Ideas from 'Logic for Philosophy' by Theodore Sider [2010], 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
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13689
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'Theorems' are formulas provable from no premises at all
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4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
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13705
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Truth tables assume truth functionality, and are just pictures of truth functions
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D
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13706
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Intuitively, deontic accessibility seems not to be reflexive, but to be serial
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13710
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In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / f. System B
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13711
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System B introduces iterated modalities
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
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13708
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S5 is the strongest system, since it has the most valid formulas, because it is easy to be S5-valid
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4. Formal Logic / D. Modal Logic ML / 5. Epistemic Logic
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13712
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Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and K¬K)
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4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
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13714
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We can treat modal worlds as different times
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4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
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13720
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Converse Barcan Formula: □∀αφ→∀α□φ
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13718
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The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic
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13723
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System B is needed to prove the Barcan Formula
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4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
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13715
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You can employ intuitionist logic without intuitionism about mathematics
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5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
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13678
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The most popular account of logical consequence is the semantic or model-theoretic one
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13680
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Maybe logical consequence is a primitive notion
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13682
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Maybe logical consequence is impossibility of the premises being true and the consequent false
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13679
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Maybe logical consequence is more a matter of provability than of truth-preservation
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5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
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13722
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A 'theorem' is an axiom, or the last line of a legitimate proof
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5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
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13696
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When a variable is 'free' of the quantifier, the result seems incapable of truth or falsity
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5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
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13700
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A 'total' function must always produce an output for a given domain
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5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction
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13703
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λ can treat 'is cold and hungry' as a single predicate
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5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
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13688
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Good axioms should be indisputable logical truths
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13687
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No assumptions in axiomatic proofs, so no conditional proof or reductio
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5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
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13691
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Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step'
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13690
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Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms
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5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
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13685
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Natural deduction helpfully allows reasoning with assumptions
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5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
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13686
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We can build proofs just from conclusions, rather than from plain formulae
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5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
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13697
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Valuations in PC assign truth values to formulas relative to variable assignments
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5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
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13684
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The semantical notion of a logical truth is validity, being true in all interpretations
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13704
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It is hard to say which are the logical truths in modal logic, especially for iterated modal operators
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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13724
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In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation
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5. Theory of Logic / K. Features of Logics / 4. Completeness
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13698
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In a complete logic you can avoid axiomatic proofs, by using models to show consequences
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5. Theory of Logic / K. Features of Logics / 6. Compactness
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13699
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Compactness surprisingly says that no contradictions can emerge when the set goes infinite
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
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13701
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A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically
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7. Existence / D. Theories of Reality / 10. Vagueness / f. Supervaluation for vagueness
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13693
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A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification
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13692
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A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation
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13694
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We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings
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13695
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Supervaluational logic is classical, except when it adds the 'Definitely' operator
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8. Modes of Existence / A. Relations / 1. Nature of Relations
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13683
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A relation is a feature of multiple objects taken together
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9. Objects / F. Identity among Objects / 7. Indiscernible Objects
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13702
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The identity of indiscernibles is necessarily true, if being a member of some set counts as a property
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10. Modality / A. Necessity / 3. Types of Necessity
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13721
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'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist
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10. Modality / A. Necessity / 5. Metaphysical Necessity
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13707
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Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible
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10. Modality / A. Necessity / 6. Logical Necessity
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13709
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Logical truths must be necessary if anything is
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10. Modality / B. Possibility / 8. Conditionals / b. Types of conditional
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13716
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'If B hadn't shot L someone else would have' if false; 'If B didn't shoot L, someone else did' is true
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10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
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13717
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Transworld identity is not a problem in de dicto sentences, which needn't identify an individual
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10. Modality / E. Possible worlds / 3. Transworld Objects / e. Possible Objects
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13719
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Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost
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