Example

The argument “when I went to the store last week I forgot my purse, and when I went today I forgot my purse. I always forget my purse when I go the store” is an inductive argument. The premises are:

I forgot my purse last week I forgot my purse today

The conclusion is:

I always forget my purse

Notice that the premises are specific situations, while the conclusion is a general statement. In this case, this is a fairly weak argument, since it is based on only two instances.

Example

The argument “every day for the past year, a plane flies over my house at 2pm. A plane will fly over my house every day at 2pm” is a stronger inductive argument, since it is based on a larger set of evidence.

Example

The argument “All cats are mammals and a tiger is a cat, so a tiger is a mammal” is a valid deductive argument. The premises are:

All cats are mammals A tiger is a cat

The conclusion is:

A tiger is a mammal

Both the premises are true. To see that the premises must logically lead to the conclusion, one approach would be use a Venn diagram. From the first premise, we can conclude that the set of cats is a subset of the set of mammals. From the second premise, we are told that a tiger lies within the set of cats. From that, we can see in the Venn diagram that the tiger also lies inside the set of mammals, so the conclusion is valid.

Example

Premise:	All firefighters know CPR
Premise:	Jill knows CPR
Conclusion:	Jill is a firefighter

From the first premise, we know that firefighters all lie inside the set of those who know CPR. From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know if she also is a member of the smaller subset that is firefighters. Since the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether Jill actually is a firefighter.

Example

Premise:	If you live in Seattle, you live in Washington.
Premise:	Marcus does not live in Seattle.
Conclusion:	Marcus does not live in Washington.

From the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether or not Marcus lives in Washington or not. This is an invalid argument.

Example

Consider the argument “You are a married man, so you must have a wife.”

Answer: This is an invalid argument, since there are, at least in parts of the world, men who are married to other men, so the premise not insufficient to imply the conclusion.

Example

Consider the argument:

Premise:	If you bought bread, then you went to the store
Premise:	You bought bread
Conclusion:	You went to the store

Answer: While this example is hopefully fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then look at the implication that the premises together imply the conclusion. If the truth table is a tautology (always true), then the argument is valid. We’ll get B represent “you bought bread” and S represent “you went to the store”. Then the argument becomes:

Premise:	[latex]B{\rightarrow}S[/latex]
Premise:	[latex]B[/latex]
Conclusion:	[latex]S[/latex]

To test the validity, we look at whether the combination of both premises implies the conclusion; is it true that [latex]\left[\left(B{\rightarrow}S\right){\wedge}B\right]{\rightarrow}S[/latex]?

[latex]B[/latex]	[latex]S[/latex]	[latex]B{\rightarrow}S[/latex]	[latex]\left(B{\rightarrow}S\right){\wedge}B[/latex]	[latex]\left[\left(B{\rightarrow}S\right){\wedge}B\right]{\rightarrow}S[/latex]
T	T	T	T	T
T	F	F	F	T
F	T	T	F	T
F	F	T	F	T

Since the truth table for [latex]\left[\left(B{\rightarrow}S\right){\wedge}B\right]{\rightarrow}S[/latex] is always true, this is a valid argument.

Example

Premise:	If I go to the mall, then I’ll buy new jeans.
Premise:	If I buy new jeans, I’ll buy a shirt to go with it.
Conclusion:	If I got to the mall, I’ll buy a shirt.

Let M = I go to the mall, J = I buy jeans, and S = I buy a shirt. The premises and conclusion can be stated as:

Premise:	[latex]M{\rightarrow}J[/latex]
Premise:	[latex]J{\rightarrow}S[/latex]
Conclusion:	[latex]M{\rightarrow}S[/latex]

We can construct a truth table for [latex]\left[\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)\right]{\rightarrow}\left(M{\rightarrow}S\right)[/latex]

[latex]M[/latex]	[latex]J[/latex]	[latex]S[/latex]	[latex]M{\rightarrow}J[/latex]	[latex]J{\rightarrow}S[/latex]	[latex]\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)[/latex]	[latex]M{\rightarrow}S[/latex]	[latex]\left[\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)\right]{\rightarrow}\left(M{\rightarrow}S\right)[/latex]
T	T	T	T	T	T	T	T
T	T	F	T	F	F	F	T
T	F	T	F	T	F	T	T
T	F	F	F	T	F	F	T
F	T	T	T	T	T	T	T
F	T	F	T	F	F	T	T
F	F	T	T	T	T	T	T
F	F	F	T	T	T	T	T

From the truth table, we can see this is a valid argument.

Example

Premise:	If I work hard, I’ll get a raise.
Premise:	If I get a raise, I’ll buy a boat.
Conclusion:	If I don’t buy a boat, I must not have worked hard.

Answer: If we let W = working hard, R = getting a raise, and B = buying a boat, then we can represent our argument symbolically:

Premise:	[latex]H{\rightarrow}R[/latex]
Premise:	[latex]R{\rightarrow}B[/latex]
Conclusion:	[latex]\sim{B}{\rightarrow}{\sim}H[/latex]

We could construct a truth table for this argument, but instead, we will use the notation of the contrapositive we learned earlier to note that the implication [latex]{\sim}B{\rightarrow}{\sim}H[/latex] is equivalent to the implication [latex]H{\rightarrow}B[/latex]. Rewritten, we can see that this conclusion is indeed a logical syllogism derived from the premises.

Example

Solve the puzzle. In other words, find a logical conclusion from these premises. All babies are illogical. Nobody who can manage a crocodile is despised. Illogical persons are despised.

Answer: Let B = is a baby, D = is despised, I = is illogical, and M = can manage a crocodile. Then we can write the premises as:

[latex]B{\rightarrow}I\\M{\rightarrow}{\sim}D\\I{\rightarrow}D[/latex]

From the first and third premises, we can conclude that [latex]B{\rightarrow}D[/latex]; that babies are despised. Using the contrapositive of the second premised, [latex]D{\rightarrow}{\sim}M[/latex], we can conclude that [latex]B\rightarrow\sim{M}[/latex]; that babies cannot manage crocodiles. While silly, this is a logical conclusion from the given premises.