Logic

formalisation

translation into formal language using dictionary

validity of an english argument

valid in L iff it has a valid L formalisation

logical truth of english sentences

logically true in L iff it has L formalisation logical truth

consistency of sets of english sentences

consistent in L iff all its L formalisations are consistent

truth-functional connectives

iff truth value of compound sentences it forms cannot be changed by replacing one of the sentence it combines with another having the same truth value in L1 formalisation

^

and, but, although

V

or, unless, either or

→

if then, provided that, only if, necessary condition

←>

if and only if, just if, exactly if

proofs

finite sequence of obvious arguments

provability

if there is a proof system S with premise L and conclusion Q, we write L |-s Q (Q is provable from L in S)

soundness

iff it only proves valid arguments (is a must)

completeness

iff it proves all valid arguments (is a plus)

Equivalent

if S1 and S2 are both sound and complete in L1

provability

Q is provable from L in natural deduction iff there is a proof of Q whose undischarged assumptions are all sentences in L

rules for natural deduction

assumption rule, connective rule, nothing else is a proof

intro rule for conjunction

may append Q ^ P to a proof of Q and of P

elim rule for conjunction

may append Q or P to a proof of Q ^ P

intro rule for material implication

may append Q → P to a proof of P and discharge all assumptions of Q in the proof

elim rule for material implication

may append Q to a proof of P and a proof of Q → P

intro rule for disjunction

may append Q V P to a proof of Q or of P

elim rule for disjunction

may append X to a proof of Q V P, a proof of X and another X, discharging all assumptions of Q in the second proof and all assumptions of P in the third proof

intro rule for negation

may append ¬Q to a proof of ¬P discharging all assumptions of Q in the proof

elim rule for negation

may append Q to a proof of P and a proof of ¬P discharging all assumptions of ¬Q in the proof

intro rule for material equivalence

may append Q ←> P

elim rule for material equivalence

may append P or Q to a proof of Q or P and a proof of Q ←> P

structure interpretation

giving truth value to sentence letter

partial structures

shows only truth value of letters that appear in argument

compositionality

meaning of whole is function of meaning of parts

|Ωb| = T

Ω is true in structure b

conjunction

T, F, F, F

disjunction

T, T, T, F

material implication

T, F, T, T

material equivalence

T, F, F, T

tautologies - logically true

iff true in all L1 structures

contradictions - logically false

iff false in all L1 structures

Logically equivalent sentences

iff truth values are the same in every L1 structure

L |= Q

L entails Q is valid

|= Q

Q is a logical truth

validity

iff set of premises with ¬conclusion is inconsistent

if … then …

->

… and …

^

… or …

v

it is not the case that

¬

… if and only if …

double arrow

propositional letters

replace english sentences with arbitrary letters, uniformity must be respected

modus ponens

P → Q, P | Q

modus tollens

P → Q, ¬Q | ¬P

Syntax

expressions

Semantics

meaning of expressions

Formal language L1 - Vocabulary

sentence letters and connectives

Formal language L1 - Syntax

all sentences are sentences of L1, if P and Q are sentences then so are ¬P or (P^Q), nothing else is a sentence of L1

Formal language L1 - formalisation

going from metalanguage (Ω) to object language (P^Q)

conventions for dropping brackets

outermost can be dropped, drop in left associated series (for ^ and V), ^ and V binds more than → and ←>

main connective

that which is applied last

Argument

set of declarative sentences

declarative sentences

sentence that can be true or false

validity

iff no uniform interpretation of subject-specific expressions make all premises true and the conclusion false

consistency

set of sentences for which there is at least one interpretation that makes them all true

logical truth

a statement that is true under all interpretations

logical falsehood

a statement that is false under all interpretations

logically contingent

if true under some but not all interpretations

logical equivalence

iff no interpretation of two sentences makes one true and the other false