Extension of a sentence is its truth-value, and the intension of a word is the part of a word that goes beyond the extension, ie. Non-truth functional. This is basically the same as Frege’s referens and sense.
How to show that a connective is not truth-functional:
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Find an α and a β which by themselves both have a truth-value of 1 on their own.
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Make or use a given connective in front of the two variables (remember that the connective has to create a wff).
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See whether each point of a truth-table has its own unique truth-value.
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If not, the connective in question is not truth-functional.
Tautology, always true in all cases
Contradiction, always false in all cases
In general there are 2(2^n) different truth-functional n-ary connectives.
We can visualise truth-functions in a geometrical fashion:
eg. the unary truth functions:
We can view them as coordinates on a cartesian plane
Thus, the unary connectives would make a square.
The most extreme inputs, ie. Those that are constant in the sense that they are only made up of 1s, or 0s, are sometimes excluded due to being vacuos and trivial.
So rather than the unary connectives forming a square, they actually form a line, if we view it non-trivially.
This is useful eg. for the binary connectives as they form a hyper-cube, but if we remove the trivial functions, we can smash the hyper-cube into a rhombic dodecahedron, and thus map it in a way that makes sense for 3-Dimensional space.
One hundred prisoners and a light bulb
Consider wff -(p^-q)
p q -( p ^ - q)
1 0 1 1 1 0
1 1 0 1 10 1
0 0 1 0 1 0
0 1 1 0 0 1
The number of rows in a truth table is a power of 2. eg. 23 in the case that we have 3 atomic propositions.
Expressive adequacy.
-a^-b ≈ -(avb)
≈ = equivalence
They are semantically the same, however, syntactically different.
They mean the same thing, but are said in different ways.
It means more or less that they have the same truth-tables
-a^-b ≈ -(avb)
-(a^b) ≈ -av-b
avb ≈ -(-a^-b)
a^b ≈ -(-av-b)
This is called either duality laws (esp. 3 and 4), or De Morgan laws (esp. 1 and 2).
They were however at least as old as William of Ockham and not necessarily found by De Morgan.
We are calling it De Morgan laws.