The temptation to treat the operator because as truth functional is a good enough reason to remind ourselves periodically why it is not. The fact that because is not truth-functional follows directly from the definition of a function.
A function is a relation which assigns a member of the domain x one and only one member of the range f(x). A function may take more than one input x to produce the same output, but this much is certain: for each input there is only one output. How does that bear on because?
For comparison, let us start with the conjunction and, as in: p • q. We let p be ‘It is cloudy’ and q ‘it is raining’. The outcome: 'It is cloudy and it is raining', is true iff p is true and q is true. Our output sentence is true irrespective of whether there is a connection between clouds and rain. Thus, two truths produce a truth if the connective is and. If we let p be ‘An isosceles triangle has two equal sides’, which is true, and q ‘A prime number is a whole number larger than 1 that is divisible only by 1 and itself’, which is also true, our reason for uttering: 'An isosceles triangle has two equal sides and a prime number is a whole number larger than 1 that is divisible only by 1 and itself’ may not be immediately clear, but there is nothing illegitimate about the sentence. In both cases, the input is two T’s (two true sentences), and in both cases the final sentence comes out true.
It is easy to see that this is not the case with because. Let p ‘The apple dropped down’ be true, and q ‘the apple was overripe’ be true too. Our output sentence is: ‘The apple dropped down because it was overripe,’ is true, and we attribute the truth to the connection between the apple falling and its being overripe. Two true sentences produce one true one. But what about the next pair of sentences above? To say that ‘An isosceles triangle has two equal sides because a prime number is a whole number larger than 1 that is divisible only by 1 and itself’ is definitely not true. One thing has nothing to do with the other. Thus, two T’s (two true sentences) do not always produce a true output sentence when connected via because.
This violates our definition of a function, which assigns one and only one member of the range to each member of the domain. The examples above show that because assigns once T (true) once F (false) to the same value of the input. The truth value of the compound is not completely determined by the truth values of the component sentences alone. We look to our knowledge of the external world for the answer to whether the compound sentence is true or not.
The operator because is variously handled in first order logic by and or if, and the sentence it introduces is absorbed into a subordinate clause.
A function is a relation which assigns a member of the domain x one and only one member of the range f(x). A function may take more than one input x to produce the same output, but this much is certain: for each input there is only one output. How does that bear on because?
For comparison, let us start with the conjunction and, as in: p • q. We let p be ‘It is cloudy’ and q ‘it is raining’. The outcome: 'It is cloudy and it is raining', is true iff p is true and q is true. Our output sentence is true irrespective of whether there is a connection between clouds and rain. Thus, two truths produce a truth if the connective is and. If we let p be ‘An isosceles triangle has two equal sides’, which is true, and q ‘A prime number is a whole number larger than 1 that is divisible only by 1 and itself’, which is also true, our reason for uttering: 'An isosceles triangle has two equal sides and a prime number is a whole number larger than 1 that is divisible only by 1 and itself’ may not be immediately clear, but there is nothing illegitimate about the sentence. In both cases, the input is two T’s (two true sentences), and in both cases the final sentence comes out true.
It is easy to see that this is not the case with because. Let p ‘The apple dropped down’ be true, and q ‘the apple was overripe’ be true too. Our output sentence is: ‘The apple dropped down because it was overripe,’ is true, and we attribute the truth to the connection between the apple falling and its being overripe. Two true sentences produce one true one. But what about the next pair of sentences above? To say that ‘An isosceles triangle has two equal sides because a prime number is a whole number larger than 1 that is divisible only by 1 and itself’ is definitely not true. One thing has nothing to do with the other. Thus, two T’s (two true sentences) do not always produce a true output sentence when connected via because.
This violates our definition of a function, which assigns one and only one member of the range to each member of the domain. The examples above show that because assigns once T (true) once F (false) to the same value of the input. The truth value of the compound is not completely determined by the truth values of the component sentences alone. We look to our knowledge of the external world for the answer to whether the compound sentence is true or not.
The operator because is variously handled in first order logic by and or if, and the sentence it introduces is absorbed into a subordinate clause.
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