Sentences considered in propositional logic are not arbitrary sentences but are the ones that are either true or false, but not both. This kind of sentences are called propositions. If a proposition is true, then we say it has a truth value of "true"; if a proposition is false, its truth value is "false".

For example, "Grass is green", and "2 + 5 = 5" are propositions. The first proposition has the truth value of "true" and the second "false".

But "Close the door", and "Is it hot outside?" are not propositions. Also "x is greater than 2", where x is a variable representing a number, is not a proposition, because unless a specific value is given to x we can not say whether it is true or false, nor do we know what x represents.

Similarly "x = x" is not a proposition because we don't know what "x" represents hence what "=" means. For example, while we understand what "3 = 3" means, what does "Air is equal to air" or "Water is equal to water" mean? Does it mean a mass of air is equal to another mass or the concept of air is equal to the concept of air? We don't quite know what "x = x" mean. Thus we can not say whether it is true or not. Hence it is not a proposition.

Simple sentences which are true or false are basic propositions. Larger and more complex sentences are constructed from basic propositions by combining them with connectives. Thus propositions and connectives are the basic elements of propositional logic. Though there are many connectives, we are going to use the following five basic connectives here:

NOT, AND, OR, IF_THEN (or IMPLY), IF_AND_ONLY_IF.

They are also denoted by the symbols: ¬, ⋀,⋁,→,↔ ,

respectively.

Often we want to discuss properties/relations common to all propositions. In such a case rather than stating them for each individual proposition we use variables representing an arbitrary proposition and state properties/relations in terms of those variables. Those variables are called a propositional variable. Propositional variables are also considered a proposition and called a proposition since they represent a proposition hence they behave the same way as propositions. A proposition in general contains a number of variables. For example (P ⋁Q) contains variables P and Q each of which represents an arbitrary proposition. Thus a proposition takes different values depending on the values of the constituent variables. This relationship of the value of a proposition and those of its constituent variables can be represented by a table. It tabulates the value of a proposition for all possible values of its variables and it is called a truth table.

For example the following table shows the relationship between the values of P, Q and P⋁Q:

In the table, F represents truth value false and T true. This table shows that P⋁Q is false if P and Q are both false, and it is true in all the other cases.

Let us define the meaning of the five connectives by showing the relationship between the truth value (i.e. true or false) of composite propositions and those of their component propositions. They are going to be shown using truth table. In the tables P and Q represent arbitrary propositions, and true and false are represented by T and F, respectively.

This table shows that if P is true, then (¬P) is false, and that if P is false, then (¬P) is true.

This table shows that (P⋀Q) is true if both P and Q are true, and that it is false in any other case. Similarly for the rest of the tables.

When P→Q is always true, we express that by P ⇒Q. That is P ⇒Q is used when proposition P always implies proposition Q regardless of the value of the variables in them.

When P ↔Q is always true, we express that by P ⇔Q. That is ⇔is used when two propositions always take the same value regardless of the value of the variables in them.

First it is informally shown how complex propositions are constructed from simple ones. Then more general way of constructing propositions is given.

In everyday life we often combine propositions to form more complex propositions without paying much attention to them. For example combining "Grass is green", and "The sun is red" we say something like "Grass is green and the sun is red", "If the sun is red, grass is green", "The sun is red and the grass is not green" etc. Here "Grass is green", and "The sun is red" are propositions, and form them using connectives "and", "if... then ..." and "not" a little more complex propositions are formed. These new propositions can in turn be combined with other propositions to construct more complex propositions. They then can be combined to form even more complex propositions. This process of obtaining more and more complex propositions can be described more generally as follows:

Let X and Y represent arbitrary propositions. Then [¬X], [X⋀Y], [X⋁Y], [X→Y], and [X↔Y] are propositions.

Note that X and Y here represent an arbitrary proposition. This is actually a part of more rigorous definition of proposition which we see later.

