Combining Philosophers

All the ideas for Peter B. Lewis, Friedrich Schiller and E.J. Lemmon

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56 ideas

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
'Contradictory' propositions always differ in truth-value [Lemmon]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / a. Symbols of PL
The sign |- may be read as 'therefore' [Lemmon]
We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon]
We write the 'negation' of P (not-P) as [Lemmon]
We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P) [Lemmon]
That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon]
If A and B are 'interderivable' from one another we may write A -||- B [Lemmon]
That proposition that both P and Q is their 'conjunction', written P∧Q [Lemmon]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon]
'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon]
A 'well-formed formula' follows the rules for variables, , →, ∧, ∨, and ↔ [Lemmon]
The 'scope' of a connective is the connective, the linked formulae, and the brackets [Lemmon]
A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon]
'Contrary' propositions are never both true, so that (A∧B) is a tautology [Lemmon]
A wff is 'inconsistent' if all assignments to variables result in the value F [Lemmon]
A 'substitution-instance' is a wff formed by consistent replacing variables with wffs [Lemmon]
A wff is 'contingent' if produces at least one T and at least one F [Lemmon]
Two propositions are 'equivalent' if they mirror one another's truth-value [Lemmon]
A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
∧E: Given A∧B, we may derive either A or B separately [Lemmon]
A: we may assume any proposition at any stage [Lemmon]
MTT: Given B and A→B, we derive A [Lemmon]
∧I: Given A and B, we may derive A∧B [Lemmon]
CP: Given a proof of B from A as assumption, we may derive A→B [Lemmon]
DN: Given A, we may derive A [Lemmon]
MPP: Given A and A→B, we may derive B [Lemmon]
RAA: If assuming A will prove B∧B, then derive A [Lemmon]
∨I: Given either A or B separately, we may derive A∨B [Lemmon]
∨E: Derive C from A∨B, if C can be derived both from A and from B [Lemmon]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
'Modus tollendo ponens' (MTP) says P, P ∨ Q |- Q [Lemmon]
'Modus ponendo tollens' (MPT) says P, (P ∧ Q) |- Q [Lemmon]
We can change conditionals into negated conjunctions with P→Q -||- (P ∧ Q) [Lemmon]
We can change conditionals into disjunctions with P→Q -||- P ∨ Q [Lemmon]
We can change conjunctions into negated conditionals with P→Q -||- (P → Q) [Lemmon]
De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions [Lemmon]
The Distributive Laws can rearrange a pair of conjunctions or disjunctions [Lemmon]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Truth-tables are good for showing invalidity [Lemmon]
A truth-table test is entirely mechanical, but this won't work for more complex logic [Lemmon]
4. Formal Logic / B. Propositional Logic PL / 4. Soundness of PL
If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology [Lemmon]
4. Formal Logic / B. Propositional Logic PL / 5. Completeness of PL
Propositional logic is complete, since all of its tautologous sequents are derivable [Lemmon]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / a. Symbols of PC
The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E [Lemmon]
Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....' [Lemmon]
'Gm' says m has property G, and 'Pmn' says m has relation P to n [Lemmon]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / b. Terminology of PC
Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers' [Lemmon]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / c. Derivations rules of PC
Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules [Lemmon]
Universal elimination if you start with the universal, introduction if you want to end with it [Lemmon]
Universal Elimination (UE) lets us infer that an object has F, from all things having F [Lemmon]
With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro [Lemmon]
UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one [Lemmon]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers [Lemmon]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional → [Lemmon]
5. Theory of Logic / B. Logical Consequence / 8. Material Implication
The paradoxes of material implication are P |- Q → P, and P |- P → Q [Lemmon]
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / d. Absolute idealism
Fichte, Schelling and Hegel rejected transcendental idealism [Lewis,PB]
Fichte, Hegel and Schelling developed versions of Absolute Idealism [Lewis,PB]
21. Aesthetics / C. Artistic Issues / 7. Art and Morality
Beauty motivates morality, by harmonising feeling and reason [Schiller, by Pinkard]
25. Society / C. Social Justice / 3. Social Freedom / e. Freedom of lifestyle
Schiller speaks obsessively of freedom throughout his works [Schiller, by Berlin]