Your question is basically the same as this one: What is the logical form of the definition of validity? . And my answer is a less formal version of what Hunan is telling you.

an argument is **valid** if having its premises be true *necessarily* leads to a true conclusion.

The necessarily / must element in the definition makes it so that we are not looking at whether the claims are in fact true but rather whether the *forms* of the claims are such that their truth implies the truth of the conclusion. Thus, we need to check to see if there is any truth value for the variable involved whether or not it is possible that the premises end up being true and the conclusion being false.

To do so involves several steps and there are multiple methods.

"All cats are mammals, All tigers are mammals, Therefore all tigers are cats".

This gives us three statements and three variables. To make it first order logic, we need understand "all" to mean if it is an A, then it is a B:

```
(1) C -> M
(2) T -> M
Therefore (3) T -> C
```

As you rightly point out, all 3 claims turn out to be true (assuming by "cats" we mean something other than felinis familiaris).

Why then is the argument invalid? The key is that the validity looks at if it is *possible* for the argument to have true premises and a false conclusion.

# Test Method #1

We can test this with several methods of varying difficulty to grasp. One of the easier ones to understand for many people is the exhaustive truth table. Here, we are going to create rows where we assign whether variables are true or false and look to see if the claims are true or false. If we end up with a situation where the premises are true and the conclusion is false, then the argument is invalid.

In our case, we have three variables. Per the law of the excluded middle, each variable can be true or false. Thus, either it is true that it is a cat or it is false that it is a cat. [etc] This will give us 8 rows as follows:

```
C T M
-----
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
```

This represents every way that these variables could be related -- regardless of how they are related in this world. We then look at whether each claim is true. A conditional is true when the antecedent (left part) is true and the consequent (right part) is true. OR when the left part is false, it is true regardless of the right part. Or to put it another way, it is only false when the antecedent is true and the consequent is false.

```
C T M C -> M T->M T->C
-----
T T T T T T
T T F F F T
T F T T T T
T F F F T T
F T T T T F
F T F T F F
F F T T T T
F F F T T T
```

If you look the fifth line has true premises and a false conclusion. Thus it is possible for the argument to have had true premises and a false conclusion. It turns out that in our world the premises and conclusion are true, but the logic behind the premises does not compel the conclusion we are drawing. So the argument is **invalid**.

# Test Method 2

There's a faster way called the short circuit method where you accomplish the same thing as the above method but cheat. Instead of making every row, we just set the conclusion to false and figure out how we can make the premises true if that's the case. If we can make all of the premises true, we've proven it is invalid.o

So we begin like this:

```
C T M C -> M T->M T->C
-----
F
```

We then ask what it takes for T -> C to be false. The answer is that T must be true and C must be false. (due to the way conditionals work).

```
C T M C -> M T->M T->C
-----
F T F
```

If C is false, then C -> M is true regardless of the value of M.

```
C T M C -> M T->M T->C
-----
F T T F
```

The question then is if we can make T -> M true with these values for C and T already set. The answer is that we can -- if we set M to true.

```
C T M C -> M T->M T->C
-----
F T T T T F
```

Thus, we've shown invalidity -- because we *can have* true premises and a false conclusion. Note again, this does not mean that we do.

# Test Method 3

Finally, we can show the same thing using rules of inference (which I will leave out here but is probably the most common method in philosophy and math).

Is invalid not a synonym for false? – Neil Meyer – 2014-10-17T10:03:46.493