The idea that it is impossible for an argument to be valid if the premises are true and the conclusion is false is central to all logic and mathematics. So if a sequence takes us from truth to falsehood, we can suspect an error in reasoning. All other combinations are perfectly acceptable, including from falsehood to truth, from falsehood to falsehood, and from truth to truth, of course.
The rules of deduction place three restrictions on us. The first is easy to state and quick to dispatch: an assumption must not be extended to all items of a certain kind until it is discharged. Suppose we assume that a car picked at random is of the colour brown. At this stage we can’t say that every car is brown – this would be making an unjustified leap in our reasoning. However, once we get to a stage, through valid reasoning, where we can say: ‘If a car picked at random is brown, then it is made in the US,’ by which we have discharged our assumption, we can say that ‘All cars which are brown are made in the US.’
The second restriction says that if, in the course of our reasoning, we introduce an individual specified by name next to an individual which we have earlier picked at random (strictly in this order), then, likewise, we can’t extend the relation between the two to a relation between all individuals and the one we have specified by name. Thus, by proceeding from ‘Every car in the car park belongs to someone,’ (true) through ‘A car in the car park picked at random belongs to Jamie,’ (possibly true), we come to ‘All cars in the car park belong to Jamie,’ (possibly false) and eventually ‘Someone is the owner of all cars in the car park,’ (possibly false), we have made an error in reasoning. We’ve gone from a true premise to a false conclusion.
The last restriction is hardest to state in plain English, but it is quite straightforward. Suppose that, figuratively speaking, we put into the same box in our head the things that we pick at random and that bear a relationship to one another. We can’t then, in the course of our reasoning, break the bond that connects them and put one thing into one box and the other into another box. Thus, from ‘Every car is identical to itself,’ (true) we can go to ‘A car picked at random is identical to a car picked at random’ (possibly true), and back to ‘Every car is identical to itself,’ but not to ‘Every car is identical to every car,’ as the latter is definitely false.
We can of course rehearse the last argument in a language approaching real life situations, as done earlier, for example, ‘Every driver admires himself,’ where, again, it would be incorrect to conclude that ‘Every driver admires every driver (that is, every other driver and himself too), but with such sentences, of course, we can’t easily demonstrate the truthfulness of the premise(s) or the falsity of the conclusion.
I will run through this again using mathematical examples another time.
The rules of deduction place three restrictions on us. The first is easy to state and quick to dispatch: an assumption must not be extended to all items of a certain kind until it is discharged. Suppose we assume that a car picked at random is of the colour brown. At this stage we can’t say that every car is brown – this would be making an unjustified leap in our reasoning. However, once we get to a stage, through valid reasoning, where we can say: ‘If a car picked at random is brown, then it is made in the US,’ by which we have discharged our assumption, we can say that ‘All cars which are brown are made in the US.’
The second restriction says that if, in the course of our reasoning, we introduce an individual specified by name next to an individual which we have earlier picked at random (strictly in this order), then, likewise, we can’t extend the relation between the two to a relation between all individuals and the one we have specified by name. Thus, by proceeding from ‘Every car in the car park belongs to someone,’ (true) through ‘A car in the car park picked at random belongs to Jamie,’ (possibly true), we come to ‘All cars in the car park belong to Jamie,’ (possibly false) and eventually ‘Someone is the owner of all cars in the car park,’ (possibly false), we have made an error in reasoning. We’ve gone from a true premise to a false conclusion.
The last restriction is hardest to state in plain English, but it is quite straightforward. Suppose that, figuratively speaking, we put into the same box in our head the things that we pick at random and that bear a relationship to one another. We can’t then, in the course of our reasoning, break the bond that connects them and put one thing into one box and the other into another box. Thus, from ‘Every car is identical to itself,’ (true) we can go to ‘A car picked at random is identical to a car picked at random’ (possibly true), and back to ‘Every car is identical to itself,’ but not to ‘Every car is identical to every car,’ as the latter is definitely false.
We can of course rehearse the last argument in a language approaching real life situations, as done earlier, for example, ‘Every driver admires himself,’ where, again, it would be incorrect to conclude that ‘Every driver admires every driver (that is, every other driver and himself too), but with such sentences, of course, we can’t easily demonstrate the truthfulness of the premise(s) or the falsity of the conclusion.
I will run through this again using mathematical examples another time.
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