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Chapter 2 — Truth functions—or not? - Coggle Diagram
Chapter 2 — Truth functions—or not?
Whether or not the rules of validity are hard-wired into us, we all have pretty strong intuitions about the validity or otherwise of various inferences.
:check:
She's a woman and a banker; so she's a banker.
:red_cross:
He's a carpenter; so he's a carpenter and plays baseball.
But our intuitions can get us into trouble sometimes.
The Queen is rich.
The Queen isn't rich.
:arrow_right:
Pigs can fly.
It certainly doesn't seem valid. The wealth of the Queen—great or not—would seem to have no bearing on the aviatory abilities of pigs.
But what do you think of the following two inferences?
Either the Queen is rich or pigs can fly.
The Queen isn't rich.
:arrow_right: Pigs can fly.
The second inference also seems valid. If one or other of two claims is true and of these isn't, the other must be.
The Queen is rich.
:arrow_right:
Either the Queen is rich or pigs can fly.
The first of these seems valid. Consider its conclusion. Logicians call sentences like this a
disjunction
; and the clauses on either side of the 'or' are called
disjuncts
.
Now, what does it take for a disjunction to be true? Just that one or other of the disjuncts is true. So in any situation where the premiss is true, so is the conclusion.
Now, the trouble is that by putting these two apparently valid inferences together, we get the apparently invalid inference, like this:
The Queen is rich.
Either the Queen is rich or pigs can fly. The Queen isn't rich.
:arrow_right:
Pigs can fly.
This can't be right. Chaining valid inferences together in this way can't give you an invalid inference. If all the premisses are true in any situation, then so are their conclusions, the conclusions that follow from these; and so on, till we reach the final conclusion. What has gone wrong?
To give an orthodox answer to this question, let us focus a bit more on the details. For a start, let's write the sentence
'Pigs can fly'
as
p
, and the sentence '
The Queen is rich
as
q
.
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