if there exists a possible situation in
which the premises are true and the
f conclusion is false = inference invalid
possible
& sound
Validity is a hypothetical concept inference conclusion can never be false
;
three steps ?
1/ syntactically define formal language
2/ say what a model for that language is & what it means for a formal statement to be true in a model
3/determining a proof system (formulate inference rules that allows to derive the
conclusion from the premises in all valid inferences)
-> should be purely syntactic
Proof system we’ll be using: semantic tableaux
After laying down a set of inference rules, we show that:
- we can derive the conclusion from the premises only if the inference is valid
soundness theorem + F -
- we can derive the conclusion from the premises whenever the inference is valid
completeness theorem F D+ -
(! Proving these theorems will be our most important mathematical result)
If conclusion is false and at least one of the premises is false too =inference is valid
A model for a formal language decides for each statement in a language wether
it’s true or false
Valid
-Every possible situation where premises are true, conclusion is true as well
- -If conclusion is false and at least one of the premises is false too
-Every inference that has inconsistent premises
⑳ -Inferences whose conclusion says that something is the case or not
-If an inference can’t be invalid it has to be valid
-Iff set of premises and negation of the conclusion is unsatisfiable
Invalid
-If there exists a possible situation in which premises are true and conclusion is false
, midterm
⑫ Propositional logic contains sentential connectives:
voor
~
ex. not
and
or
if…
then…
Syntax, semantics, snd proof theory for first-order logic are extensions of those
for the propositional logic.
First-order logic deals w/ inferences of propositional logic + inferences
involving generality
3 provides the model
for logical reasoning
For dealing with general claims, quantifiers are needed.
ex. for all, there exists
⑬ Classical logic = Standard logic
Characteristics:
consistency assumption,
completeness assumption (each statement is true or not =valid),
? Every inference with inconsistent premises is valid = principle of ex falso luit het valse kun
je alles afleiden)
quodlibet (a law of Classical logic)
verum ex quodlibet (betrekking op completeness assumption) in situation each statement is either
every
true , or not .
Statements that are true in all situations = logical thruths the ball is red not or
↳Contradicting statements = logical falsehoods the ball is both red and not red
every inference whose conclusion says that something is the case or not = valid
Paraconsistent logics = logics which don’t share the consistency assumptions
Paracomplete logics = logics without completeness assumptions
Our definition of truth in a model:
Bivalence = for every model of a language, every statement is either true or false in the model, and never both
⑭ Classical propositionaloflogic
assumption
↳i consitency
conjunction completeness
is decidable (possible to write an algorithm that can determine(after
finitely many steps) if an inference is valid) It’s provable by using models or proof systems
First-order logic is not decidable
Semantic tableaux are essentially theorem provers for Propositional logic (Even if an inference is
valid, a proof may not be found quickly)
Point of completeness: derive all valid inferences
A proof system is sound and complete iff the valid inferences are precisely the ones whose conclusion
can be derived from the premises
, Aantekeningen Lecture 1
8 September 2022
Allgemeen
-
Slides belangrinste punten
:
-Bihorende opdrachten voor
eind werkcollege opdrachten
↳
eerstvolgende
feedback
college :
maken
inleveren +
werk
2 (wetenschap redeneren)
Logica- studie gelelige inferentiel
van
van
①
↳ Stelsel van uitspraken bestaand
Inferentie/gevolgtrekking wit permissent conclusie
Permisse 1+ Permisse 2 >
-
conclusie
A d.
.
V .
Slides :
① geldig-checkslides
② geldig
③
ongelding Spermissewaar ,
conclusie onwaar)
geldigheid
②
inductief -
permissen maken de conclusie
waarschighly
ll
l
deductief -
conclusie
volgt onvermydelijk uit de permissen
dan slechts dan als sitf
inferen tie-geldig &
% dat
↳D I desda het
onmogelijk is
waar permissen
en de conclusie onwaar
zijn
Geldigheid :
Als A dan B ,
A - dus B
geldig .
↳ Abstraheren van
logische irrelevante nitdrukkingen .
XRA- dus Dx
ongeldig .
Materieel :
op basis van de betekenis van
begrippen
Formeel ·alleen op basis van logische vorm
which the premises are true and the
f conclusion is false = inference invalid
possible
& sound
Validity is a hypothetical concept inference conclusion can never be false
;
three steps ?
1/ syntactically define formal language
2/ say what a model for that language is & what it means for a formal statement to be true in a model
3/determining a proof system (formulate inference rules that allows to derive the
conclusion from the premises in all valid inferences)
-> should be purely syntactic
Proof system we’ll be using: semantic tableaux
After laying down a set of inference rules, we show that:
- we can derive the conclusion from the premises only if the inference is valid
soundness theorem + F -
- we can derive the conclusion from the premises whenever the inference is valid
completeness theorem F D+ -
(! Proving these theorems will be our most important mathematical result)
If conclusion is false and at least one of the premises is false too =inference is valid
A model for a formal language decides for each statement in a language wether
it’s true or false
Valid
-Every possible situation where premises are true, conclusion is true as well
- -If conclusion is false and at least one of the premises is false too
-Every inference that has inconsistent premises
⑳ -Inferences whose conclusion says that something is the case or not
-If an inference can’t be invalid it has to be valid
-Iff set of premises and negation of the conclusion is unsatisfiable
Invalid
-If there exists a possible situation in which premises are true and conclusion is false
, midterm
⑫ Propositional logic contains sentential connectives:
voor
~
ex. not
and
or
if…
then…
Syntax, semantics, snd proof theory for first-order logic are extensions of those
for the propositional logic.
First-order logic deals w/ inferences of propositional logic + inferences
involving generality
3 provides the model
for logical reasoning
For dealing with general claims, quantifiers are needed.
ex. for all, there exists
⑬ Classical logic = Standard logic
Characteristics:
consistency assumption,
completeness assumption (each statement is true or not =valid),
? Every inference with inconsistent premises is valid = principle of ex falso luit het valse kun
je alles afleiden)
quodlibet (a law of Classical logic)
verum ex quodlibet (betrekking op completeness assumption) in situation each statement is either
every
true , or not .
Statements that are true in all situations = logical thruths the ball is red not or
↳Contradicting statements = logical falsehoods the ball is both red and not red
every inference whose conclusion says that something is the case or not = valid
Paraconsistent logics = logics which don’t share the consistency assumptions
Paracomplete logics = logics without completeness assumptions
Our definition of truth in a model:
Bivalence = for every model of a language, every statement is either true or false in the model, and never both
⑭ Classical propositionaloflogic
assumption
↳i consitency
conjunction completeness
is decidable (possible to write an algorithm that can determine(after
finitely many steps) if an inference is valid) It’s provable by using models or proof systems
First-order logic is not decidable
Semantic tableaux are essentially theorem provers for Propositional logic (Even if an inference is
valid, a proof may not be found quickly)
Point of completeness: derive all valid inferences
A proof system is sound and complete iff the valid inferences are precisely the ones whose conclusion
can be derived from the premises
, Aantekeningen Lecture 1
8 September 2022
Allgemeen
-
Slides belangrinste punten
:
-Bihorende opdrachten voor
eind werkcollege opdrachten
↳
eerstvolgende
feedback
college :
maken
inleveren +
werk
2 (wetenschap redeneren)
Logica- studie gelelige inferentiel
van
van
①
↳ Stelsel van uitspraken bestaand
Inferentie/gevolgtrekking wit permissent conclusie
Permisse 1+ Permisse 2 >
-
conclusie
A d.
.
V .
Slides :
① geldig-checkslides
② geldig
③
ongelding Spermissewaar ,
conclusie onwaar)
geldigheid
②
inductief -
permissen maken de conclusie
waarschighly
ll
l
deductief -
conclusie
volgt onvermydelijk uit de permissen
dan slechts dan als sitf
inferen tie-geldig &
% dat
↳D I desda het
onmogelijk is
waar permissen
en de conclusie onwaar
zijn
Geldigheid :
Als A dan B ,
A - dus B
geldig .
↳ Abstraheren van
logische irrelevante nitdrukkingen .
XRA- dus Dx
ongeldig .
Materieel :
op basis van de betekenis van
begrippen
Formeel ·alleen op basis van logische vorm
