### 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)