12TRUTH IN A MODEL

Paal Antonsenantonsp@tcd.ie

https://sites.google.com/site/paalantonsen/teaching/logica

Formal Logic

The story so far. . .

I We’ve started describing the logical system we called predicate logic.

Logical systems A description of a logical system comes in three parts:

(A) Grammar: A description of what counts as a formula.

(B) Semantics: A definition of truth on an evaluation (or truth in a model);and derivatively validity and related concepts.

(C) Proofs: A description of what counts as a proof.

GrammarATOMIC If P is a predicate with n-places and a1, . . . an are names

then Pa1. . .an is a formula.QUANTIFIERS If A is a formula then (∃x)A(a := x) and (∀x)A(a := x)

formulas, where A(a := x) is the result of replacing alloccurences of a with x .

CONNECTIVES If A is a formula then ∼A is a formula.If A and B are formulas then (A & B), (A ∨ B), (A ⊃ B)and (A ≡ B) are formulas.

CLOSURE Nothing else counts as a formula.

The story so far. . .

I We’ve started describing the logical system we called predicate logic.

Logical systems A description of a logical system comes in three parts:

(A) Grammar: A description of what counts as a formula.

(B) Semantics: A definition of truth on an evaluation (or truth in a model);and derivatively validity and related concepts.

(C) Proofs: A description of what counts as a proof.

GrammarATOMIC If P is a predicate with n-places and a1, . . . an are names

then Pa1. . .an is a formula.QUANTIFIERS If A is a formula then (∃x)A(a := x) and (∀x)A(a := x)

formulas, where A(a := x) is the result of replacing alloccurences of a with x .

CONNECTIVES If A is a formula then ∼A is a formula.If A and B are formulas then (A & B), (A ∨ B), (A ⊃ B)and (A ≡ B) are formulas.

CLOSURE Nothing else counts as a formula.

Truth conditions

I How are we to give a semantics for predicate logic?

I We constructed a semantics for propositional logic by giving a definitionof truth on an evaluation. We understood an evaluation to be a possibleassignment of truth values to formulas – formally, we characterized it asa function from formulas to the set {1, 0}.

I For an atomic p we said that either v (p) = 1 or v (p) = 0.

I We gave a truth functional characterization of the logical connectives, i.e.a set of rules that showed how the truth values of complex formulas wasdetermined by the truth values of their constituent formulas. E.g.:

v (∼A) ={

1 if v (A) = 00 otherwise

v (A & B) ={

1 if v (A) = 1 and v (B) = 10 otherwise

Truth conditions

I How are we to give a semantics for predicate logic?

I We constructed a semantics for propositional logic by giving a definitionof truth on an evaluation. We understood an evaluation to be a possibleassignment of truth values to formulas – formally, we characterized it asa function from formulas to the set {1, 0}.

I For an atomic p we said that either v (p) = 1 or v (p) = 0.

I We gave a truth functional characterization of the logical connectives, i.e.a set of rules that showed how the truth values of complex formulas wasdetermined by the truth values of their constituent formulas. E.g.:

v (∼A) ={

1 if v (A) = 00 otherwise

v (A & B) ={

1 if v (A) = 1 and v (B) = 10 otherwise

Truth conditions

I How are we to give a semantics for predicate logic?

I We constructed a semantics for propositional logic by giving a definitionof truth on an evaluation. We understood an evaluation to be a possibleassignment of truth values to formulas – formally, we characterized it asa function from formulas to the set {1, 0}.

I For an atomic p we said that either v (p) = 1 or v (p) = 0.

I We gave a truth functional characterization of the logical connectives, i.e.a set of rules that showed how the truth values of complex formulas wasdetermined by the truth values of their constituent formulas. E.g.:

v (∼A) ={

1 if v (A) = 00 otherwise

v (A & B) ={

1 if v (A) = 1 and v (B) = 10 otherwise

Truth conditions

I Turning to predicate logic, the situation is somewhat more complicated.We cannot start with a stipulation that we simply assign a truth value toevery atomic formula. Instead we have to specify how the elements ofthe vocabulary contribute to determining their truth value.

The basic semantic idea(i) The truth value of Pa is determined by whether the individual picked

out by a has the property expressed by P.

(ii) The truth value of Rab is determined by whether the individual pickedout by a stands in the relation expressed by R to b.

(iii) The truth value of (∃x)Px is determined by whether there is someindividual that has the property expressed by P.

(iv) The truth value of (∀x)Px is determined by whether every individualhas the property expressed by P.

I We must find a way to specify what individuals are picked out names andpredicates, and what individuals we are quantifying over.

Truth conditions

I Turning to predicate logic, the situation is somewhat more complicated.We cannot start with a stipulation that we simply assign a truth value toevery atomic formula. Instead we have to specify how the elements ofthe vocabulary contribute to determining their truth value.

The basic semantic idea(i) The truth value of Pa is determined by whether the individual picked

out by a has the property expressed by P.

(ii) The truth value of Rab is determined by whether the individual pickedout by a stands in the relation expressed by R to b.

(iii) The truth value of (∃x)Px is determined by whether there is someindividual that has the property expressed by P.

(iv) The truth value of (∀x)Px is determined by whether every individualhas the property expressed by P.

I We must find a way to specify what individuals are picked out names andpredicates, and what individuals we are quantifying over.

Truth conditions

I Turning to predicate logic, the situation is somewhat more complicated.We cannot start with a stipulation that we simply assign a truth value toevery atomic formula. Instead we have to specify how the elements ofthe vocabulary contribute to determining their truth value.

The basic semantic idea(i) The truth value of Pa is determined by whether the individual picked

out by a has the property expressed by P.

(ii) The truth value of Rab is determined by whether the individual pickedout by a stands in the relation expressed by R to b.

(iii) The truth value of (∃x)Px is determined by whether there is someindividual that has the property expressed by P.

(iv) The truth value of (∀x)Px is determined by whether every individualhas the property expressed by P.

I We must find a way to specify what individuals are picked out names andpredicates, and what individuals we are quantifying over.

Introducing models

I The notion of truth simpliciter won’t do, because that assumes oneintended reading. Rather, we need to talk about the things picked outrelative to a model.

The task of providing a semantics for predicate logic comes down todefining truth in a model for formulas of predicate logic.

I What do we mean by a model?

A modelM can be represented as a pair 〈domain, intepretation〉.

I Where A is a formula of predicate logic, A is true in the modelM (or truerelative toM) iff A is true relative to the domain and interpretation ofM.What we need to do is to define the conditions under which formulas ofpredicate logic are true relative to a model.

Introducing models

I The notion of truth simpliciter won’t do, because that assumes oneintended reading. Rather, we need to talk about the things picked outrelative to a model.

The task of providing a semantics for predicate logic comes down todefining truth in a model for formulas of predicate logic.

I What do we mean by a model?

A modelM can be represented as a pair 〈domain, intepretation〉.

I Where A is a formula of predicate logic, A is true in the modelM (or truerelative toM) iff A is true relative to the domain and interpretation ofM.What we need to do is to define the conditions under which formulas ofpredicate logic are true relative to a model.

Introducing models

I The notion of truth simpliciter won’t do, because that assumes oneintended reading. Rather, we need to talk about the things picked outrelative to a model.

The task of providing a semantics for predicate logic comes down todefining truth in a model for formulas of predicate logic.

I What do we mean by a model?

A modelM can be represented as a pair 〈domain, intepretation〉.

I Where A is a formula of predicate logic, A is true in the modelM (or truerelative toM) iff A is true relative to the domain and interpretation ofM.What we need to do is to define the conditions under which formulas ofpredicate logic are true relative to a model.

Introducing models

I The notion of truth simpliciter won’t do, because that assumes oneintended reading. Rather, we need to talk about the things picked outrelative to a model.

The task of providing a semantics for predicate logic comes down todefining truth in a model for formulas of predicate logic.

I What do we mean by a model?

A modelM can be represented as a pair 〈domain, intepretation〉.

I Where A is a formula of predicate logic, A is true in the modelM (or truerelative toM) iff A is true relative to the domain and interpretation ofM.What we need to do is to define the conditions under which formulas ofpredicate logic are true relative to a model.

Introducing models

I The notion of truth simpliciter won’t do, because that assumes oneintended reading. Rather, we need to talk about the things picked outrelative to a model.

The task of providing a semantics for predicate logic comes down todefining truth in a model for formulas of predicate logic.

I What do we mean by a model?

A modelM can be represented as a pair 〈domain, intepretation〉.

I Where A is a formula of predicate logic, A is true in the modelM (or truerelative toM) iff A is true relative to the domain and interpretation ofM.What we need to do is to define the conditions under which formulas ofpredicate logic are true relative to a model.

Introducing models: domains

I What do you mean by a domain?

A domain D is a non-empty set of objects.

A set is an arbitrary collection of arbitrary objects.

I If a set is finite and small, we can list all its member.e.g. The set {Yoda, Jabba, Luke}

I We can specify some condition or property that gives a set.e.g. The set of all individuals that are students at Trinity College Dublin

We write this as {x : x is a student at Trinity College Dublin}

I We can specify a rule that generates a set.e.g. The set of the natural numbers N can be specified as:

(i) 0 is a member of N ;(ii) If n is a member of N then n + 1 is a member of N

Introducing models: domains

I What do you mean by a domain?

A domain D is a non-empty set of objects.

A set is an arbitrary collection of arbitrary objects.

I If a set is finite and small, we can list all its member.e.g. The set {Yoda, Jabba, Luke}

I We can specify some condition or property that gives a set.e.g. The set of all individuals that are students at Trinity College Dublin

We write this as {x : x is a student at Trinity College Dublin}

I We can specify a rule that generates a set.e.g. The set of the natural numbers N can be specified as:

(i) 0 is a member of N ;(ii) If n is a member of N then n + 1 is a member of N

Introducing models: domains

I What do you mean by a domain?

A domain D is a non-empty set of objects.

A set is an arbitrary collection of arbitrary objects.

I If a set is finite and small, we can list all its member.e.g. The set {Yoda, Jabba, Luke}

I We can specify some condition or property that gives a set.e.g. The set of all individuals that are students at Trinity College Dublin

We write this as {x : x is a student at Trinity College Dublin}

I We can specify a rule that generates a set.e.g. The set of the natural numbers N can be specified as:

(i) 0 is a member of N ;(ii) If n is a member of N then n + 1 is a member of N

Introducing models: domains

I What do you mean by a domain?

A domain D is a non-empty set of objects.

A set is an arbitrary collection of arbitrary objects.

I If a set is finite and small, we can list all its member.e.g. The set {Yoda, Jabba, Luke}

I We can specify some condition or property that gives a set.e.g. The set of all individuals that are students at Trinity College Dublin

We write this as {x : x is a student at Trinity College Dublin}

I We can specify a rule that generates a set.e.g. The set of the natural numbers N can be specified as:

(i) 0 is a member of N ;(ii) If n is a member of N then n + 1 is a member of N

Introducing models: domains

I What do you mean by a domain?

A domain D is a non-empty set of objects.

A set is an arbitrary collection of arbitrary objects.

I If a set is finite and small, we can list all its member.e.g. The set {Yoda, Jabba, Luke}

I We can specify some condition or property that gives a set.e.g. The set of all individuals that are students at Trinity College Dublin

We write this as {x : x is a student at Trinity College Dublin}

I We can specify a rule that generates a set.e.g. The set of the natural numbers N can be specified as:

(i) 0 is a member of N ;(ii) If n is a member of N then n + 1 is a member of N

Introducing models: domains

I What do you mean by a domain?

A domain D is a non-empty set of objects.

A set is an arbitrary collection of arbitrary objects.

I If a set is finite and small, we can list all its member.e.g. The set {Yoda, Jabba, Luke}

I We can specify some condition or property that gives a set.e.g. The set of all individuals that are students at Trinity College Dublin

We write this as {x : x is a student at Trinity College Dublin}

I We can specify a rule that generates a set.e.g. The set of the natural numbers N can be specified as:

(i) 0 is a member of N ;(ii) If n is a member of N then n + 1 is a member of N

Introducing models: domains

I Let’s introduce some useful terminology:

Where S1 and S2 are sets, we write:x ∈ S x is a member of SS1 is a subset of S2 (S1 ⊆ S2) for all x, if x ∈ S1 then x ∈ S2

S1 is a proper subset of S2 (S1 ⊂ S2) for all x, if x ∈ S1 then x ∈ S2

and some y, y /∈ S1 and y ∈ S2

S1 is identical to S2 (S1 = S2) for all x, x ∈ S1 iff x ∈ S2

The empty set ∅ for any x, x /∈ ∅

I Sets are exhaustively determined by their members and the order inwhich members are written is irrelevant. Given the above definitions,{Lysander, Hermina} = {Hermia, Lysander}.

I When we want to talk about ordered sets, such as pairs, we use thesharp brackets 〈 〉. 〈Lysander, Hermina〉 6= 〈Hermina, Lysander〉.

Introducing models: domains

I Let’s introduce some useful terminology:

Where S1 and S2 are sets, we write:x ∈ S x is a member of SS1 is a subset of S2 (S1 ⊆ S2) for all x, if x ∈ S1 then x ∈ S2

S1 is a proper subset of S2 (S1 ⊂ S2) for all x, if x ∈ S1 then x ∈ S2

and some y, y /∈ S1 and y ∈ S2

S1 is identical to S2 (S1 = S2) for all x, x ∈ S1 iff x ∈ S2

The empty set ∅ for any x, x /∈ ∅

I Sets are exhaustively determined by their members and the order inwhich members are written is irrelevant. Given the above definitions,{Lysander, Hermina} = {Hermia, Lysander}.

I When we want to talk about ordered sets, such as pairs, we use thesharp brackets 〈 〉. 〈Lysander, Hermina〉 6= 〈Hermina, Lysander〉.

Introducing models: domains

I Let’s introduce some useful terminology:

Where S1 and S2 are sets, we write:x ∈ S x is a member of SS1 is a subset of S2 (S1 ⊆ S2) for all x, if x ∈ S1 then x ∈ S2

S1 is a proper subset of S2 (S1 ⊂ S2) for all x, if x ∈ S1 then x ∈ S2

and some y, y /∈ S1 and y ∈ S2

S1 is identical to S2 (S1 = S2) for all x, x ∈ S1 iff x ∈ S2

The empty set ∅ for any x, x /∈ ∅

I Sets are exhaustively determined by their members and the order inwhich members are written is irrelevant. Given the above definitions,{Lysander, Hermina} = {Hermia, Lysander}.

I When we want to talk about ordered sets, such as pairs, we use thesharp brackets 〈 〉. 〈Lysander, Hermina〉 6= 〈Hermina, Lysander〉.

Introducing models: interpretations

I What do you mean by an interpretation?

An interpretation I is an assignment of extensions to expressions.

I What extensions are assigned to what expressions depend on theirgrammatical category. We characterize it in the following way:

Let an interpretation I be a function, such that it

(i) maps names to individuals in the domain

(ii) maps one-place predicates to sets individuals in the domain

(iii) maps two-place predicates to pairs of individuals in the domain...

I We will write I(e) to denote the extension of the expression e on theinterpretation I

Introducing models: interpretations

I What do you mean by an interpretation?

An interpretation I is an assignment of extensions to expressions.

I What extensions are assigned to what expressions depend on theirgrammatical category. We characterize it in the following way:

Let an interpretation I be a function, such that it

(i) maps names to individuals in the domain

(ii) maps one-place predicates to sets individuals in the domain

(iii) maps two-place predicates to pairs of individuals in the domain...

I We will write I(e) to denote the extension of the expression e on theinterpretation I

Introducing models: interpretations

I What do you mean by an interpretation?

An interpretation I is an assignment of extensions to expressions.

I What extensions are assigned to what expressions depend on theirgrammatical category. We characterize it in the following way:

Let an interpretation I be a function, such that it

(i) maps names to individuals in the domain

(ii) maps one-place predicates to sets individuals in the domain

(iii) maps two-place predicates to pairs of individuals in the domain...

I We will write I(e) to denote the extension of the expression e on theinterpretation I

Introducing models: interpretations

I What do you mean by an interpretation?

An interpretation I is an assignment of extensions to expressions.

I What extensions are assigned to what expressions depend on theirgrammatical category. We characterize it in the following way:

Let an interpretation I be a function, such that it

(i) maps names to individuals in the domain

(ii) maps one-place predicates to sets individuals in the domain

(iii) maps two-place predicates to pairs of individuals in the domain...

I We will write I(e) to denote the extension of the expression e on theinterpretation I

Truth in a model: definition

I We are now in a positiont to define truth in a model. We start with theatomic cases and quantified formulas.

Pa is true at 〈D, I〉 iff I(a) ∈ I(P)Pa1. . .an is true at 〈D, I〉 iff 〈I(a1), . . . I(an)〉 ∈ I(P)

(∃x)A is true at 〈D, I〉 iff A(a:= x) is true at some 〈D, I*〉,where I∗ differs from I at mostin the extension assigned to a.

(∀x)A is true at 〈D, I〉 iff A(a:= x) is true at every 〈D, I*〉,where I∗ differs from I at mostin the extension assigned to a.

I The clauses for the quantified formulas are a mouthful. You can think ofit in a much more common sense way, e.g.

(∃x)Px is true at 〈D, I〉 iff there is a member of D that is in I(P)(∀x)Px is true at 〈D, I〉 iff every member of D is in I(P)

Truth in a model: definition

I We are now in a positiont to define truth in a model. We start with theatomic cases and quantified formulas.

Pa is true at 〈D, I〉 iff I(a) ∈ I(P)Pa1. . .an is true at 〈D, I〉 iff 〈I(a1), . . . I(an)〉 ∈ I(P)

(∃x)A is true at 〈D, I〉 iff A(a:= x) is true at some 〈D, I*〉,where I∗ differs from I at mostin the extension assigned to a.

(∀x)A is true at 〈D, I〉 iff A(a:= x) is true at every 〈D, I*〉,where I∗ differs from I at mostin the extension assigned to a.

I The clauses for the quantified formulas are a mouthful. You can think ofit in a much more common sense way, e.g.

(∃x)Px is true at 〈D, I〉 iff there is a member of D that is in I(P)(∀x)Px is true at 〈D, I〉 iff every member of D is in I(P)

Truth in a model: definition

I We are now in a positiont to define truth in a model. We start with theatomic cases and quantified formulas.

Pa is true at 〈D, I〉 iff I(a) ∈ I(P)Pa1. . .an is true at 〈D, I〉 iff 〈I(a1), . . . I(an)〉 ∈ I(P)

(∃x)A is true at 〈D, I〉 iff A(a:= x) is true at some 〈D, I*〉,where I∗ differs from I at mostin the extension assigned to a.

(∀x)A is true at 〈D, I〉 iff A(a:= x) is true at every 〈D, I*〉,where I∗ differs from I at mostin the extension assigned to a.

I The clauses for the quantified formulas are a mouthful. You can think ofit in a much more common sense way, e.g.

(∃x)Px is true at 〈D, I〉 iff there is a member of D that is in I(P)(∀x)Px is true at 〈D, I〉 iff every member of D is in I(P)

Truth in a model: definition

I The logical connectives can be handled in the familiar way:

∼A is true at 〈D, I〉 iff A is not true 〈D, I〉

(A & B) is true at 〈D, I〉 iff A true at 〈D, I〉and B is true at 〈D, I〉

(A ∨ B) is true at 〈D, I〉 iff A true at 〈D, I〉or B is true at 〈D, I〉

(A ⊃ B) is true at 〈D, I〉 iff A not true at 〈D, I〉or B is true at 〈D, I〉

(A ≡ B) is true at 〈D, I〉 iff A and B are true at 〈D, I〉or B and B are not true at 〈D, I〉

I Together this gives us a way of handling any type of formula in predicatelogic, and we have fixed their truth conditions in a way that ultimatelydepends on the extension of the names and predicates.

Truth in a model: definition

I The logical connectives can be handled in the familiar way:

∼A is true at 〈D, I〉 iff A is not true 〈D, I〉

(A & B) is true at 〈D, I〉 iff A true at 〈D, I〉and B is true at 〈D, I〉

(A ∨ B) is true at 〈D, I〉 iff A true at 〈D, I〉or B is true at 〈D, I〉

(A ⊃ B) is true at 〈D, I〉 iff A not true at 〈D, I〉or B is true at 〈D, I〉

(A ≡ B) is true at 〈D, I〉 iff A and B are true at 〈D, I〉or B and B are not true at 〈D, I〉

I Together this gives us a way of handling any type of formula in predicatelogic, and we have fixed their truth conditions in a way that ultimatelydepends on the extension of the names and predicates.

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

Pa is true inMBecause I(a) is in I(P)

Qb is false inMBecause I(b) is not in I(Q)

Pb ∨ Qb is true inMBecause I(b) is in I(P)

Qb ⊃ Qa is true inMBecause I(b) is not in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

Pa is true inMBecause I(a) is in I(P)

Qb is false inMBecause I(b) is not in I(Q)

Pb ∨ Qb is true inMBecause I(b) is in I(P)

Qb ⊃ Qa is true inMBecause I(b) is not in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

Pa is true inMBecause I(a) is in I(P)

Qb is false inMBecause I(b) is not in I(Q)

Pb ∨ Qb is true inMBecause I(b) is in I(P)

Qb ⊃ Qa is true inMBecause I(b) is not in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