Example : [ P → [Q ⋁ R] ] is a proposition and it is obtained by first constructing [Q ⋁ R] by applying [X ⋁ Y] to propositions Q and R considering them as X and Y, respectively, then by applying [ X→Y ] to the two propositions P and [Q ⋁ R] considering them as X and Y, respectively.

Note: Rigorously speaking X and Y above are place holders for propositions, and so they are not exactly a proposition. They are called a propositional variable, and propositions formed from them using connectives are called a propositional form. However, we are not going to distinguish them here, and both specific propositions such as "2 is greater than 1" and propositional forms such as (P ⋁Q) are going to be called a proposition.

For the proposition P→Q, the proposition Q→P is called its converse, and the proposition ¬ Q→ ¬ P is called its contrapositive.

For example for the proposition "If it rains, then I get wet",

Converse: If I get wet, then it rains.

Contrapositive: If I don't get wet, then it does not rain.

The converse of a proposition is not necessarily logically equivalent to it, that is they may or may not take the same truth value at the same time.

On the other hand, the contrapositive of a proposition is always logically equivalent to the proposition. That is, they take the same truth value regardless of the values of their constituent variables. Therefore, "If it rains, then I get wet." and "If I don't get wet, then it does not rain." are logically equivalent. If one is true then the other is also true, and vice versa.

If-then statements appear in various forms in practice. The following list presents some of the variations. These are all logically equivalent, that is as far as true or false of statement is concerned there is no difference between them. Thus if one is true then all the others are also true, and if one is false all the others are false.

For instance, instead of saying "If she smiles then she is happy", we can say "If she smiles, she is happy", "She is happy whenever she smiles", "She smiles only if she is happy" etc. without changing their truth values.

"Only if" can be translated as "then". For example, "She smiles only if she is happy" is equivalent to "If she smiles, then she is happy".

Note that "She smiles only if she is happy" means "If she is not happy, she does not smile", which is the contrapositive of "If she smiles, she is happy". You can also look at it this way: "She smiles only if she is happy" means "She smiles only when she is happy". So any time you see her smile you know she is happy. Hence "If she smiles, then she is happy". Thus they are logically equivalent.

Also "If she smiles, she is happy" is equivalent to "It is necessary for her to smile that she is happy". For "If she smiles, she is happy" means "If she smiles, she is always happy". That is, she never fails to be happy when she smiles. "Being happy" is inevitable consequence/necessity of "smile". Thus if "being happy" is missing, then "smile" can not be there either. "Being happy" is necessary "for her to smile" or equivalently "It is necessary for her to smile that she is happy".

As we are going to see in the next section, reasoning is done on propositions using inference rules. For example, if the two propositions "if it snows, then the school is closed", and "it snows" are true, then we can conclude that "the school is closed" is true. In everyday life, that is how we reason.

To check the correctness of reasoning, we must check whether or not rules of inference have been followed to draw the conclusion from the premises. However, for reasoning in English or in general for reasoning in a natural language, that is not necessarily straightforward and it often encounters some difficulties. Firstly, connectives are not necessarily easily identified as we can get a flavor of that from the previous topic on variations of if_then statements. Secondly, if the argument becomes complicated involving many statements in a number of different forms twisted and tangled up, it can easily get out of hand unless it is simplified in some way.

One solution for that is to use symbols (and mechanize it). Each sentence is represented by symbols representing building block sentences, and connectives. For example, if P represents "it snows" and Q represents "the school is closed", then the previous argument can be expressed as

[ [ P → Q ] ⋀ P ] → Q,

or

P → Q

P

-----------------------------

Q This representation is concise, much simpler and much easier to deal with. In addition today there are a number of automatic reasoning systems and we can verify our arguments in symbolic form using them. One such system called TPS is used for reasoning exercises in this course. For example, we can check the correctness of our argument using it.

To convert English statements into a symbolic form, we restate the given statements using the building block sentences, those for which symbols are given, and the connectives of propositional logic (not, and, or, if_then, if_and_only_if), and then substitute the symbols for the building blocks and the connectives. For example, let P be the proposition "It is snowing", Q be the proposition "I will go the beach", and R be the proposition "I have time".

