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In propositional logic, if $Q$ is a theorem, then so is $P\to Q$, for any choice of $P$. It helps to think of "$P\to Q$" as saying "knowing that $P$ is true is sufficient to know that $Q$ is true". It's not saying anything about whether $P$ "causes" $Q$, it's just describing the pairs of truth values that we consider possible. We think of each proposition as a fully-qualified sentence about the world, so no proposition is fundamentally "conditionally true" or true "because of" another proposition; each one is just either a true statement about the world or a false one.

First-order logic (propositional logic with variables and quantifiers) is a little more expressive: you can have $P(t)\to Q(t)$ be true for a particular term $t$ while the "for all" sentence $\forall x(P(x)\to Q(x))$ is false. In this case, you might think of the former sentence as an "accidental" implication and the latter as stating more of a "real" one. (But note that $\forall x(P(x)\to Q(x))$ is derivable from $\forall xQ(x)$, so it can still be "accidentally true".)

In propositional logic, if $Q$ is a theorem, then so is $P\to Q$, for any choice of $P$. It helps to think of "$P\to Q$" as saying "knowing that $P$ is true is sufficient to know that $Q$ is true". It's not saying anything about whether $P$ "causes" $Q$, it's just describing the pairs of truth values that we consider possible. We think of each proposition as a fully-qualified sentence about the world, so no proposition is fundamentally "conditionally true" or true "because of" another proposition; each one is just either a true statement about the world or a false one.

First-order logic (propositional logic with variables and quantifiers) is a little more expressive: you can have $P(t)\to Q(t)$ be true for a particular term $t$ while the "for all" sentence $\forall x(P(x)\to Q(x))$ is false. In this case, you might think of the former sentence as an "accidental" implication and the latter as stating more of a "real" one. (But note that $\forall x(P(x)\to Q(x))$ is derivable from $\forall xQ(x)$, so it can still be "accidentally true".)

You might also be interested in modal logic, which formalizes the notions of "necessary" and "possible" propositions.

In propositional logic, if $Q$ is a theorem, then so is $P\to Q$, for any choice of $P$. It helps to think of "$P\to Q$" as saying "knowing that $P$ is true is sufficient to know that $Q$ is true". It's not saying anything about whether $P$ "causes" $Q$, it's just describing the pairs of truth values that we consider possible. We think of each proposition as a fully-qualified sentence about the world, so there are no "sometimes true, sometimes false" propositions, or propositions that are true "because of" one another; each one is just either a true statement about the world or a false one.

On the other hand, in first-order logic (propositional logic with variables and quantifiers), it's possible to have $P(t)\to Q(t)$ be true for a particular term $t$ while the "for all" sentence $\forall x(P(x)\to Q(x))$ is false. In this case, you might think of the former sentence as an "accidental" implication and the latter as stating more of a "real" one. (But note that $\forall x(P(x)\to Q(x))$ is derivable from $\forall xQ(x)$, so it can still be "accidentally true".)

In propositional logic, if $Q$ is a theorem, then so is $P\to Q$, for any choice of $P$. It helps to think of "$P\to Q$" as saying "knowing that $P$ is true is sufficient to know that $Q$ is true". It's not saying anything about whether $P$ "causes" $Q$, it's just describing the pairs of truth values that we consider possible. We think of each proposition as a fully-qualified sentence about the world, so no proposition is fundamentally "conditionally true" or true "because of" another proposition; each one is just either a true statement about the world or a false one.

First-order logic (propositional logic with variables and quantifiers) is a little more expressive: you can have $P(t)\to Q(t)$ be true for a particular term $t$ while the "for all" sentence $\forall x(P(x)\to Q(x))$ is false. In this case, you might think of the former sentence as an "accidental" implication and the latter as stating more of a "real" one. (But note that $\forall x(P(x)\to Q(x))$ is derivable from $\forall xQ(x)$, so it can still be "accidentally true".)

In propositional logic, if $Q$ is a theorem, then so is $P\to Q$, for any choice of $P$. It helps to think of "$P\to Q$" as saying "knowing that $P$ is true is sufficient to know that $Q$ is true". It's not saying anything about whether $P$ "causes" $Q$, it's just describing the pairs of truth values that we consider possible. We think of each proposition as a fully-qualified sentence about the world, so there are no "sometimes true, sometimes false" propositions.

On the other hand, in first-order logic (propositional logic with variables and quantifiers), it's possible to have $P(t)\to Q(t)$ be true for a particular term $t$ while the "for all" sentence $\forall x(P(x)\to Q(x))$ is false. In this case, you might think of the former sentence as an "accidental" implication and the latter as stating more of a "real" one. (But note that $\forall x(P(x)\to Q(x))$ is derivable from $\forall xQ(x)$, so it can still be "accidentally true".)

In propositional logic, if $Q$ is a theorem, then so is $P\to Q$, for any choice of $P$. It helps to think of "$P\to Q$" as saying "knowing that $P$ is true is sufficient to know that $Q$ is true". It's not saying anything about whether $P$ "causes" $Q$, it's just describing the pairs of truth values that we consider possible. We think of each proposition as a fully-qualified sentence about the world, so there are no "sometimes true, sometimes false" propositions, or propositions that are true "because of" one another; each one is just either a true statement about the world or a false one.

On the other hand, in first-order logic (propositional logic with variables and quantifiers), it's possible to have $P(t)\to Q(t)$ be true for a particular term $t$ while the "for all" sentence $\forall x(P(x)\to Q(x))$ is false. In this case, you might think of the former sentence as an "accidental" implication and the latter as stating more of a "real" one. (But note that $\forall x(P(x)\to Q(x))$ is derivable from $\forall xQ(x)$, so it can still be "accidentally true".)