Pa is true inMBecause I(a) is in I(P)

Qb is false inMBecause I(b) is not in I(Q)

Pb ∨ Qb is true inMBecause I(b) is in I(P)

Qb ⊃ Qa is true inMBecause I(b) is not in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

Pa is true inMBecause I(a) is in I(P)

Qb is false inMBecause I(b) is not in I(Q)

Pb ∨ Qb is true inMBecause I(b) is in I(P)

Qb ⊃ Qa is true inMBecause I(b) is not in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

Pa is true inMBecause I(a) is in I(P)

Qb is false inMBecause I(b) is not in I(Q)

Pb ∨ Qb is true inMBecause I(b) is in I(P)

Qb ⊃ Qa is true inMBecause I(b) is not in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

Pa is true inMBecause I(a) is in I(P)

Qb is false inMBecause I(b) is not in I(Q)

Pb ∨ Qb is true inMBecause I(b) is in I(P)

Qb ⊃ Qa is true inMBecause I(b) is not in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

Pa is true inMBecause I(a) is in I(P)

Qb is false inMBecause I(b) is not in I(Q)

Pb ∨ Qb is true inMBecause I(b) is in I(P)

Qb ⊃ Qa is true inMBecause I(b) is not in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

Pa is true inMBecause I(a) is in I(P)

Qb is false inMBecause I(b) is not in I(Q)

Pb ∨ Qb is true inMBecause I(b) is in I(P)

Qb ⊃ Qa is true inMBecause I(b) is not in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

(∃x)Px is true inMThere is a member of D that is in I(P)

(∀x)Qx is false inMNot all members of D are in I(Q)

(∃x)(Px ∨ Qx) is true inMSome member of D is both in I(P) and in I(Q)

(∀x)(Px ⊃ Qx) is false inMNot all members of D that are in I(P) are in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

(∃x)Px is true inMThere is a member of D that is in I(P)

(∀x)Qx is false inMNot all members of D are in I(Q)

(∃x)(Px ∨ Qx) is true inMSome member of D is both in I(P) and in I(Q)

(∀x)(Px ⊃ Qx) is false inMNot all members of D that are in I(P) are in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

(∃x)Px is true inMThere is a member of D that is in I(P)

(∀x)Qx is false inMNot all members of D are in I(Q)

(∃x)(Px ∨ Qx) is true inMSome member of D is both in I(P) and in I(Q)

(∀x)(Px ⊃ Qx) is false inMNot all members of D that are in I(P) are in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

(∃x)Px is true inMThere is a member of D that is in I(P)

(∀x)Qx is false inMNot all members of D are in I(Q)

(∃x)(Px ∨ Qx) is true inMSome member of D is both in I(P) and in I(Q)

(∀x)(Px ⊃ Qx) is false inMNot all members of D that are in I(P) are in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

(∃x)Px is true inMThere is a member of D that is in I(P)

(∀x)Qx is false inMNot all members of D are in I(Q)

(∃x)(Px & Qx) is true inMSome member of D is both in I(P) and in I(Q)

(∀x)(Px ⊃ Qx) is false inMNot all members of D that are in I(P) are in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

(∃x)Px is true inMThere is a member of D that is in I(P)

(∀x)Qx is false inMNot all members of D are in I(Q)

(∃x)(Px & Qx) is true inMSome member of D is both in I(P) and in I(Q)

(∀x)(Px ⊃ Qx) is false inMNot all members of D that are in I(P) are in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

(∃x)Px is true inMThere is a member of D that is in I(P)

(∀x)Qx is false inMNot all members of D are in I(Q)

(∃x)(Px & Qx) is true inMSome member of D is both in I(P) and in I(Q)

(∀x)(Px ⊃ Qx) is false inMNot all members of D that are in I(P) are in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).

Truth in a model: example

MD = {a, b}I(a) = aI(b) = bI(P) = {a, b}I(Q) = {a}

(∃x)Px is true inMThere is a member of D that is in I(P)

(∀x)Qx is false inMNot all members of D are in I(Q)

(∃x)(Px & Qx) is true inMSome member of D is both in I(P) and in I(Q)

(∀x)(Px ⊃ Qx) is false inMNot all members of D that are in I(P) are in I(Q)

When making a model, we need to provide a domain (a set of individuals)and add an interpretation that assigns extensions to the names andpredicates. For intuitive purposes you can think of a model as arepresentation of a way the world could have been (or a part of it).