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I have been trying to teach myself philosophical logic, but my mind is completely stuck. I cannot seem to understand the abstractions, which is unusual for me given that I have taken very advanced mathematics.

Taken from the book "A Short Introduction to Logic":

"Suppose we think of the relevant domain of objects as causes and effects, and write "x is caused by y" as xCy:

1). ∀x ∃y xCy 2).∃y ∀x xCy

Can someone rewrite this in plain English? I'm not sure what's happening here. Where do I plug in words like "and," "or," "true," "if," etc.? What does this sentence mean step by step?

I know that it is very simple, but I'm lost without the other logical symbols.

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1). ∀x ∃y xCy 

Literally: for every x, there exists some y such that x is caused by y. This means that every object x has at least one object y that causes it.

2).∃y ∀x xCy

Literally: there exists some y such that every x is caused by y. This means that there is some object y that is the cause of every object x.

I'm not sure what your text is asking for here. Does it want you to explain these phrases in natural language? Does it want you to evaluate them? Does it want you to combine them somehow, or prove a result? Evaluation would lead you to true/false conditions; combining them would involve 'and,' 'or,' and other logical connectors; trying to prove some result from these premises would likely involve both. What does the text want?

  • Whoops, misread it. easy enough to fix, though... – Ted Wrigley Jul 14 at 19:37
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This is a formal description of causality.

The text specifies a "domain of discourse" or Universe (U).

U = {The set of causes and effects}

This isn't explicitly written but then the universe is basically divided into two sets of objects.

X = {the set of effects}
Y = {the set of causes}

Note: the difference in case is important.

x = an arbitrary element of X
y = an arbitrary element of Y

Then it defines a relationship between those sets.

C ≔ "is caused by"

So in plain English, the statements mean:

1. For all x, there exists a y, such that x is caused by y.
2. There exists a y, such that for all x, x is caused by y.

Upon drawing them out, you'd realize that they have the same meaning.

Conditionals or Boolean operators not required.

  • Actually the second part says "there is some y that is caused by every x, – o.m. Jul 15 at 5:39

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