nazmi

Logic — 2026-10-23

lecture 9
chapter 6

logic validity

Logical Validity

necessary truth preservation based on formal grounds

ann is a widow, so ann is a woman
deductively valid
not logically valid
if ann is a widow, then ann is a woman
deductively necessary
but not logically necessary
ann is a widow, so ann is a widow
logically valid
deductively valid
if ann is a widow, then ann is a widow
logically necessary
deductively necessary

ann is a widow, so ann is sad
deductively invalid
logically invalid
enthymeme?

ann is a widow, so ann is a woman
deductively valid, but logically invalid
ann is a widow, [all widows are women], so ann is a woman
now we attributed being a woman
logically valid thus deductively valid
the hidden premise is itself deductively necessary

it is deductively valid at first because we're assuming we know what widow means, dont see it as a hidden premise

every deductively valid argument can be turned into a logically valid argument by adding a premise that is deductively necessary

logic is risky business, do not touch students. -actual quote

the boundaries of logical validity

all $F$ are $G$, some $H$ are $F$, so some $H$ are $G$
$n$ is a widow, so $n$ is a woman

• ‘widow’, ‘woman’: not topic-neutral
• ‘all’, ‘some’: topic-neutral

is there a criterion for topic-neutrality?

if not:
- the notion of topic-neutrality has vague boundaries
- the notion of logical validity has vague boundaries

syncategorematic - categorematic

ann is taller than bob, bob is taller than cath, so ann is taller than cath
it is certainly deductively valid but, what is the argument form here? is it logically valid?
“taller than” is not topic-neutral

a is taller than b, b is taller than c, so a is taller than c. loud incorrect buzzer
now very flamboyantly:
$a$ is more $F$ than $b$, $b$ is more $F$ than $c$, so $a$ is more $F$ than $c$.
- this looks very topic-neutral!

some other instantiations:

2 is more even than 4, 4 is more even than 6, so 2 is more even than 6

problem :D

being tall/heavy comes in degrees
being even/prime is all or nothing

when instantiating the argument form, we can only fill in adjectives that are about stuff

which words count as topic-neutral and which ones dont idk

a final worry

deductive validiy, necessary truth preservation
logical validity, necessary truth preservation on purely logical grounds

the moon is made of cheese, so its raining or it is not raining

deductively valid
- there does not exist a situation in which its conclusion is false
logically valid
- $A$, so $B$ or not $B$
- only topic-neutral vocab

lorenz is a logician, he’s not a logician, so the moon is made of cheese

it looks weird to call these arguments valid. (relevance?)
answer 1:
- this is so bad that we have to start all over again
- we paid too much attention to the idea of truth-preservation
- we ignored the intuition that premises should be relevant for conclusion
- this can be done, but the math will be harder

answer 2:
- this is a bullet that we are willing to bite, arguments like above are limit cases that are irrelevant in daily life
- it is impossible to catch all of our pretheoretical ntuitions about validity in one encompassing, fully precise notion
- in this way we can also keep the math quite easy

three connectives

chapter 8, we're skipping 7

relinde is in the library or she is at home
relinde isnt in the library
so she is at home

$A$ or $B$
not $A$
so $B$

its not the case that jack attended the party and did well on the exam, jack attended the party, so jack did not do well on the exam

not $(AandB)$
$A$
so not $B$

‘and’ connects two expressions of the same grammatical category
lorenz is a logician and a father
lorenz is walking and speaking
lorenz talks quickly and loudly
lorenz is smart and self-centered

most importantly it connects two sentences
brussels is the capital of belgium and logic is interesting
lorenz is a logician and lorenz is a father

one problem:
lorenz and roger can carry the piano upstairs $\ne$ lorenz can carry the piano upstairs and roger can carry the piano upstairs

$A$ and $B$: the conjunction of $A$ and $B$
and is a binary connective
and is a propositional connective

truth conditions:

A	B	A and B
1	1	1
1	0	0
0	1	0
0	0	0

formalized as: $A\wedge B$

the symbol $\wedge$ can be used to formalize natural language ‘and’
but also to formalize other natural language words, e.g. ‘but’ (and with contrast)

to be continued monday