Then first "I will go to the beach if it is not snowing" is restated as "If it is not snowing, I will go to the beach". Then symbols P and Q are substituted for the respective sentences to obtain ~P → Q.

Similarly, "It is not snowing and I have time only if I will go to the beach" is restated as "If it is not snowing and I have time, then I will go to the beach", and it is translated as (~P ⋀ R ) → Q.

Logical reasoning is the process of drawing conclusions from premises using rules of inference. Here we are going to study reasoning with propositions. Later we are going to see reasoning with predicate logic, which allows us to reason about individual objects. However, inference rules of propositional logic are also applicable to predicate logic and reasoning with propositions is fundamental to reasoning with predicate logic.

These inference rules are results of observations of human reasoning over centuries. Though there is nothing absolute about them, they have contributed significantly in the scientific and engineering progress the mankind have made. Today they are universally accepted as the rules of logical reasoning and they should be followed in our reasoning.

Since inference rules are based on identities and implications, we are going to study them first. We start with three types of proposition which are used to define the meaning of "identity" and "implication".

Some propositions are always true regardless of the truth value of its component propositions. For example (P ⋁¬P) is always true regardless of the value of the proposition P.

A proposition that is always true called a tautology.

There are also propositions that are always false such as (P ⋀¬P). Such a proposition is called a contradiction.

A proposition that is neither a tautology nor a contradiction is called a contingency. For example (P ⋁Q) is a contingency.

These types of propositions play a crucial role in reasoning. In particular every inference rule is a tautology as we see in identities and implications.

Let X be a proposition involving only ¬, ⋀, and ⋁ as a connective. Let X* be the proposition obtained from X by replacing ⋀ with ⋁, ⋁with ⋀, T with F, and F with T. Then X* is called the dual of X.

For example, the dual of [P ⋀Q ] ⋁P is [P ⋁Q ] ⋀P, and the dual of [¬ P ⋀Q] ⋁¬[ T ⋀¬R] is [¬ P ⋁Q] ⋀¬[ F ⋁¬R] .

Property of Dual: If two propositions P and Q involving only ¬, ⋀, and ⋁ as connectives are equivalent, then their duals P* and Q* are also equivalent.

Here a few examples are presented to show how the identities in section Identities can be used to prove some useful results.

1. ¬( P →Q ) ⇔( P ⋀¬Q )

What this means is that the negation of "if P then Q" is "P but not Q". For example, if you said to someone "If I win a lottery, I will give you $100,000." and later that person says "You lied to me." Then what that person means is that you won the lottery but you did not give that person $100,000 you promised.

To prove this, first let us get rid of → using one of the identities: (P→Q ) ⇔( ¬P ⋁Q).

That is, ¬( P →Q ) ⇔¬( ¬P ⋁Q ).

Then by De Morgan, it is equivalent to ¬¬P ⋀¬Q , which is equivalent to P ⋀¬Q, since the double negation of a proposition is equivalent to the original proposition as seen in the identities.

2. P ⋁( P ⋀Q ) ⇔P --- Absorption

What this tells us is that P ⋁( P ⋀Q ) can be simplified to P, or if necessary P can be expanded into P ⋁( P ⋀Q ) .

To prove this, first note that P ⇔( P ⋀T ).

Hence

P ⋁( P ⋀Q )

⇔( P ⋀T ) ⋁( P ⋀Q )

⇔P ⋀( T ⋁Q ) , by the distributive law.

⇔( P ⋀T ) , since ( T ⋁Q ) ⇔T.

⇔P , since ( P ⋀T ) ⇔P.

Note that by the duality

P ⋀( P ⋁Q ) ⇔P also holds.

The following implications are some of the relationships between propositions that can be derived from the definitions (meaning) of connectives. ⇒ below corresponds to → and it means that the implication always holds. That is it is a tautology.

These implications are used in logical reasoning. When the right hand side of these implications is substituted for the left hand side appearing in a proposition, the resulting proposition is implied by the original proposition, that is, one can deduce the new proposition from the original one.

