Logic 28-10-24
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who drinks, sleeps.
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who sleeps, does not sin
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who does not sin is holy
therefore, who drinks is holy.
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if A then B
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if B then C
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if C then D
therefore, if A then D
A = Drinking
B = Sleeping
C = not Sinning
D = being Holy
C could potentially mean two things, there’s a scope ambiguity.
C1 not sinning
C2 not ever sinning
If we insantiate C1, it seems improbable that me not sinning right now can implicate that I am holy. However in C2 it seems improbable that ”sleeping now” can imply that I will not ever sin. Does none of the scopes of C can be true, and thus the argument is not sound, however, still valid.
World Knowledge, encyclopedic knowledge. But world knowledge also means that you know how to live in the world and that’s kind of a necessity for making sense of things logically.
We would however want to utilise the principle of charity in analysing this kind of argument. Thus when resolving scope ambiguities we should choose the ones which make the argument as sound as possible, even though they might still sometimes not be sound, nor valid.
Syntax, the study of language as a closed system, lacking meaning and relations to the world.
Semantics, the study of language as it relates to the world, and focuses on the actual meanings that are conveyed by language.
When logicians talk of neuroscience it is always wrong.
The area of the brain which is used to deal with syntax is called Broca’s area. The area that allows us to deal with semantics is Wernicke’s area.
Thus one can be destroyed without the other one being completely destroyed or damaged.
If eg. the Broca’s area is destroyed or damaged, you will likely receive Broca’s aphasia.
Likewise with the Wernicke’s area.
In PL the syntax is going to be okay if and only if it is semantically okay.
”The colorless green ideas sleep furiously”
Still synatictically okay.
Four letters, or more.: P Q R S
The prime symbol: ’
Three connectives, ^,v, -
brackets to remove scope ambiguities: ()
inference marker: three dots
comma: ,
We use Greek letters, but these are not part of the alphabet.
They are instead variables to talk about a language, they are a meta language.
Every PL language has some (finite) number of atomic propositions.
If we need more letters than PQRS we use the ’-symbol to denote more variables.
Each atomic proposition consis of one of the four letters followed by zero or more primes.
Well-formed formulas
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Suppose that we have specified a finite number of atomic propositions
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The well-formed formulas (wff) are defined as follows:
All atomic propositions are well-formed
if α is a wff and β is wff then α^β is also a wff
if α is a wff and β is a wff, then αvβ is also a wff
if α is a wff, then - α is also a wff
Suppose we work with PQR
let’s show in full detail that ’-(pvq)^-(-QvR)) is a wff
PQR are atomic propositions
Therefore they are all wffs
Given that, ’(PVQ)’ is a wff (from 4,5,W3)
’-Q’ is a wff (from 5,W4)
’(-QVR)’ is a wff
thus ’-(pvq)^-(-QvR))’ is a wff
This method can be done for every wff
maybe a bit too much detail (and redundancy)
A more compact representation: (add picture later)
Every Wff has a unique parse tree
but eg. PVQVR is not a wff because you cannot make a proper parse tree out of it, because it will not become unique.
Suppose that α is a molecular wff
the main connective of α is the first connective that is removed if you read the parse tree for α from top to bottom
equivalentely: it’s the last connective that is added, if you read the parse tree for α from the bottom to the top
β is a subformula of a iff β occurs somewhere on the parse tree of a
note so a is also a subformula of itself (sometimes in limit cases)
example: what are all the subformulas of ((PVQ) ^ R)?
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(PVR)
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R
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P
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Q
All subformulas consist of every part that can be on a parse tree.
The scope of an occurence of a connective in a wff α is the lowermost subforumula in the parse tree of α that contains that connective.
What the scope of some connective thus becomes a question that we can feasibly answer.
Different occurrences of a single connective have different scopes.
eg. ((P^Q)^(R^P))
the scope of the first ^ is (P^Q), the scope of the second ((P^Q)^(R^P)), and the scope of the third is (R^P).
for any wff α, the scope of the main connective of α is the highest formula in a parse tree.
Thus we can also understand what a main connective really is in practice.
α is a schematic variable over wffs (reminder).