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Single Idea 9524
[filed under theme 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
]
Full Idea
The proof that P→Q -||- ¬P ∨ Q is useful for enabling us to change conditionals into disjunctions.
Gist of Idea
We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q
Source
E.J. Lemmon (Beginning Logic [1965], 2.2)
Book Ref
Lemmon,E.J.: 'Beginning Logic' [Nelson 1979], p.62
The
52 ideas
from 'Beginning Logic'
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13901
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Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules
[Lemmon]
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9508
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The sign |- may be read as 'therefore'
[Lemmon]
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9511
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We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q
[Lemmon]
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9512
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We write the 'negation' of P (not-P) as ¬
[Lemmon]
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9510
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That proposition that either P or Q is their 'disjunction', written P∨Q
[Lemmon]
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9509
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That proposition that both P and Q is their 'conjunction', written P∧Q
[Lemmon]
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9513
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We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P)
[Lemmon]
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9396
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DN: Given A, we may derive ¬¬A
[Lemmon]
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9399
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∧E: Given A∧B, we may derive either A or B separately
[Lemmon]
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9401
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∨E: Derive C from A∨B, if C can be derived both from A and from B
[Lemmon]
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9395
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MTT: Given ¬B and A→B, we derive ¬A
[Lemmon]
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9514
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If A and B are 'interderivable' from one another we may write A -||- B
[Lemmon]
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9402
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RAA: If assuming A will prove B∧¬B, then derive ¬A
[Lemmon]
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9397
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CP: Given a proof of B from A as assumption, we may derive A→B
[Lemmon]
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9393
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A: we may assume any proposition at any stage
[Lemmon]
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9400
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∨I: Given either A or B separately, we may derive A∨B
[Lemmon]
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9398
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∧I: Given A and B, we may derive A∧B
[Lemmon]
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9394
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MPP: Given A and A→B, we may derive B
[Lemmon]
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9517
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The 'scope' of a connective is the connective, the linked formulae, and the brackets
[Lemmon]
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9516
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A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔
[Lemmon]
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9518
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A 'theorem' is the conclusion of a provable sequent with zero assumptions
[Lemmon]
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9524
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We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q
[Lemmon]
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9525
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We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q)
[Lemmon]
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9521
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'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q
[Lemmon]
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9519
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A 'substitution-instance' is a wff formed by consistent replacing variables with wffs
[Lemmon]
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9522
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'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q
[Lemmon]
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9526
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We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q)
[Lemmon]
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9527
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The Distributive Laws can rearrange a pair of conjunctions or disjunctions
[Lemmon]
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9523
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De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions
[Lemmon]
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9520
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The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q
[Lemmon]
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9529
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A wff is 'inconsistent' if all assignments to variables result in the value F
[Lemmon]
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9534
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Two propositions are 'equivalent' if they mirror one another's truth-value
[Lemmon]
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9532
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'Subcontrary' propositions are never both false, so that A∨B is a tautology
[Lemmon]
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9531
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'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology
[Lemmon]
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9535
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'Contradictory' propositions always differ in truth-value
[Lemmon]
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9528
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A wff is a 'tautology' if all assignments to variables result in the value T
[Lemmon]
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9530
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A wff is 'contingent' if produces at least one T and at least one F
[Lemmon]
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9533
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A 'implies' B if B is true whenever A is true (so that A→B is tautologous)
[Lemmon]
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9537
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Truth-tables are good for showing invalidity
[Lemmon]
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9536
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If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology
[Lemmon]
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9539
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Propositional logic is complete, since all of its tautologous sequents are derivable
[Lemmon]
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9538
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A truth-table test is entirely mechanical, but this won't work for more complex logic
[Lemmon]
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13909
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Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....'
[Lemmon]
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13902
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'Gm' says m has property G, and 'Pmn' says m has relation P to n
[Lemmon]
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13910
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Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers'
[Lemmon]
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13900
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'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional →
[Lemmon]
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13905
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If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers
[Lemmon]
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13904
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Universal Elimination (UE) lets us infer that an object has F, from all things having F
[Lemmon]
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13903
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Universal elimination if you start with the universal, introduction if you want to end with it
[Lemmon]
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13906
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With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro
[Lemmon]
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13908
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UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one
[Lemmon]
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13911
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The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E
[Lemmon]
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