First the implications are listed, then examples to illustrate them are given. List of Implications:

1. P ⇒(P ⋁Q) ----- addition

2. (P ⋀Q) ⇒P ----- simplification

3. [P ⋀(P →Q] ⇒Q ----- modus ponens

4. [(P →Q) ⋀¬Q] ⇒¬P ----- modus tollens

5. [ ¬P ⋀(P ⋁Q] ⇒Q ----- disjunctive syllogism

6. [(P →Q) ⋀(Q→R)] ⇒(P→R) ----- hypothetical syllogism

7. (P→Q) ⇒[(Q→R)→(P→R)]

8. [(P→Q) ⋀(R→S)] ⇒[(P ⋀R)→(Q ⋀S)]

9. [(P ↔Q) ⋀(Q ↔R)] ⇒(P ↔R)

Examples:

1. P ⇒(P ⋁Q) ----- addition

For example, if the sun is shining, then certainly the sun is shining or it is snowing. Thus

"if the sun is shining, then the sun is shining or it is snowing." "If 0 < 1, then 0 ≤1 or a similar statement is also often seen.

2. (P ⋀Q) ⇒P ----- simplification

For example, if it is freezing and (it is) snowing, then certainly it is freezing. Thus "If it is freezing and (it is) snowing, then it is freezing."

3. [P ⋀(P →Q] ⇒Q ----- modus ponens

For example, if the statement "If it snows, the schools are closed" is true and it actually snows, then the schools are closed.

This implication is the basis of all reasoning. Theoretically, this is all that is necessary for reasoning. But reasoning using only this becomes very tedious.

4. [(P →Q) ⋀¬Q] ⇒¬P ----- modus tollens

For example, if the statement "If it snows, the schools are closed" is true and the schools are not closed, then one can conclude that it is not snowing. Note that this can also be looked at as the application of the contrapositive and modus ponens. That is, (P→Q) is equivalent to ( ¬Q )→( ¬P ). Thus if in addition ¬Q holds, then by the modus ponens, ¬P is concluded.

5. [ ¬P ⋀(P ⋁Q] ⇒Q ----- disjunctive syllogism

For example, if the statement "It snows or (it) rains." is true and it does not snow, then one can conclude that it rains.

6. [(P→Q) ⋀(Q→R)] ⇒(P→R) ----- hypothetical syllogism

For example, if the statements "If the streets are slippery, the school buses can not be operated." and "If the school buses can not be operated, the schools are closed." are true, then the statement "If the streets are slippery, the schools are closed." is also true.

7. (P→Q) ⇒[(Q→R)→(P→R)]

This is actually the hypothetical syllogism in another form. For by considering (P→Q) as a proposition S, (Q→R) as a proposition T, and (P→R) as a proposition U in the hypothetical syllogism above, and then by applying the "exportation" from the identities, this is obtained.

8. [(P→Q) ⋀(R→S)] ⇒[(P ⋀R)→(Q ⋀S)]

For example, if the statements "If the wind blows hard, the beach erodes." and "If it rains heavily, the streets get flooded." are true, then the statement "If the wind blows hard and it rains heavily, then the beach erodes and the streets get flooded." is also true.

9. [(P ↔Q) ⋀(Q ↔R)] ⇒(P ↔R)

This just says that the logical equivalence is transitive, that is, if P and Q are equivalent, and if Q and R are also equivalent, then P and R are equivalent.

Logical reasoning is the process of drawing conclusions from premises using rules of inference. The basic inference rule is modus ponens. It states that if both P→Q and P hold, then Q can be concluded, and it is written as

P

P →Q

-----------------

Q

Here the lines above the dotted line are premises and the line below it is the conclusion drawn from the premises.

For example if "if it rains, then the game is not played" and "it rains" are both true, then we can conclude that the game is not played.

In addition to modus ponens, one can also reason by using identities and implications.

If the left (right) hand side of an identity appearing in a proposition is replaced by the right(left) hand side of the identity, then the resulting proposition is logically equivalent to the original proposition. Thus the new proposition is deduced from the original proposition. For example in the proposition P ⋀(Q→R), (Q→R) can be replaced with (¬Q ⋁R) to conclude P ⋀(¬Q ⋁R), since (Q→R) ⇔(¬Q ⋁R)

Similarly if the left (right) hand side of an implication appearing in a proposition is replaced by the right(left) hand side of the implication, then the resulting proposition is logically implied by the original proposition. Thus the new proposition is deduced from the original proposition.

The tautologies listed as "implications" can also be considered inference rules as shown below.

Consider the following argument:

1. Today is Tuesday or Wednesday.

2. But it can't be Wednesday, since the doctor's office is open today, and that office is always closed on Wednesdays.

3. Therefore today must be Tuesday.

This sequence of reasoning (inferencing) can be represented as a series of application of modus ponens to the corresponding propositions as follows.

The modus ponens is an inference rule which deduces Q from P → Q and P.

T: Today is Tuesday.

W: Today is Wednesday.

D: The doctor's office is open today.

C: The doctor's office is always closed on Wednesdays.

The above reasoning can be represented by propositions as follows.

1. T ⋁ W

2. D

C

------------

~W

------------

3. T

To see if this conclusion T is correct, let us first find the relationship among C, D, and W:

C can be expressed using D and W. That is, restate C first as the doctor's office is always closed if it is Wednesday. Then C ↔ (W → ~D) Thus substituting (W → ~D) for C, we can proceed as follows.

D

W → ~D

------------

~W

which is correct by modus tollens.

From this ~W combined with T V W of 1. above,

~W

T ⋁ W

------------

T

which is correct by disjunctive syllogism.

Thus we can conclude that the given argument is correct.

To save space we also write this process as follows eliminating one of the ~W's:

D

W → ~D

------------

~W

T ⋁ W

------------

T

All the identities in Section Identities can be proven to hold using truth tables as follows. In general two propositions are logically equivalent if they take the same value for each set of values of their variables. Thus to see whether or not two propositions are equivalent, we construct truth tables for them and compare to see whether or not they take the same value for each set of values of their variables.

For example consider the commutativity of ⋁:

(P ⋁Q) ⇔(Q ⋁P).

To prove that this equivalence holds, let us construct a truth table for each of the proposition (P ⋁Q) and (Q ⋁P).

A truth table for (P ⋁Q) is, by the definition of ⋁,

A truth table for (Q ⋁P) is, by the definition of ⋁,

As we can see from these tables (P ⋁Q) and (Q ⋁P) take the same value for the same set of value of P and Q. Thus they are (logically) equivalent.

We can also put these two tables into one as follows:

Using this convention for truth table we can show that the first of De Morgan's Laws also holds.

By comparing the two right columns we can see that ¬(P ⋁Q) and ¬P ⋀¬Q are equivalent.

1. All the implications in Section Implications can be proven to hold by constructing truth tables and showing that they are always true.

For example consider the first implication "addition": P ⇒ (P ⋁ Q).

To prove that this implication holds, let us first construct a truth table for the proposition P ⋁ Q.

Then by the definition of →, we can add a column for P → (P ⋁ Q) to obtain the following truth table.

The first row in the rightmost column results since P is false, and the others in that column follow since (P ⋁ Q) is true.

The rightmost column shows that P → (P ⋁ Q) is always true.

2. Some of the implications can also be proven by using identities and implications that have already been proven.

For example suppose that the identity "exportation":

[(X ⋀Y) →Z] ⇔[X →(Y→Z)] ,

and the implication "hypothetical syllogism":

[(P→Q) ⋀(Q→R)] ⇒(P→R)

have been proven. Then the implication No. 7:

(P→Q) ⇒[(Q→R)→(P→R)]

can be proven by applying the "exportation" to the "hypothetical syllogism" as follows:

Consider (P→Q) , (Q→R) , and (P→R) in the "hypothetical syllogism" as X, Y and Z of the "exportation", respectively.

Then since [ (X ⋀Y )→Z ] ⇔[ X→( Y→Z ) ] implies [ ( X ⋀Y )→Z ] ⇒[ X→(Y→Z ) ] , the implication of No. 7 follows.

Similarly the modus ponens (implication No. 3) can be proven as follows:

Noting that ( P→Q ) ⇔( ¬P ⋁Q ) ,

P ⋀( P→Q )

⇔P ⋀( ¬P ⋁Q )

⇔( P ⋀¬P ) ⋁( P ⋀Q ) --- by the distributive law

⇔F ⋁( P ⋀Q )

⇔( P ⋀Q )

⇒Q

Also the exportation (identity No. 20), ( P→( Q→R ) ) ⇔ ( P ⋀Q )→R ) can be proven using identities as follows:

( P→( Q→R ) ) ⇔ ¬P ⋁( Q→R )

⇔ ¬P ⋁( ¬Q ⋁R )

⇔ ( ¬P ⋁¬Q ) ⋁R

⇔ ¬( P ⋀Q ) ⋁R

⇔ ( P ⋀Q )→R

3. Some of them can be proven by noting that a proposition in an implication can be replaced by an equivalent proposition without affecting its value.

For example by substituting ( ¬Q→¬P ) for ( P→Q ) , since they are equivalent being contrapositive to each other, modus tollens (the implication No. 4): [ ( P→Q ) ⋀¬Q ] ⇒ ¬P , reduces to the modus ponens: [ X ⋀( X→Y ) ] ⇒Y. Hence if the modus ponens and the "contrapositive" in the "Identities" have been proven, then the modus tollens follows from them.

To cope with deficiencies of propositional logic we introduce two new features: predicates and quantifiers.

A predicate is a verb phrase template that describes a property of objects, or a relationship among objects represented by the variables.

For example, the sentences "The car Tom is driving is blue", "The sky is blue", and "The cover of this book is blue" come from the template "is blue" by placing an appropriate noun/noun phrase in front of it. The phrase "is blue" is a predicate and it describes the property of being blue. Predicates are often given a name. For example any of "is_blue", "Blue" or "B" can be used to represent the predicate "is blue" among others. If we adopt B as the name for the predicate "is_blue", sentences that assert an object is blue can be represented as "B(x)", where x represents an arbitrary object. B(x) reads as "x is blue".

Similarly the sentences "John gives the book to Mary", "Jim gives a loaf of bread to Tom", and "Jane gives a lecture to Mary" are obtained by substituting an appropriate object for variables x, y, and z in the sentence "x gives y to z". The template "... gives ... to ..." is a predicate and it describes a relationship among three objects. This predicate can be represented by Give( x, y, z ) or G( x, y, z ), for example.

Note: The sentence "John gives the book to Mary" can also be represented by another predicate such as "gives a book to". Thus if we use B( x, y ) to denote this predicate, "John gives the book to Mary" becomes B( John, Mary ). In that case, the other sentences, "Jim gives a loaf of bread to Tom", and "Jane gives a lecture to Mary", must be expressed with other predicates.

A predicate with variables is not a proposition. For example, the statement x > 1 with variable x over the universe of real numbers is neither true nor false since we don't know what x is. It can be true or false depending on the value of x.

For x > 1 to be a proposition either we substitute a specific number for x or change it to something like "There is a number x for which x > 1 holds", or "For every number x, x > 1 holds".

More generally, a predicate with variables (called an atomic formula) can be made a proposition by applying one of the following two operations to each of its variables:

1. assign a value to the variable

2. quantify the variable using a quantifier (see below).

For example, x > 1 becomes 3 > 1 if 3 is assigned to x, and it becomes a true statement, hence a proposition.

In general, a quantification is performed on formulas of predicate logic (called wff), such as x > 1 or P(x), by using quantifiers on variables. There are two types of quantifiers: universal quantifier and existential quantifier.

