Inclusion and exclusion are critical for reasoning, and these two words are supposed to make it easier to factor things in or out. It is not always so. The prepositions ‘besides’ and ‘except’ present both grammatical and logical challenges.
‘Besides’ adds, ‘except’ subtracts:
Everyone besides the vicar gambles.
Everyone except the vicar gambles.
This distinction seems to break down in negative E-type sentences where both words exclude:
No one besides / except the vicar gambles.
Various other combinations of ‘besides’ and ‘except’ with A, E, I, and O-type sentences produce more or less felicitous outcomes, including anomalies such as:
Someone except for the vicar gambles.
Some people except for the vicar gamble.
Logically speaking, that is when we take logic to be common sense, the first sentence is odd. The second sentence holds up, especially when we use intonation to dramatic effect (successful replacement candidates for ‘except’ would be ‘but not’ or ‘other than’). On the strict understanding of ‘some’ in logic though, both sentences are OK. Each says: there is at least one person who gambles and that person is not the vicar.
If we give the vicar a name, for example: Some people except for Michael gamble, then the translation into FOL comes out as:
(∃x)[¬(x = m) • Gx] • ¬ Gm
This says: Some people other than Michael gamble, and Michael doesn’t gamble. We need the part: ¬ Gm, to show that ‘m’ has the subject property but not the predicate property (we could have fleshed out our symbolization by saying that ‘x’ is a person, or ‘Px’ if we wanted to – it would have set off the meaning but crowded the formula).
We ought to be able to show something similar for:
Some people except for the vicar gamble.
The difference this time is that we have to accommodate a definite description. It would be very awkward to maneuver a definite description into our sentence so that ‘some people’ would have scope over it. My suggestion is: Some people who are not a vicar gamble, and the vicar doesn’t gamble. Again, breaking it down this way shows that ‘x’ has the subject property but does not have the predicate property. The truth value is also preserved. Net loss: we are saying that some people who are not a or any vicar, rather than the vicar from our example, gamble.
(∃x)( ¬Vx • Gx) • (∃x)[Vx • (y)(Vy ⊃y = x) • ¬ Gy]
The corresponding sentence for inclusion: Some people besides the vicar gamble, would then be:
(∃x)( ¬Vx • Gx) • (∃x)[Vx • (y)(Vy ⊃y = x) • Gy]
‘Besides’ adds, ‘except’ subtracts:
Everyone besides the vicar gambles.
Everyone except the vicar gambles.
This distinction seems to break down in negative E-type sentences where both words exclude:
No one besides / except the vicar gambles.
Various other combinations of ‘besides’ and ‘except’ with A, E, I, and O-type sentences produce more or less felicitous outcomes, including anomalies such as:
Someone except for the vicar gambles.
Some people except for the vicar gamble.
Logically speaking, that is when we take logic to be common sense, the first sentence is odd. The second sentence holds up, especially when we use intonation to dramatic effect (successful replacement candidates for ‘except’ would be ‘but not’ or ‘other than’). On the strict understanding of ‘some’ in logic though, both sentences are OK. Each says: there is at least one person who gambles and that person is not the vicar.
If we give the vicar a name, for example: Some people except for Michael gamble, then the translation into FOL comes out as:
(∃x)[¬(x = m) • Gx] • ¬ Gm
This says: Some people other than Michael gamble, and Michael doesn’t gamble. We need the part: ¬ Gm, to show that ‘m’ has the subject property but not the predicate property (we could have fleshed out our symbolization by saying that ‘x’ is a person, or ‘Px’ if we wanted to – it would have set off the meaning but crowded the formula).
We ought to be able to show something similar for:
Some people except for the vicar gamble.
The difference this time is that we have to accommodate a definite description. It would be very awkward to maneuver a definite description into our sentence so that ‘some people’ would have scope over it. My suggestion is: Some people who are not a vicar gamble, and the vicar doesn’t gamble. Again, breaking it down this way shows that ‘x’ has the subject property but does not have the predicate property. The truth value is also preserved. Net loss: we are saying that some people who are not a or any vicar, rather than the vicar from our example, gamble.
(∃x)( ¬Vx • Gx) • (∃x)[Vx • (y)(Vy ⊃y = x) • ¬ Gy]
The corresponding sentence for inclusion: Some people besides the vicar gamble, would then be:
(∃x)( ¬Vx • Gx) • (∃x)[Vx • (y)(Vy ⊃y = x) • Gy]
A few more examples:
Everyone besides the vicar gambles.
(x)(¬ Vx ⊃Gx) • (∃x)[Vx • (y)(Vy ⊃y = x) • Gy]
Everyone except the vicar gambles.
(x)(¬ Vx ⊃Gx) • (∃x)[Vx • (y)(Vy ⊃y = x) • ¬ Gy]
No one besides / except the vicar gambles.
(∃x){Gx • (y)(Gy ⊃ y = x) • (∃z)[Vz • (w)(Vw ⊃ w = z) • z = x]}
The last statement is of course equivalent to: The only person who gambles is the vicar.
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