2. Use a tree to test whether the following two propositions are equivalent. (Present the tree, and say whether they’re equivalent or not.) If they’re not equivalent, read off from your tree a model on which the propositions have different truth values (and indicate the path from which you are reading it off). ∀x(Ax → Gx) ∃x(Ax → Gx)
Question
- Use a tree to test whether the following two propositions are equivalent. (Present the tree, and say whether they’re equivalent or not.) If they’re not equivalent, read off from your tree a model on which the propositions have different truth values (and indicate the path from which you are reading it off). ∀x(Ax → Gx) ∃x(Ax → Gx)
Solution
The two propositions are not equivalent. Here's why:
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Let's start with the first proposition, ∀x(Ax → Gx). This is a universal quantifier, meaning that for all x, if Ax is true then Gx is also true.
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Now let's look at the second proposition, ∃x(Ax → Gx). This is an existential quantifier, meaning that there exists at least one x such that if Ax is true then Gx is also true.
The difference between the two is that the first proposition requires the implication to be true for all x, while the second only requires it to be true for at least one x.
To illustrate this with a tree:
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For the first proposition, we would have a root node labeled ∀x(Ax → Gx), and every branch would have a node labeled (Ax → Gx).
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For the second proposition, we would have a root node labeled ∃x(Ax → Gx), and at least one branch would have a node labeled (Ax → Gx).
A model in which the propositions have different truth values could be one where there are multiple instances of x, and at least one of them makes Ax true but Gx false. This would make the first proposition false (because not all x satisfy the implication), but the second proposition would still be true (because there exists at least one x that satisfies the implication). This path can be read off from the branch of the tree where Ax is true and Gx is false.
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