### Ideas from 'Logic for Philosophy' by Theodore Sider , by Theme Structure

#### [found in 'Logic for Philosophy' by Sider,Theodore [OUP 2010,978-0-19-957558-9]].

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###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
 13689 'Theorems' are formulas provable from no premises at all
###### 4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
 13705 Truth tables assume truth functionality, and are just pictures of truth functions
###### 4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D
 13706 Intuitively, deontic accessibility seems not to be reflexive, but to be serial
 13710 In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary
###### 4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / f. System B
 13711 System B introduces iterated modalities
###### 4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
 13708 S5 is the strongest system, since it has the most valid formulas, because it is easy to be S5-valid
###### 4. Formal Logic / D. Modal Logic ML / 5. Epistemic Logic
 13712 Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and K¬K)
###### 4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
 13714 We can treat modal worlds as different times
###### 4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
 13718 The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic
 13720 Converse Barcan Formula: □∀αφ→∀α□φ
 13723 System B is needed to prove the Barcan Formula
###### 4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
 13715 You can employ intuitionist logic without intuitionism about mathematics
###### 5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
 13680 Maybe logical consequence is a primitive notion
 13678 The most popular account of logical consequence is the semantic or model-theoretic one
 13679 Maybe logical consequence is more a matter of provability than of truth-preservation
 13682 Maybe logical consequence is impossibility of the premises being true and the consequent false
###### 5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
 13722 A 'theorem' is an axiom, or the last line of a legitimate proof
###### 5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
 13696 When a variable is 'free' of the quantifier, the result seems incapable of truth or falsity
###### 5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
 13700 A 'total' function must always produce an output for a given domain
###### 5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction
 13703 λ can treat 'is cold and hungry' as a single predicate
###### 5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
 13687 No assumptions in axiomatic proofs, so no conditional proof or reductio
 13688 Good axioms should be indisputable logical truths
###### 5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
 13690 Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms
 13691 Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step'
###### 5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
 13685 Natural deduction helpfully allows reasoning with assumptions
###### 5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
 13686 We can build proofs just from conclusions, rather than from plain formulae
###### 5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
 13697 Valuations in PC assign truth values to formulas relative to variable assignments
###### 5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
 13684 The semantical notion of a logical truth is validity, being true in all interpretations
 13704 It is hard to say which are the logical truths in modal logic, especially for iterated modal operators
###### 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
 13724 In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation
###### 5. Theory of Logic / K. Features of Logics / 4. Completeness
 13698 In a complete logic you can avoid axiomatic proofs, by using models to show consequences
###### 5. Theory of Logic / K. Features of Logics / 6. Compactness
 13699 Compactness surprisingly says that no contradictions can emerge when the set goes infinite
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
 13701 A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically
###### 7. Existence / D. Theories of Reality / 9. Vagueness / f. Supervaluation for vagueness
 13692 A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation
 13693 A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification
 13694 We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings
 13695 Supervaluational logic is classical, except when it adds the 'Definitely' operator
###### 8. Modes of Existence / A. Relations / 1. Nature of Relations
 13683 A relation is a feature of multiple objects taken together
###### 9. Objects / F. Identity among Objects / 7. Indiscernible Objects
 13702 The identity of indiscernibles is necessarily true, if being a member of some set counts as a property
###### 10. Modality / A. Necessity / 3. Types of Necessity
 13721 'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist
###### 10. Modality / A. Necessity / 5. Metaphysical Necessity
 13707 Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible
###### 10. Modality / A. Necessity / 6. Logical Necessity
 13709 Logical truths must be necessary if anything is
###### 10. Modality / B. Possibility / 8. Conditionals / b. Types of conditional
 13716 'If B hadn't shot L someone else would have' if false; 'If B didn't shoot L, someone else did' is true
###### 10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
 13717 Transworld identity is not a problem in de dicto sentences, which needn't identify an individual
###### 10. Modality / E. Possible worlds / 3. Transworld Objects / e. Possible Objects
 13719 Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost