display all the ideas for this combination of texts
14 ideas
| 13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock] |
| Full Idea: Venn Diagrams are a traditional method to test validity of syllogisms. There are three interlocking circles, one for each predicate, thus dividing the universe into eight possible basic elementary quantifications. Is the conclusion in a compartment? | |
| From: David Bostock (Intermediate Logic [1997], 3.8) |
| 13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock] |
| Full Idea: 'Disjunctive Normal Form' (DNF) is rearranging the occurrences of ∧ and ∨ so that no conjunction sign has any disjunction in its scope. This is achieved by applying two of the distribution laws. | |
| From: David Bostock (Intermediate Logic [1997], 2.6) |
| 13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock] |
| Full Idea: 'Conjunctive Normal Form' (CNF) is rearranging the occurrences of ∧ and ∨ so that no disjunction sign has any conjunction in its scope. This is achieved by applying two of the distribution laws. | |
| From: David Bostock (Intermediate Logic [1997], 2.6) |
| 13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock] |
| Full Idea: The Principle of Disjunction says that Γ,φ∨ψ |= iff Γ,φ |= and Γ,ψ |=. | |
| From: David Bostock (Intermediate Logic [1997], 2.5.G) | |
| A reaction: That is, a disjunction leads to a contradiction if they each separately lead to contradictions. |
| 13350 | 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock] |
| Full Idea: The Principle of Assumptions says that any formula entails itself, i.e. φ |= φ. The principle depends just upon the fact that no interpretation assigns both T and F to the same formula. | |
| From: David Bostock (Intermediate Logic [1997], 2.5.A) | |
| A reaction: Thus one can introduce φ |= φ into any proof, and then use it to build more complex sequents needed to attain a particular target formula. Bostock's principle is more general than anything in Lemmon. |
| 13351 | 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock] |
| Full Idea: The Principle of Thinning says that if a set of premisses entails a conclusion, then adding further premisses will still entail the conclusion. It is 'thinning' because it makes a weaker claim. If γ|=φ then γ,ψ|= φ. | |
| From: David Bostock (Intermediate Logic [1997], 2.5.B) | |
| A reaction: It is also called 'premise-packing'. It is the characteristic of a 'monotonic' logic - where once something is proved, it stays proved, whatever else is introduced. |
| 13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock] |
| Full Idea: The Conditional Principle says that Γ |= φ→ψ iff Γ,φ |= ψ. With the addition of negation, this implies φ,φ→ψ |= ψ, which is 'modus ponens'. | |
| From: David Bostock (Intermediate Logic [1997], 2.5.H) | |
| A reaction: [Second half is in Ex. 2.5.4] |
| 13352 | 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock] |
| Full Idea: The Principle of Cutting is the general point that entailment is transitive, extending this to cover entailments with more than one premiss. Thus if γ |= φ and φ,Δ |= ψ then γ,Δ |= ψ. Here φ has been 'cut out'. | |
| From: David Bostock (Intermediate Logic [1997], 2.5.C) | |
| A reaction: It might be called the Principle of Shortcutting, since you can get straight to the last conclusion, eliminating the intermediate step. |
| 13353 | 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock] |
| Full Idea: The Principle of Negation says that Γ,¬φ |= iff Γ |= φ. We also say that φ,¬φ |=, and hence by 'thinning on the right' that φ,¬φ |= ψ, which is 'ex falso quodlibet'. | |
| From: David Bostock (Intermediate Logic [1997], 2.5.E) | |
| A reaction: That is, roughly, if the formula gives consistency, the negation gives contradiction. 'Ex falso' says that anything will follow from a contradiction. |
| 13354 | 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock] |
| Full Idea: The Principle of Conjunction says that Γ |= φ∧ψ iff Γ |= φ and Γ |= ψ. This implies φ,ψ |= φ∧ψ, which is ∧-introduction. It is also implies ∧-elimination. | |
| From: David Bostock (Intermediate Logic [1997], 2.5.F) | |
| A reaction: [Second half is Ex. 2.5.3] That is, if they are entailed separately, they are entailed as a unit. It is a moot point whether these principles are theorems of propositional logic, or derivation rules. |
| 13610 | A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock] |
| Full Idea: For ¬,→ Schemas: (A1) |-φ→(ψ→φ), (A2) |-(φ→(ψ→ξ)) → ((φ→ψ)→(φ→ξ)), (A3) |-(¬φ→¬ψ) → (ψ→φ), Rule:DET:|-φ,|-φ→ψ then |-ψ | |
| From: David Bostock (Intermediate Logic [1997], 5.2) | |
| A reaction: A1 says everything implies a truth, A2 is conditional proof, and A3 is contraposition. DET is modus ponens. This is Bostock's compact near-minimal axiom system for proposition logic. He adds two axioms and another rule for predicate logic. |
| 13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
| Full Idea: A 'free' logic is one in which names are permitted to be empty. A 'universally free' logic is one in which the domain of an interpretation may also be empty. | |
| From: David Bostock (Intermediate Logic [1997], 8.6) |
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |