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.

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    $\begingroup$ Your mistake is assuming that all logical formulas have a truth value. The formula $x=2$, for example, is a perfectly well-formed logical formula, but it does not have a truth value. In fact, any formula with a free variable does not have a truth value. To bind the variables, you need to introduce quantifiers; then you get not just a formula but a statement. The formula $x=2$ is neither true nor false, but the statement $(\forall x)(x=2)$ is false. $\endgroup$ – symplectomorphic Sep 3 '19 at 8:39
  • $\begingroup$ Does this mean that the initial example is incorrect? I.e. x=2 --> x^2 > 6 is not a well formed statement? $\endgroup$ – masiewpao Sep 3 '19 at 10:26
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    $\begingroup$ It’s a well-formed formula, but it’s not a statement. $\endgroup$ – symplectomorphic Sep 3 '19 at 14:14
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    $\begingroup$ It's probably worth being clear that the terminology here varies with author. While I think most authors would agree on "formula" (in a generic sense), for closed formulas, i.e. ones with no free variables, authors say "sentence", "statement", "closed formula", "proposition", and no doubt others. Meanwhile, some of these terms, such as "statement" and "proposition" may be used as synonyms for arbitrary formulas. There are other distinctions the terminology may be indicating as well which can also vary with author. $\endgroup$ – Derek Elkins left SE Sep 4 '19 at 2:48

"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.

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 $Q$.

  • $\begingroup$ Hi thank you very much! I just have a small clarification to ask, which is whether x=2 -> x^2 > 6 an imprecise statement? Would it have to have context, for example x = 2 where x is 2, implies x^2 > 6 where x is 2? $\endgroup$ – masiewpao Sep 3 '19 at 10:07
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    $\begingroup$ @masiewpao - "imprecise" is not the correct one; it is a "propositional function" i.e. a lingustic entity depending on a parameter $x$. When we define a "context" (through an interpretation and the assignment of a value to the variable $x$) the expression becomes a statment with a definite meaning and truth value. If we assume the arithmetical interpretation and we let $x=2$, the result will be the statement $(2=2 \to 2^2 > 6)$, which is FALSE. $\endgroup$ – Mauro ALLEGRANZA Sep 3 '19 at 11:08
  • $\begingroup$ Thank you so much, that's made it very clear! $\endgroup$ – masiewpao Sep 3 '19 at 13:45

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