How is the definition of material implication derived from the definition of valid argument?

Question

According to my book

An argument is valid if and only if it's impossible that the premises are true and the conclusion false.

So if I have the argument $\{p_{1}∧p_{2}∧...∧p_{t}\}⇒q$ and I want to prove that it's valid, I suppose that all premises are true and the conclusion is false, and if I get to a contradiction, then I have proven that it's impossible for the premises to be true and the conclusion false, hence making it a valid argument.

What I want to understand is how the definition of the truth table of the material implication "models"/"agrees"/"is derived", etc from this definition.

When I want to prove if $\{p_{1}∧p_{2}∧...∧p_{t}\}⇒q$ is valid, I assume that all the premises are true and the conclusion is false, right? like this : $\{V_{1}∧V_{2}∧...∧V_{t}\}⇒F$, which makes the conditional false (according to the third line of the pic of the truth table).

Again the definition of a valid argument is

An argument is valid if and only if it's impossible that the premises are true and the conclusion false.

It doesn't say anything about arguments with true premises and true conclusions, or arguments with false premises and true conclusions, or arguments with false premises and false conclusions. I assume that this are valid arguments. Coincidentally these are also the ones which are true in the truth table.

The definition only refers to the case in which the premises are true and the conclusion false, as an invalid argument. Coincidentally the only one which is false in the truth table.

Now, my teacher says that an argument is neither true nor false, but valid or invalid, while the ones that are either true or false are propositions. However, it seems to me, from what I have said, that there is a certain relation between the falsehood or veracity of a material conditional, and the validity of the argument it models; however this doesn't completely "click" in my brain, and I don't know if there are any mistakes in my reasoning.

I also want to know how this relates with the definition of a valid argument in propositional logic.

$\{p_{1}∧p_{2}∧...∧p_{t}\}⇒q$ is valid if and only if $\{p_{1}∧p_{2}∧...∧p_{t}\}⇒q$ is a tautology.

I'm pretty sure it follows from all the aforementioned, but it still doesn't completely "click" in my brain.

Answer 1

First off, if we have an argument with multiple premises, then we don't have a conjunction assumed. If we had a conjunction assumed, we would have only one premise instead of many premises. So, instead of having {p1∧p2∧...∧pt}⇒q, we have {p1,p2,...,pt}⇒q, where each pn is a member of the set of premises. Additionally, some propositional calculi don't even have a conjunction connective.

Suppose that it is impossible that all of the premises are true, and the conclusion false. Suppose also that there exists but one premise p, and a conclusion q. By the definition you've given, the argument p⇒q is valid. There's a meta-theorem of propositional logic, called the deduction theorem, which says that if p⇒q, then ⇒(p$\rightarrow$q), where $\rightarrow$ corresponds to your horseshoe symbol. ⇒(p$\rightarrow$q) is another way of saying that (p$\rightarrow$q) is true. The only case excluded was when p was true and q false. So for any valid argument with premise p and conclusion q, it follows that if p is false, and q is false, (p$\rightarrow$q) is true. If p is false, and q is true, then (p$\rightarrow$q) is true. If p is true, and q is true, then (p$\rightarrow$q) is true. And there's the definition of the truth table, given that propositions can only take on two truth values.

If we don't have "if p⇒q, then ⇒(p$\rightarrow$q)", nor have something like a truth table for $\rightarrow$, then it's probably not clear what $\rightarrow$ means in the first place. We can't define $\rightarrow$ merely by modus ponens, since logical equivalence also has a rule of modus ponens formally speaking.

Answer 2

A fundamental distinction here is that an argument is often about many similar situations that the argument treats uniformly. Often this can be seen because some variable appears both in the premise and the conclusion:

Assume that $x$ is an even number.
Then $x^2$ is an even number.

Here we can see the $x$ on both sides, and we can apply your definition directly: The argument is valid because it is impossible that $x$ is even and $x^2$ is not even.

