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
'Theorems' are formulas provable from no premises at all
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
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
Intuitively, deontic accessibility seems not to be reflexive, but to be serial
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
System B introduces iterated modalities
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
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
Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and KČK)
4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
We can treat modal worlds as different times
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
Converse Barcan Formula: □∀αφ→∀α□φ
System B is needed to prove the Barcan Formula
The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
You can employ intuitionist logic without intuitionism about mathematics
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Maybe logical consequence is impossibility of the premises being true and the consequent false
The most popular account of logical consequence is the semantic or model-theoretic one
Maybe logical consequence is more a matter of provability than of truth-preservation
Maybe logical consequence is a primitive notion
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
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
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
A 'total' function must always produce an output for a given domain
5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction
λ can treat 'is cold and hungry' as a single predicate
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
No assumptions in axiomatic proofs, so no conditional proof or reductio
Good axioms should be indisputable logical truths
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step'
Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Natural deduction helpfully allows reasoning with assumptions
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
We can build proofs just from conclusions, rather than from plain formulae
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Valuations in PC assign truth values to formulas relative to variable assignments
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
It is hard to say which are the logical truths in modal logic, especially for iterated modal operators
The semantical notion of a logical truth is validity, being true in all interpretations
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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
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
Compactness surprisingly says that no contradictions can emerge when the set goes infinite
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2nd-order
A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically
7. Existence / D. Theories of Reality / 9. Vagueness / e. Supervaluation for vagueness
A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation
Supervaluational logic is classical, except when it adds the 'Definitely' operator
A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification
We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings
8. Modes of Existence / A. Relations / 1. Nature of Relations
A relation is a feature of multiple objects taken together
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
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
'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist
10. Modality / A. Necessity / 5. Metaphysical Necessity
Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible
10. Modality / A. Necessity / 6. Logical Necessity
Logical truths must be necessary if anything is
10. Modality / B. Possibility / 8. Conditionals / b. Types of conditional
'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
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
Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost