Ideas from 'Logical Consequence' by JC Beall / G Restall [2005], by Theme Structure

[found in 'Stanford Online Encyclopaedia of Philosophy' (ed/tr Stanford University) [plato.stanford.edu ,-]].

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4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic

10688

'Equivocation' is when terms do not mean the same thing in premises and conclusion

			Full Idea: 'Equivocation' is when the terms do not mean the same thing in the premises and in the conclusion.
			From: JC Beall / G Restall (Logical Consequence [2005], Intro)

5. Theory of Logic / A. Overview of Logic / 4. Pure Logic

10690

Formal logic is invariant under permutations, or devoid of content, or gives the norms for thought

			Full Idea: Logic is purely formal either when it is invariant under permutation of object (Tarski), or when it has totally abstracted away from all contents, or it is the constitutive norms for thought.
			From: JC Beall / G Restall (Logical Consequence [2005], 2)
			A reaction: [compressed] The third account sounds rather woolly, and the second one sounds like a tricky operation, but the first one sounds clear and decisive, so I vote for Tarski.

5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence

10691

Logical consequence needs either proofs, or absence of counterexamples

			Full Idea: Technical work on logical consequence has either focused on proofs, where validity is the existence of a proof of the conclusions from the premises, or on models, which focus on the absence of counterexamples.
			From: JC Beall / G Restall (Logical Consequence [2005], 3)

5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=

10695

Logical consequence is either necessary truth preservation, or preservation based on interpretation

			Full Idea: Two different views of logical consequence are necessary truth-preservation (based on modelling possible worlds; favoured by Realists), or truth-preservation based on the meanings of the logical vocabulary (differing in various models; for Anti-Realists).
			From: JC Beall / G Restall (Logical Consequence [2005], 2)
			A reaction: Thus Dummett prefers the second view, because the law of excluded middle is optional. My instincts are with the first one.

5. Theory of Logic / B. Logical Consequence / 8. Material Implication

10689

A step is a 'material consequence' if we need contents as well as form

			Full Idea: A logical step is a 'material consequence' and not a formal one, if we need the contents as well as the structure or form.
			From: JC Beall / G Restall (Logical Consequence [2005], 2)

5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth

10696

A 'logical truth' (or 'tautology', or 'theorem') follows from empty premises

			Full Idea: If a conclusion follows from an empty collection of premises, it is true by logic alone, and is a 'logical truth' (sometimes a 'tautology'), or, in the proof-centred approach, 'theorems'.
			From: JC Beall / G Restall (Logical Consequence [2005], 4)
			A reaction: These truths are written as following from the empty set Φ. They are just implications derived from the axioms and the rules.

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

10693

Models are mathematical structures which interpret the non-logical primitives

			Full Idea: Models are abstract mathematical structures that provide possible interpretations for each of the non-logical primitives in a formal language.
			From: JC Beall / G Restall (Logical Consequence [2005], 3)

6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics

10692

Hilbert proofs have simple rules and complex axioms, and natural deduction is the opposite

			Full Idea: There are many proof-systems, the main being Hilbert proofs (with simple rules and complex axioms), or natural deduction systems (with few axioms and many rules, and the rules constitute the meaning of the connectives).
			From: JC Beall / G Restall (Logical Consequence [2005], 3)