Combining Philosophers

All the ideas for Peter Geach, Nicholas Hill and E.J. Lemmon

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80 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
We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon]
That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon]
That proposition that both P and Q is their 'conjunction', written P∧Q [Lemmon]
The sign |- may be read as 'therefore' [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]
If A and B are 'interderivable' from one another we may write A -||- B [Lemmon]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
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 'substitution-instance' is a wff formed by consistent replacing variables with wffs [Lemmon]
A wff is 'inconsistent' if all assignments to variables result in the value F [Lemmon]
'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology [Lemmon]
Two propositions are 'equivalent' if they mirror one another's truth-value [Lemmon]
A wff is 'contingent' if produces at least one T and at least one F [Lemmon]
'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon]
A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon]
A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon]
A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
∧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]
MPP: Given A and A→B, we may derive B [Lemmon]
∨E: Derive C from A∨B, if C can be derived both from A and from B [Lemmon]
DN: Given A, we may derive ¬¬A [Lemmon]
A: we may assume any proposition at any stage [Lemmon]
∧E: Given A∧B, we may derive either A or B separately [Lemmon]
RAA: If assuming A will prove B∧¬B, then derive ¬A [Lemmon]
MTT: Given ¬B and A→B, we derive ¬A [Lemmon]
∨I: Given either A or B separately, we may derive A∨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]
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]
We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q) [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
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]
The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E [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
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]
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]
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]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
Are 'word token' and 'word type' different sorts of countable objects, or two ways of counting? [Geach, by Perry]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
Abstraction from objects won't reveal an operation's being performed 'so many times' [Geach]
8. Modes of Existence / B. Properties / 10. Properties as Predicates
Attributes are functions, not objects; this distinguishes 'square of 2' from 'double of 2' [Geach]
9. Objects / A. Existence of Objects / 6. Nihilism about Objects
We should abandon absolute identity, confining it to within some category [Geach, by Hawthorne]
9. Objects / C. Structure of Objects / 2. Hylomorphism / b. Form as principle
Form is the principle that connects a thing's constitution (rather than being operative) [Hill,N]
9. Objects / F. Identity among Objects / 3. Relative Identity
Denial of absolute identity has drastic implications for logic, semantics and set theory [Wasserman on Geach]
Identity is relative. One must not say things are 'the same', but 'the same A as' [Geach]
9. Objects / F. Identity among Objects / 8. Leibniz's Law
Leibniz's Law is incomplete, since it includes a non-relativized identity predicate [Geach, by Wasserman]
9. Objects / F. Identity among Objects / 9. Sameness
Being 'the same' is meaningless, unless we specify 'the same X' [Geach]
15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
A big flea is a small animal, so 'big' and 'small' cannot be acquired by abstraction [Geach]
We cannot learn relations by abstraction, because their converse must be learned too [Geach]
15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind
If concepts are just recognitional, then general judgements would be impossible [Geach]
17. Mind and Body / B. Behaviourism / 2. Potential Behaviour
You can't define real mental states in terms of behaviour that never happens [Geach]
17. Mind and Body / B. Behaviourism / 4. Behaviourism Critique
Beliefs aren't tied to particular behaviours [Geach]
18. Thought / D. Concepts / 2. Origin of Concepts / a. Origin of concepts
The mind does not lift concepts from experience; it creates them, and then applies them [Geach]
18. Thought / D. Concepts / 3. Ontology of Concepts / b. Concepts as abilities
For abstractionists, concepts are capacities to recognise recurrent features of the world [Geach]
18. Thought / D. Concepts / 5. Concepts and Language / c. Concepts without language
If someone has aphasia but can still play chess, they clearly have concepts [Geach]
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
'Abstractionism' is acquiring a concept by picking out one experience amongst a group [Geach]
18. Thought / E. Abstraction / 8. Abstractionism Critique
'Or' and 'not' are not to be found in the sensible world, or even in the world of inner experience [Geach]
We can't acquire number-concepts by extracting the number from the things being counted [Geach]
Abstractionists can't explain counting, because it must precede experience of objects [Geach]
The numbers don't exist in nature, so they cannot have been abstracted from there into our languages [Geach]
Blind people can use colour words like 'red' perfectly intelligently [Geach]
If 'black' and 'cat' can be used in the absence of such objects, how can such usage be abstracted? [Geach]
We can form two different abstract concepts that apply to a single unified experience [Geach]
The abstractionist cannot explain 'some' and 'not' [Geach]
Only a judgement can distinguish 'striking' from 'being struck' [Geach]
22. Metaethics / C. The Good / 1. Goodness / a. Form of the Good
'Good' is an attributive adjective like 'large', not predicative like 'red' [Geach, by Foot]