”Logic gets quite repetetive, it’s a way of saying the same things over and over again.”
Consistency and inconsistency
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A number of propositions is jointly consistent if and only if these propositions can be true together.
→ A number of propositions is jointly consistent if and only if if there exists a situation in which all these propositions are true
eg. {all humans are mammal, Lorenz is a crocodile, Leuven is the capital of Belgium}
Inconsistency: the negation of consistency.
{all humans are mammal, Lorenz is a human, Lorenz is not a human}
Equivalence:
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Two propositions are equivalent to each other if and only if they are true in exactly the same possible situations.
Eg. Pete Kisses Mary = Mary is kissed by Pete
Not equivalent:
Lorenz is a human =/ Lorenz is a mammal
→ the propositions A and B are equivalent to each other if and only if the argument ’A therefore B’ and ’B, therefore A’ are both Valid.
The above examples formulates two possible valid arguments, equivalent to the schema drawn above:
Pete kisses Mary, so Mary is kissed by Pete
&
Mary is kissed by Pete, so Pete kisses Mary.
5 = 3 + 2
5 ≤ 3 + 2
3 + 2 ≤ 5
What doesn’t exist:
A valid argument with true premises and a false conclusion.
In other words:
If an argument has true premises and a false conclusion, then that argument is invalid.
Eg. Brussel is the capital of Belgium, so Paris is the capital of France.
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true premise, true conclusion – invalid argument
Validity is a property of arguments!!!
Truth is a property of propositions!!!
The relationship between soundness and consistency.
A therefore A (a fallacy)
But a very valid argument
An argument is sound if and only if it is valid and it has true premises
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Ie. You do need encyclopedic knowledge
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Every sound argument has a true conclusion!
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If I have two arguments which are sound, the conclusions of those arguments will never be inconsistent with each other.
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No sound argument has inconsistent premises
→ What if we replace ’sound’ with ’valid’?
Basic principles thus far: (1). Truth, (2). possible situation
And from this all following notions can be derived:
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Possibility
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Necessity
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Validity
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Consisitency
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Equivalence
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Soundness
Forms of Inference
No F is G, n is F, so n is not G.
F, n and G are formal variables
Big letters are nouns, and n is a proper name.
All F are G, n is F, so n is G.
No F is G, n is F, so n is not G
no F is G, so no G is F
All F are H, no G is H, so no F is G
All F are either G or H
All G are K
All H are K
So all F are K
A valid argument can be an instance of an invalid argument.
A concrete argument exemplifies many argument forms
there is no unique argument form that is exemplified by the argument.