more on this theme
|
more from this thinker
Single Idea 9536
[filed under theme 4. Formal Logic / B. Propositional Logic PL / 4. Soundness of PL
]
Full Idea
If any application of the nine derivation rules of propositional logic is made on tautologous sequents, we have demonstrated that the result is always a tautologous sequent. Thus the system is consistent.
Gist of Idea
If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology
Source
E.J. Lemmon (Beginning Logic [1965], 2.4)
Book Ref
Lemmon,E.J.: 'Beginning Logic' [Nelson 1979], p.80
A Reaction
The term 'sound' tends to be used now, rather than 'consistent'. See Lemmon for the proofs of each of the nine rules.
The
52 ideas
from 'Beginning Logic'
|
13901
|
Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules
[Lemmon]
|
|
9508
|
The sign |- may be read as 'therefore'
[Lemmon]
|
|
9511
|
We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q
[Lemmon]
|
|
9512
|
We write the 'negation' of P (not-P) as ¬
[Lemmon]
|
|
9510
|
That proposition that either P or Q is their 'disjunction', written P∨Q
[Lemmon]
|
|
9509
|
That proposition that both P and Q is their 'conjunction', written P∧Q
[Lemmon]
|
|
9513
|
We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P)
[Lemmon]
|
|
9396
|
DN: Given A, we may derive ¬¬A
[Lemmon]
|
|
9399
|
∧E: Given A∧B, we may derive either A or B separately
[Lemmon]
|
|
9401
|
∨E: Derive C from A∨B, if C can be derived both from A and from B
[Lemmon]
|
|
9395
|
MTT: Given ¬B and A→B, we derive ¬A
[Lemmon]
|
|
9514
|
If A and B are 'interderivable' from one another we may write A -||- B
[Lemmon]
|
|
9402
|
RAA: If assuming A will prove B∧¬B, then derive ¬A
[Lemmon]
|
|
9397
|
CP: Given a proof of B from A as assumption, we may derive A→B
[Lemmon]
|
|
9393
|
A: we may assume any proposition at any stage
[Lemmon]
|
|
9400
|
∨I: Given either A or B separately, we may derive A∨B
[Lemmon]
|
|
9398
|
∧I: Given A and B, we may derive A∧B
[Lemmon]
|
|
9394
|
MPP: Given A and A→B, we may derive B
[Lemmon]
|
|
9517
|
The 'scope' of a connective is the connective, the linked formulae, and the brackets
[Lemmon]
|
|
9516
|
A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔
[Lemmon]
|
|
9518
|
A 'theorem' is the conclusion of a provable sequent with zero assumptions
[Lemmon]
|
|
9524
|
We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q
[Lemmon]
|
|
9525
|
We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q)
[Lemmon]
|
|
9521
|
'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q
[Lemmon]
|
|
9519
|
A 'substitution-instance' is a wff formed by consistent replacing variables with wffs
[Lemmon]
|
|
9522
|
'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q
[Lemmon]
|
|
9526
|
We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q)
[Lemmon]
|
|
9527
|
The Distributive Laws can rearrange a pair of conjunctions or disjunctions
[Lemmon]
|
|
9523
|
De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions
[Lemmon]
|
|
9520
|
The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q
[Lemmon]
|
|
9529
|
A wff is 'inconsistent' if all assignments to variables result in the value F
[Lemmon]
|
|
9534
|
Two propositions are 'equivalent' if they mirror one another's truth-value
[Lemmon]
|
|
9532
|
'Subcontrary' propositions are never both false, so that A∨B is a tautology
[Lemmon]
|
|
9531
|
'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology
[Lemmon]
|
|
9535
|
'Contradictory' propositions always differ in truth-value
[Lemmon]
|
|
9528
|
A wff is a 'tautology' if all assignments to variables result in the value T
[Lemmon]
|
|
9530
|
A wff is 'contingent' if produces at least one T and at least one F
[Lemmon]
|
|
9533
|
A 'implies' B if B is true whenever A is true (so that A→B is tautologous)
[Lemmon]
|
|
9537
|
Truth-tables are good for showing invalidity
[Lemmon]
|
|
9536
|
If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology
[Lemmon]
|
|
9539
|
Propositional logic is complete, since all of its tautologous sequents are derivable
[Lemmon]
|
|
9538
|
A truth-table test is entirely mechanical, but this won't work for more complex logic
[Lemmon]
|
|
13909
|
Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....'
[Lemmon]
|
|
13902
|
'Gm' says m has property G, and 'Pmn' says m has relation P to n
[Lemmon]
|
|
13910
|
Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers'
[Lemmon]
|
|
13900
|
'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional →
[Lemmon]
|
|
13905
|
If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers
[Lemmon]
|
|
13904
|
Universal Elimination (UE) lets us infer that an object has F, from all things having F
[Lemmon]
|
|
13903
|
Universal elimination if you start with the universal, introduction if you want to end with it
[Lemmon]
|
|
13906
|
With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro
[Lemmon]
|
|
13908
|
UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one
[Lemmon]
|
|
13911
|
The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E
[Lemmon]
|