# Chapter 32 Semantic concepts

## 32.1 Validities, entailment, etc.

Defining truth in FOL was quite fiddly. But now that we are done, we can define various other central logical notions. These definitions will look very similar to those for TFL, from chapter 12. However, remember that they concern interpretations, rather than valuations.

We will use the symbol ‘$\vDash$’ for FOL much as we did for TFL. So:

$\mathscr{A}_{1},\mathscr{A}_{2},\ldots,\mathscr{A}_{n}\vDash\mathscr{C}$

means that there is no interpretation in which all of $\mathscr{A}_{1}$, $\mathscr{A}_{2}$, …, $\mathscr{A}_{n}$ are true and in which $\mathscr{C}$ is false. Derivatively,

$\vDash\mathscr{A}$

means that $\mathscr{A}$ is true in every interpretation.

The other logical notions also have corresponding definitions in FOL:

• An FOL sentence $\mathscr{A}$ is a validity if⁠f $\mathscr{A}$ is true in every interpretation; i.e., $\vDash\mathscr{A}$.

• $\mathscr{A}$ is a contradiction if⁠f $\mathscr{A}$ is false in every interpretation; i.e., $\vDash\neg\mathscr{A}$.

• $\mathscr{A}_{1},\mathscr{A}_{2},\ldots,\mathscr{A}_{n}\therefore\mathscr{C}$ is valid in FOL if⁠f there is no interpretation in which all of the premises are true and the conclusion is false; i.e., $\mathscr{A}_{1},\mathscr{A}_{2},\ldots,\mathscr{A}_{n}\vDash\mathscr{C}$. It is invalid in FOL otherwise.

• Two FOL sentences $\mathscr{A}$ and $\mathscr{B}$ are equivalent if⁠f they are true in exactly the same interpretations as each other; i.e., both $\mathscr{A}\vDash\mathscr{B}$ and $\mathscr{B}\vDash\mathscr{A}$.

• The FOL sentences $\mathscr{A}_{1}$, $\mathscr{A}_{2}$, …, $\mathscr{A}_{n}$ are jointly satisfiable if⁠f some interpretation makes all of them true. They are jointly unsatisfiable if⁠f there is no such interpretation.

Note that we use the standard term ‘validity’ for sentences that are true in every interpretation. Validities are to FOL what tautologies are to TFL.

## 32.2 Expressibility

The concept of an object satisfying a formula with one free variable introduced in section 31.3 can also be extended to formulas with more than one free variable. If we have a formula $\mathscr{A}(\mathscr{x},\mathscr{y})$ with two free variables $\mathscr{x}$ and $\mathscr{y}$, then we can say that a pair of objects $\langle a,b\rangle$ satisfies $\mathscr{A}(\mathscr{x},\mathscr{y})$ if⁠f $\mathscr{A}(\mathscr{c},\mathscr{d})$ is true in the interpretation extended by two names $\mathscr{c}$ and $\mathscr{d}$, where $\mathscr{c}$ names $a$ and $\mathscr{d}$ names $b$. So, for instance, $\langle\text{Socrates},\text{Plato}\rangle$ satisfies $R(x,y)$ since $R(c,d)$ is true in the interpretation:

domain:

all people born before 2000 ce

$a$:

Aristotle

$b$:

Beyoncé

$c$:

Socrates

$d$:

Plato

$P(x)$:

blankx is a philosopher

$R(x,y)$:

blankx was born before blanky

For atomic formulas, the objects, pairs of objects, etc., that satisfy them are exactly the extension of the predicate given in the interpretation. But the notion of satisfaction also applies to non-atomic formulas, e.g., the formula $P(x)\land R(x,b)$ is satisfied by all philosophers born before Beyoncé. It even applies to formulas involving quantifiers, e.g., $P(x)\wedge\lnot\exists y(P(y)\land R(y,x))$ is satisfied by all people who are philosophers and for whom it is true that no philosopher was born before them—in other words, it is true of the first philosopher.

By considering formulas (possibly involving quantifiers) with two free variables, we can express relations for which we do not have dedicated predicate symbols in our interpretation or symbolization key. Consider the formula $R(x,y)$. It expresses the relation ‘blankx was born before blanky’, since that is how we have specified its extension. What happens if we switch the variables, i.e., consider ‘$R(y,x)$’? A pair of objects $\mathopen{\langle}$$\text{y},\text{x}$$\mathclose{\rangle}$ in the domain (i.e., a pair of people) satisfies $R(y,x)$ if, and only if, the reverse pair $\mathopen{\langle}$$\text{x},\text{y}$$\mathclose{\rangle}$ satisfies $R(x,y)$. In other words, $R(y,x)$ expresses the relation ‘blankx was born after blanky’. Or suppose we add to our interpretation a predicate for ‘teacher of’.

$T(x,y)$:

blankx was a teacher of blanky

Then the formula ‘$\exists z(T(z,x)\land T(z,y))$’ is satisfied by x and y if, and only if, some person z was a teacher of both x and y, i.e., it expresses ‘blankx and blanky have a teacher in common’. Similarly, ‘$\forall z(T(x,z)\leftrightarrow T(y,z))$’ expresses ‘blankx and blanky taught the same people’.

The take-home message of these examples is that some English predicates, such as ‘blankx and blanky have a teacher in common’, can be sometimes be expressed in an interpretation even if there is no explicit predicate symbol available for them. If that is the case, you can use a formula that expresses them (such as ‘$\exists z(T(z,x)\land T(z,y))$’) to symbolize English sentences involving the predicate.

## Practice exercises

A. Given the following interpretation:

domain:

people

$W(x)$:

blankx is a woman (or girl)

$M(x)$:

blankx is a man (or boy)

$Y(x,y)$:

blankx is younger than blanky

$O(x,y)$:

blankx is an offspring of blanky

$d$:

Dana

$a$:

Alex

express the following relations:

1. 1.

blankx is older than blanky

2. 2.

blankx is blanky’s mother

3. 3.

blankx and blanky are siblings (note that you can’t be your own sibling)

4. 4.

blankx is blanky’s brother

B. Using the symbolization key from the previous exercise, symbolize the following sentences:

1. 1.

Alex and Dana are sisters.

2. 2.

Fathers are older than their children.

3. 3.

Alex’s parents are the same age.