proof-related notion of demonstrability, coincide - but they are distinct concept. In some logics they come apart, in intension and extension. | |
From: John P. Burgess (Philosophical Logic [2009], 3.3) | |
A reaction: They coincide in classical logic because it is sound and complete. This strikes me as the correct approach to logical necessity, tying it to the actual nature of logic, rather than some handwavy notion of just 'true in all possible worlds'. |
15422 | Three conditionals theories: Materialism (material conditional), Idealism (true=assertable), Nihilism (no truth) |
Full Idea: Three main theories of the truth of indicative conditionals are Materialism (the conditions are the same as for the material conditional), Idealism (identifying assertability with truth-value), and Nihilism (no truth, just assertability). | |
From: John P. Burgess (Philosophical Logic [2009], 4.3) |
15423 | It is doubtful whether the negation of a conditional has any clear meaning |
Full Idea: It is contentious whether conditionals have negations, and whether 'it is not the case that if A,B' has any clear meaning. | |
From: John P. Burgess (Philosophical Logic [2009], 4.9) | |
A reaction: This seems to be connected to Lewis's proof that a probability conditional cannot be reduced to a single proposition. If a conditional only applies to A-worlds, it is not surprising that its meaning gets lost when it leaves that world. |