The universal quantifier turns, for example, the statement x > 1 to "for every object x in the universe, x > 1", which is expressed as "∀x x > 1". This new statement is true or false in the universe of discourse. Hence it is a proposition once the universe is specified.

Similarly the existential quantifier turns, for example, the statement x > 1 to "for some object x in the universe, x > 1", which is expressed as "∃x x > 1." Again, it is true or false in the universe of discourse, and hence it is a proposition once the universe is specified.

Not all strings can represent propositions of the predicate logic. Those which produce a proposition when their symbols are interpreted must follow the rules given below, and they are called wffs (well-formed formulas) of the first order predicate logic.

A wff is, in general, not a proposition. For example, consider the wff ∀x P(x). Assume that P(x) means that x is non-negative (greater than or equal to 0). This wff is true if the universe is the set {1, 3, 5}, the set {2, 4, 6} or the set of natural numbers, for example, but it is not true if the universe is the set {-1, 3, 5}, or the set of integers, for example. Further more the wff ∀x Q(x, y), where Q(x, y) means x is greater than y, for the universe {1, 3, 5} may be true or false depending on the value of y.

As one can see from these examples, the truth value of a wff is determined by the universe, specific predicates assigned to the predicate variables such as P and Q, and the values assigned to the free variables. The specification of the universe and predicates, and an assignment of a value to each free variable in a wff is called an interpretation for the wff.

For example, specifying the set {1, 3, 5} as the universe and assigning 0 to the variable y, for example, is an interpretation for the wff ∀x Q(x, y), where Q(x, y) means x is greater than y. ∀x Q(x, y) with that interpretation reads, for example, "Every number in the set {1, 3, 5} is greater than 0".

As can be seen from the above example, a wff becomes a proposition when it is given an interpretation.

There are, however, wffs which are always true or always false under any interpretation. Those and related concepts are discussed below.

A wff is said to be satisfiable if there exists an interpretation that makes it true, that is if there are a universe, specific predicates assigned to the predicate variables, and an assignment of values to the free variables that make the wff true.

For example, ∀x N(x), where N(x) means that x is non-negative, is satisfiable. For if the universe is the set of natural numbers, the assertion ∀x N(x) is true, because all natural numbers are non-negative. Similarly ∃x N(x) is also satisfiable.

However, ∀x [N(x) ⋀¬N(x)] is not satisfiable because it can never be true. A wff is called invalid or unsatisfiable, if there is no interpretation that makes it true.

A wff is valid if it is true for every interpretation*. For example, the wff ∀x P(x) ⋁∃x ¬P(x) is valid for any predicate name P , because ∃x ¬P(x) is the negation of ∀x P(x). However, the wff ∀x N(x) is satisfiable but not valid.

Note that a wff is not valid iff it is unsatisfiable for a valid wff is equivalent to true. Hence its negation is false.

Two wffs W1 and W2 are equivalent if and only if W1 ↔W2 is valid, that is if and only if W1 ↔W2 is true for all interpretations.

For example ∀x P(x) and ¬∃x ¬P(x) are equivalent for any predicate name P. So are ∀x [ P(x) ⋀Q(x) ] and [ ∀x P(x) ⋀∀x Q(x) ] for any predicate names P and Q .

Predicate logic is more powerful than propositional logic. It allows one to reason about properties and relationships of individual objects. In predicate logic, one can use some additional inference rules, which are discussed below, as well as those for propositional logic such as the equivalences, implications and inference rules.

The following four rules describe when and how the universal and existential quantifiers can be added to or deleted from an assertion.

1. Universal Instantiation: ∀x P(x)

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P(c)

where c is some arbitrary element of the universe.

2. Universal Generalization:

P(c)

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∀x P(x)

where P(c) holds for every element c of the universe of discourse.

3. Existential Instantiation:

∃x P(x)

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P(c)

where c is some element of the universe of discourse. It is not arbitrary but must be one for which P(c) is true.

4. Existential Generalization:

P(c)

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∃x P(x)

where c is an element of the universe.