Combining Texts

All the ideas for 'Truly Understood', 'Intro to Non-Classical Logic (1st ed)' and 'Logic for Philosophy'

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78 ideas

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 [Sider]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Truth tables assume truth functionality, and are just pictures of truth functions [Sider]
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 [Sider]
In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary [Sider]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / f. System B
System B introduces iterated modalities [Sider]
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 [Sider]
4. Formal Logic / D. Modal Logic ML / 5. Epistemic Logic
Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and K¬K) [Sider]
4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
We can treat modal worlds as different times [Sider]
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
Converse Barcan Formula: □∀αφ→∀α□φ [Sider]
The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic [Sider]
System B is needed to prove the Barcan Formula [Sider]
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
You can employ intuitionist logic without intuitionism about mathematics [Sider]
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
Free logic is one of the few first-order non-classical logics [Priest,G]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets [Priest,G]
<a,b&62; is a set whose members occur in the order shown [Priest,G]
a ∈ X says a is an object in set X; a ∉ X says a is not in X [Priest,G]
{x; A(x)} is a set of objects satisfying the condition A(x) [Priest,G]
{a1, a2, ...an} indicates that a set comprising just those objects [Priest,G]
Φ indicates the empty set, which has no members [Priest,G]
{a} is the 'singleton' set of a (not the object a itself) [Priest,G]
X⊂Y means set X is a 'proper subset' of set Y [Priest,G]
X⊆Y means set X is a 'subset' of set Y [Priest,G]
X = Y means the set X equals the set Y [Priest,G]
X ∩ Y indicates the 'intersection' of sets X and Y, the objects which are in both sets [Priest,G]
X∪Y indicates the 'union' of all the things in sets X and Y [Priest,G]
Y - X is the 'relative complement' of X with respect to Y; the things in Y that are not in X [Priest,G]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
The 'relative complement' is things in the second set not in the first [Priest,G]
The 'intersection' of two sets is a set of the things that are in both sets [Priest,G]
The 'union' of two sets is a set containing all the things in either of the sets [Priest,G]
The 'induction clause' says complex formulas retain the properties of their basic formulas [Priest,G]
A 'member' of a set is one of the objects in the set [Priest,G]
An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order [Priest,G]
A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets [Priest,G]
A 'set' is a collection of objects [Priest,G]
The 'empty set' or 'null set' has no members [Priest,G]
A set is a 'subset' of another set if all of its members are in that set [Priest,G]
A 'proper subset' is smaller than the containing set [Priest,G]
A 'singleton' is a set with only one member [Priest,G]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
The empty set Φ is a subset of every set (including itself) [Priest,G]
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
The most popular account of logical consequence is the semantic or model-theoretic one [Sider]
Maybe logical consequence is more a matter of provability than of truth-preservation [Sider]
Maybe logical consequence is impossibility of the premises being true and the consequent false [Sider]
Maybe logical consequence is a primitive notion [Sider]
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
A 'theorem' is an axiom, or the last line of a legitimate proof [Sider]
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 [Sider]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'total' function must always produce an output for a given domain [Sider]
5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction
λ can treat 'is cold and hungry' as a single predicate [Sider]
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
Good axioms should be indisputable logical truths [Sider]
No assumptions in axiomatic proofs, so no conditional proof or reductio [Sider]
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 [Sider]
Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' [Sider]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Natural deduction helpfully allows reasoning with assumptions [Sider]
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
We can build proofs just from conclusions, rather than from plain formulae [Sider]
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Valuations in PC assign truth values to formulas relative to variable assignments [Sider]
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 [Sider]
It is hard to say which are the logical truths in modal logic, especially for iterated modal operators [Sider]
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 [Sider]
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 [Sider]
5. Theory of Logic / K. Features of Logics / 6. Compactness
Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider]
7. Existence / D. Theories of Reality / 10. Vagueness / f. Supervaluation for vagueness
A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation [Sider]
Supervaluational logic is classical, except when it adds the 'Definitely' operator [Sider]
A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification [Sider]
We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings [Sider]
8. Modes of Existence / A. Relations / 1. Nature of Relations
A relation is a feature of multiple objects taken together [Sider]
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 [Sider]
10. Modality / A. Necessity / 3. Types of Necessity
'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist [Sider]
10. Modality / A. Necessity / 5. Metaphysical Necessity
Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible [Sider]
10. Modality / A. Necessity / 6. Logical Necessity
Logical truths must be necessary if anything is [Sider]
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 [Sider]
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 [Sider]
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 [Sider]
18. Thought / A. Modes of Thought / 6. Judgement / a. Nature of Judgement
Concepts are distinguished by roles in judgement, and are thus tied to rationality [Peacocke]
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
A sense is individuated by the conditions for reference [Peacocke]
Fregean concepts have their essence fixed by reference-conditions [Peacocke]
18. Thought / D. Concepts / 4. Structure of Concepts / a. Conceptual structure
Concepts have distinctive reasons and norms [Peacocke]
18. Thought / D. Concepts / 4. Structure of Concepts / b. Analysis of concepts
Any explanation of a concept must involve reference and truth [Peacocke]
19. Language / C. Assigning Meanings / 4. Compositionality
Encountering novel sentences shows conclusively that meaning must be compositional [Peacocke]