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I understand why with a truth table p ∧ (p → q) and p → q are different but from a semantic point they look exactly the same to me. At a glance from a semantics point they look like they should have the same truth table but they don’t. I understand formal logic abstracts from semantics and looks only at the externality truth values. However these value patterns should hold with semantic examples.

If it’s raining then I need an umbrella. = p → q

If it’s raining then I need an umbrella and it’s raining. = (p → q) ∧ p

To summarize these statements look exactly the same to me logically but logically they have completely different truth tables so logically they must be different. If truth tables are valid which I’m sure they are then they must be different but I’m having trouble understanding how they are. How are they different logically using semantics as an example.

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    A truth table is a semantic point of view. p-->q is false only when p is true and q is false. But (p-->q)^p is false whenever p is false. There's not much more that can be said. I mean, consider maybe an example with more gravity. Surely you'd agree that "If I'm a murderer, then I should go to jail" is very different than "If I'm a murder, then I should go to jail and I'm a murderer." Surely you wouldn't hire a lawyer who didn't see a logical difference between these two.
    – transitionsynthesis
    Commented Jul 23, 2020 at 1:39
  • I think you are getting caught up on using "if, then" for the material conditional. It is, supposedly, the best we have in English, but natural language use of "if,then" is often not consistent with the truth-functional behavior of the material conditional.
    – Rob
    Commented Jul 23, 2020 at 5:59

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p → q says nothing about whether p is true or not.   It is simply a promise that should p be true, then q will be too.

(p → q) ∧ p however, affirms that p is true, and therefore q is also true.   This is equivalent to q ∧ p a much stronger statement p → q.

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