(Sometimes the shared variable is hidden in natural language: "If a number is even, then its square is even too". That makes no formal difference).

In contrast the connective $\supset$ and its truth table is in principle always something you apply to a single situations. The formula $$ x\text{ is even} \supset x^2\text{ is even} $$ does not have a truth value until you plug in a concrete value for $x$. Once you do that, you can start applying the truth table: $$ 4\text{ is even } \supset 16\text{ is even} $$ is true due to the first line of the truth table, and $$ 5\text{ is even} \supset 25\text{ is even} $$ is true due to the last line of the truth table.

So what we can say is that

• The argument "If $p(x)$ then $q(x)$" is valid, and
• The formula "$p(x)\supset q(x)$" is always true, meaning for every value of $x$.

are two ways of making the same claim. With quantifiers we can express the second of these more succinctly as

• The formula "$\forall x(p(x)\supset q(x)) $" is true. (No "always" here).

The truth table for $\supset$ can really only be understood when one knows that it is designed to be used together with an (implicit or explicit) quantification over all the situations one is interested in.

Unfortunately a large majority of introductory texts expect students to understand $\supset$ before they even begin to speak about quantifiers, and they spend huge amounts of verbiage trying to cajole the reader into thinking it makes sense even then. It doesn't really.

Answer 3

Now, my teacher says that an argument is neither true nor false, but valid or invalid

The conditional $$P_1\land\ldots \land P_n\to C$$ corresponds to the argument $$P_1,\ldots\text{ and }P_n;\text{ therefore }C.$$ The argument above is said to be valid precisely when the conditional above is logically valid (in propositional logic: when the conditional above is a tautology).

An argument is valid if and only if it's impossible that the premises are true and the conclusion false.

The definition only refers to the case in which the premises are true and the conclusion false, as an invalid argument.

No, that definition does inspect every case/row of the argument's truth table; it says that a valid argument's truth table has no row corresponding to row 3 of the table that you displayed.

Remember, every argument corresponds not to a specific row of its conditional's truth table, but to the conditional itself.

It doesn't say anything about (1) arguments with true premises and true conclusions, or (2) arguments with false premises and true conclusions, or (3) arguments with false premises and false conclusions. I assume that these are valid arguments.

You misunderstand: evaluating an argument means to evaluate not whether its corresponding conditional is true, but whether its corresponding conditional is logically valid. Here, we are concerned only with the argument's form.

Consider the following arguments:

1. an hour is longer than a minute; therefore, Earth's moon orbits around Earth;
2. an hour is shorter than a minute; therefore, Earth's moon orbits around Earth;
3. an hour is shorter than a minute; therefore, Earth orbits around Earth's moon.

All of them are invalid, because the corresponding conditionals

1. $r>n \:→\: mOe$
2. $r>n \:→\: mOe$
3. $r<n \:→\: eOm$

while all true, are all logically invalid.

Answer 4

• Let P be the conjunction of all the premises of an argument ( with a finite number of premises) and let C be its conclusion.

Let I be the set of all interpretations ( truth value assignments to the atomic sentences. of your language).

• Saying that the reasoning is valid is equivalent to saying

(1) that P logically implies C ( in symbols : $P \implies C$)

(2) there is no interpretation $i\in I$ such that ($P$ is true in $i$ and $C$ is false in $i$ ) .

(3) for all $i\in I$ it not the case that ( $P$ is true in $i$ and $C$ is false in $i$ )

(4) for all $i\in I$ ( if $P$ is true in $i$ then $C$ is true in $i$).

The " if ... then " at line (4) is a material conditional.

• From what I want to conclude that the material conditional cannot derive from logical implication ( or logical validity) since the notion of material conditional ( $\rightarrow$) is an ingredient of the definition of logical "implication" ( $\implies$).

• Note : However it might be the case that the fact we use the material conditional ( with the truth table you exhibit) in order to define logical implication is the very reason why it has been decided to call it ( material ) implication.