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
The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic
Converse Barcan Formula: □∀αφ→∀α□φ
System B is needed to prove the Barcan Formula
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 more a matter of provability than of truth-preservation
The most popular account of logical consequence is the semantic or model-theoretic one
Maybe logical consequence is impossibility of the premises being true and the consequent false
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
Good axioms should be indisputable logical truths
No assumptions in axiomatic proofs, so no conditional proof or reductio
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms
Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step'
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
The semantical notion of a logical truth is validity, being true in all interpretations
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
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
Supervaluational logic is classical, except when it adds the 'Definitely' operator
A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation
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