An argument in the form as discussed in the last section is only one example of a valid syllogism. In fact, there are 15 valid forms of arguments that have categorical predicates, negation, and two quantifiers (``all'' and ``some''). The corpus used in this section consists of these 15 forms of arguments and is listed below.
all m are p and all s are m -> all s are p
all p are m and some s are not m -> some s are not p
some m are not p and all m are s -> some s are not p
all p are m and no m are s -> no s are p
all p are m and no s are m -> no s are p
no m are p and all s are m -> no s are p
no p are m and all s are m -> no s are p
all m are p and some s are m -> some s are p
all m are p and some m are s -> some s are p
some m are p and all m are s -> some s are p
some p are m and all m are s -> some s are p
no m are p and some s are m -> some s are not p
no p are m and some s are m -> some s are not p
no p are m and some m are s -> some s are not p
no m are p and some m are s -> some s are not p
If we use this corpus to train the aforementioned quantum architecture, it has no difficulty in learning all these sentences. The absolute squares and phases are shown in Figure 7.8 and 7.9 respectively. In Figure 7.8, the absolute square of each component is represented by the area of its corresponding black square. In Figure 7.9, the phase of each component is represented by the angle of the line in the circle (as the hand of a clock). The upper rows are the target sentences and the lower the actual outputs of the system.
As all students learning elementary logic know, these forms of arguments, especially when they are thus abstracted, are difficult to follow without resorting to sophisticated reasoning (with the help of Venn diagrams, for example). The ability of the quantum computational framework to learn all these arguments is remarkable and therefore very encouraging.
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