Is logical implication always determinable from just the given statements?

Question

I am reading introductory texts on logic and am having a hard time understanding intuitively logical implication.

Specifically, I am wondering if logical implication can always be determined by two statements, P and Q alone? I'll try to give an example to clarify my question.

For example if the statement P is $x = 2$, and the statement Q is $x^2 < 6$, then determining the truth of P $\Rightarrow$ Q is fairly straightforward:

If $x$ does equal 2, then $4<6$ and Q is true, so $P \Rightarrow Q$ is true. And if $x$ does not equal 2, $P \Rightarrow Q$ is vacuously true.

In this example, with two statements P and Q, I can evaluate the truth/falsity of the statement P implies Q. However, what if the statement Q is changed such that it is now the statement $y = 5$?

So now I am trying to prove from that $$x = 2 \Rightarrow y = 5$$

Here is where things get murky for me. Say $x$ is not equal to 2, then $P \Rightarrow Q$ is vacuously true. But what about when $x$ does equal 2, now the statement Q could be true, but it could also be false, depending on some 'context'.

For example, if I said at the start: "Here are two statements, P and Q. Know that x = 2 and y = 4." Given this 'context', $P \Rightarrow Q$ is false. But if I say "Here are two statements, P and Q. Know that x = 2 and y = 5", then now $P \Rightarrow Q$ is true.

My discomfort here is that I assumed with two statements P and Q, I can determine the truth/falsity of $P \Rightarrow Q$ with just this information. Is this the wrong assumption?

As a secondary question, how can the truthfulness of the statement $P \Rightarrow Q$ be variable, depending on context? I have read on some other questions that implication can be interpreted as a promise. So how could it be that depending on some context the promise is sometimes true but sometimes false? Wouldn't this actually mean the promise is not always consistent and is therefore always false?

Some further information in case it is relevant, I recognise that in the first example, the statement Q is 'dependent' in some way on the statement P, since the object under examination in both statements is $x$, whereas in the second example there are two different objects which I am trying to make a logical connection between, which I am thinking may not be valid for some reason. I also recognise that there is still 'context' in the first example, in that we need information on whether $x$ is equal to 2 or not, however in that case it doesn't matter, since $P \Rightarrow Q$ is always true anyway.

Thanks very much in advance.

Answer 1

"Logical implication" is a potentially misleading term; it may mean the propositional connective often called Conditional.

In this case : YES, having two statements $P,Q$ we can always produce the "complex" statement $P → Q$, that reads :

"if $P$, then $Q$".

A different (but related) case is when we use "implies" to mean Logical consequence :

a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements.

In this case we use the symbol : $Γ \vDash \varphi$, that reads : "statement $\varphi$ logically follows from the set $Γ$ of statements".

Statements are $2=2$ (which is True) and $2=3$ (which is False). To evaluate the truth value of a "complex" statement (like $P → Q$) we have to start from statements having a precise truth value.

$x=2$ is not a statement : it is a formula with a variable and its truth value depends on the value assigned to variable $x$.

A different case is when we have quantifiers, like e.g. $∀x(x=2 → x>1)$. In this case there are no more free variables and the formula is a statement : if we read it as a formula about natural numbers, it has a precise truth value : it is True.

Regarding your examples, we have that $\forall x (x=2 \to x^2 < 6)$ is always True (as you say) when red as an arithmetical statement, while $\forall x \forall y (x=2 \to y=5)$ is not.

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how can the truthfulness of the statement $P \to Q$ be variable, depending on context?

$P \to Q$ is a formula of propositional calculus.

Formulas of propositional calculus are Truth functions meaning that :

a compound statement is constructed by one or two statements connected by a logical connective; if the truth value of the compound statement is determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and the logical connective is said to be truth functional.

This means exactly that, in order to evaluate the truthfulness of the statement $P \to Q$, we have to specify a "context", i.e. a truth assignment, that is a function that maps propositional variables to True or False.

In this way, given a "context" (a truth assignment), then YES : the truth value of a (truth-functional) compound statement, like e.g. the conditional $P \to Q$, is always determinable from the given statements $P$ and $Q$.