# 1.5.1: Arguments

- Page ID
- 9910

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let *p *stand for the proposition “Today is Tuesday”, and let *q *stand for the proposition “This is Belgium”. Then the above argument has the form

*p*→*q*

*\(\frac{p}{\therefore q}\)*

Now, for any propositions *p *and *q*—not just the ones in this particular argument—if*p *→ *q *is true and *p *is true, then *q *must also be true. This is easy to check in a truth table:

The only case where both *p *→ *q *and *p *are true is on the last line of the table, and in this case, *q *is also true. If you believe *p *→ *q *and *p*, you have no logical choice but to believe *q*. This applies no matter what *p *and *q *represent. For example, if you believe “If Jill is breathing, then Jill pays taxes”, and you believe that “Jill is breathing”, logic forces you to believe that “Jill pays taxes”. Note that we can’t say for sure that the conclusion is true, only that if the premises are true, then the conclusion must be true.

This fact can be rephrased by saying that ((*p *→ *q*) ∧ *p*) → *q *is a tautology. More generally, for any compound propositions P and Q, saying “P → Q is a tautology” is the same as saying that “in all cases where P is true, Q is also true”.14 We will use the notation P =⇒ Q to mean that P → Q is a tautology. Think of P as being the premise of an argument or the conjunction of several premises. To say P =⇒ Q is to say that Q follows logically from P. We will use the same notation in both propositional logic and predicate logic. (Note that the relation of =⇒ to → is the same as the relation of ≡ to↔.)

Definition 2.10.

Let P and Q be any formulas in either propositional logic or predicate logic. The notation P =⇒ Q is used to mean that P → Q is a tautology. That is, in all cases where P is true, Q is also true. We then say that Q can be **logically** deduced from P or that P **logically implies** Q.

An argument in which the conclusion follows logically from the premises is said to be a valid argument. To test whether an argument is valid, you have to replace the particular propositions or predicates that it contains with variables, and then test whether the conjunction of the premises logically implies the conclusion. We have seen that any argument of the form

*p*→*q*

*\(\frac{p}{\therefore q}\)*

is valid, since ((*p *→ *q*) ∧ *p*) → *q *is a tautology. This rule of deduction is called **modus** **ponens**. It plays a central role in logic. Another, closely related rule is **modus tollens**, which applies to arguments of the form

*p*→*q*

*\(\frac{\neg q}{\therefore \neg p}\)*

To verify that this is a valid argument, just check that ((*p *→ *q*) ∧ ¬*q*) =⇒ ¬*p*, that is, that ((*p *→ *q*) ∧ ¬*q*) → ¬*p *is a tautology. As an example, the following argument has the form of modus tollens and is therefore a valid argument:

If Feyenoord is a great team, then I’m the king of the Netherlands

I am not the king of the Netherlands

∴ Feyenoord is not a great team

You might remember this argument from page 10. You should note carefully that the validity of this argument has nothing to do with whether or not Feyenoord can play football well. The argument forces you to accept the conclusion only if you accept the premises. You can logically believe that the conclusion is false, as long as you believe that at least one of the premises is false.

Another named rule of deduction is the **Law of Syllogism**, which has the form

*p*→*q*

*\(\frac{q \rightarrow r}{\therefore p \rightarrow r}\)*

For example:

f you study hard, you do well in school

If you do well in school, you get a good job

∴ If you study hard, you get a good job

There are many other rules. Here are a few that might prove useful. Some of them might look trivial, but don’t underestimate the power of a simple rule when it is combined with other rules.

*p*∨*q p p*∧*q p*

¬*p q *∴*p *∴*p*∨*q*

∴*q *∴*p*∧*q*

Logical deduction is related to logical equivalence. We defined P and Q to be logically equivalent if P ↔ Q is a tautology. Since P ↔ Q is equivalent to (P → Q) ∧ (Q → P), we see that P ≡ Q if and only if both Q =⇒ P and P =⇒ Q. Thus, we can show that two statements are logically equivalent if we can show that each of them can be logically deduced from the other. Also, we get a lot of rules about logical deduction for free—two rules of deduction for each logical equivalence we know. For example, since ¬(*p *∧ *q*) ≡(¬*p *∨ ¬*q*), we get that ¬(*p *∧ *q*) =⇒ (¬*p *∨ ¬*q*). For example, if we know “It is not both sunny and warm”, then we can logically deduce “Either it’s not sunny or it’s not warm.” (And vice versa.)