display all the ideas for this combination of philosophers
18 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) |
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) |
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) |
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. |
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] |
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. |
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. |
18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
Full Idea: None of the classical ways of defining one logical constant in terms of others is available in intuitionist logic (and this includes the two quantifiers). | |
From: David Bostock (Philosophy of Mathematics [2009], 7.2) |
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) |
18114 | There is no single agreed structure for set theory [Bostock] |
Full Idea: There is so far no agreed set of axioms for set theory which is categorical, i.e. which does pick just one structure. | |
From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
A reaction: This contrasts with Peano Arithmetic, which is categorical in its second-order version. |
18107 | A 'proper class' cannot be a member of anything [Bostock] |
Full Idea: A 'proper class' cannot be a member of anything, neither of a set nor of another proper class. | |
From: David Bostock (Philosophy of Mathematics [2009], 5.4) |
18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
Full Idea: We could add the axiom that all sets are constructible (V = L), making the universe of sets as small as possible, or add the axiom that there is a supercompact cardinal (SC), making the universe as large as we no know how to. | |
From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
A reaction: Bostock says most mathematicians reject the first option, and are undecided about the second option. |
18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
Full Idea: The usual accounts of ZF are not restricted to subsets that we can describe, and that is what justifies the axiom of choice. | |
From: David Bostock (Philosophy of Mathematics [2009], 8.4 n36) | |
A reaction: This contrasts interestingly with predicativism, which says we can only discuss things which we can describe or define. Something like verificationism hovers in the background. |
18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
Full Idea: The Axiom of Replacement (or the Axiom of Subsets, 'Aussonderung', Fraenkel 1922) in effect enforces the idea that 'limitation of size' is a crucial factor when deciding whether a proposed set or does not not exist. | |
From: David Bostock (Philosophy of Mathematics [2009], 5.4) |