7. Lecture Mental models in cognitive science 7.1 Preliminary remarks

The status of logic in cognitive science is very controversial, in contrast to an intuitive view of logic, as the holy grail of cognitive science and artificial intelligence. The logic we used as the primary theoretical symbolic means modelling of reasoning [9.11]. But on the other hand, the analysis of human ability to use logic on elementary level [6.14], we see serious flaws.

These experimental findings are in contradiction with our general belief that man is a rational being, who wisely uses the principles of logic in drawing new information from given assumptions - knowledge and lead together with other arguments, the concept of limited rationality [1,3,6,14 ].

The aim of this paper is to present the current view of cognitive science to create a mental model of logic. Cognitive science is dealing several decades with the issue of how people use simple logical laws and what errors could occur. Innovative cognitivist perspective on this interesting problem [6.14] is that it proposes mental models of human cognitive activity to solve simple logical problems. It is required that these models are able to influence not only the right solutions but also to interpret the error in the construction of solutions. The mental model has a complexity that is variable when applied to different problems (e.g. different categorical syllogisms) [6], and we expect that the complexity of the model correlates with the occurrence of errors. Mental models (or mental modelling) were first postulated by Scottish psychologist Kenneth Craik in year 1943, who predicted that the human mind contains a small model of reality with which to visualize events, to think and explain. The model exists in working memory and is formed as a result of communication and imagination. The essential feature of mental models is that they are "isomorphistic" with the object represented (similarly as chemical molecular models of same molecules). According to Johnson-Laird, use of mental models in cognitive psychology eliminates its descriptive nature, without trying to interpret the observed results. There are two options for creating a mental model of logic:

(1) Syntactic approach. The mental model is identified with Gentzen’s natural deduction [15], which contains about a dozen rules of inference based on the elementary laws of logic with simple

and clear interpretation of their meaning. The set of assumptions 1 2 n, ,..., , using the

above-mentioned rules of the natural deduction derive the conclusion . We say that this

conclusion follows logically the assumptions (or there is evidence of logical formulas from the

set of premises 1 2 n, ,..., ), which is written using "session" as follows: � . Syntactic

approach to creating a mental model of logic is suitable for people who have already completed some basic training in logic and therefore can masterfully use the natural deduction proofs to construct more complex formulas.

(2) The semantic approach. The mental model is identified with the creation of a semantic model, a set of premises , together with the tautological consequence , written down � . The

semantic approach is based on a graphical method called semantic table. For that construction we need to know only the basic elementary concepts of the semantic interpretation of Boolean operators.

7.2 Syntax, semantics and pragmatics of propositional logic

One of the first formal systems in human history was Euclid axiomatic system of geometry, which became a model for developing other systems, especially in mathematics, computer science and logic. The theory of formal systems is usually strictly separated from each other with following three aspects:

(1) Syntax – method of the structure of language (formal) theory,

(2) Semantics - the semantic interpretation of language (various forms), and

(3) Pragmatics – change of semantics if changing environments /situations where semantic interpretation takes place as formulas.

The formal specification of the system emerges following two questions:

(1) Under what conditions is the proof of the formula - correct sentence and

(2) What methods can be used in the construction of such proof?

Theorem (theorem, formula, opinion, law, argument ...) is a statement that can be shown to be true. In this context, talking about the proof of Theorem, which consists a sequence of "intermediate step" that are derived either from a simple set of postulates called axioms, or sentences from the previous (auxiliary sentences, often called angles/edges) sequence. Complicated proofs are usually more clearly formulated, though the proof is divided on every intermediate step, which is formulated as a single sentence. This intermediate step – sentence in the sequence is generated by rules (rules of inference/deduction) that derive from a number of true statements - argument creates a new claim that is true - argument.

Proof methods of logic are not only important for making correct proofs in the logic itself, but also in mathematics and informatics. In theoretical computer science, for example studying different methods of verification of correctness in the program or the operating system is safe. In artificial intelligence to derive new facts from the knowledge base (set of propositional formulas, which is called a propositional logic theory) is important to ensure that the user database is consistent (correct), so that it follows the statement and also its negation. We therefore conclude that the management of methods of mathematical proof is important to the logic itself, as well as in mathematics and informatics.

To sum up this introductory chapter we list a few remarks on the pragmatics of formal logic. Usually this aspect is not mentioned in the specification of logic, it is believed that in the rigid formal logic system change can occur in the semantic interpretation of environmental change (e.g., social). Among logicians there is strong belief that time back to ancient Greek, logic is only one law and its use depends on the environment in which the user performs reasoning and more.

The area of pragmatic logic opened the psychology of Russian neuropsychology in the first half of last century by Alexander R.Luria [8], who at the age of 30 conducted expedition to Central Asian regions (which was Russia then) with his colleagues. They examined the cultural differences of using simple logical laws.

They let the illiterate local inhabitants to explain the first premise of the task: every animal that lives above the Arctic Circle has white fur, followed by a specific question, what colour is polar bear fur coat, who lives above the Arctic Circle, which formally can be expressed by scheme:

All animals living above the Arctic has white fur Polar bears live in the Arctic Circle

?

Most of the illiterate inhabitants replied that he does not know that they had never been above the Arctic Circle. A literate respondents (especially those who have received education in Russian language), answering differently, responded that the polar bear has white fur coat.

All animals living above the Arctic has white fur Polar bears live in the Arctic Circle Polar bears have white fur colour

A. Luria made these and similar observations and inferred that there is a cultural dependence to the use of logic reasoning schemes; that their use is dependent on education and the respondents.

These experiments are extremely interesting and were essentially done in other areas (e.g. in Africa and the Pacific).Unfortunately, there is quite much critic on the research; that it is not dependent on any cultural elemental reasoning schemes, but there is a problem of communication between the researcher and illiterate respondents, among whom there is a huge cultural barrier, and it is even questionable whether respondents correctly understand the question. Literate respondents who have mastered the basic school (reading, writing and counting), usually in Russian (foreign) language, probably has much better understanding of problem solving and are able to use the rule modus ponens for the construction of the correct answer - the solution.

7.3 Specification logic language (syntax)

Language L of propositional logic consists of propositional logic formulas, which are defined by a

set of atomic propositional variables p,q,..., p ,q ,... and set of Boolean operators , , , , .

Propositional logic formula, which form the language L are defined as recurrent minimal set that satisfies these properties:

(1) p,q,..., p ,q ,... L ,

(2) ak potomL , L ,

(3) ak potom, L , , , , L .

7.4 Specification of the importance of propositional logic (semantics)

Another important concept for the semantics of propositional logic is semantics. The term comes from the theory of natural language where semantics specifies the meaning of the sentence (which also has its syntax). In propositional logic, which deals only with truth-values of the variables and their format, semantics is not very rich. Semantics of formulas is actually a truth table values of formulas for different of the atomic statements values. Formula (elements of the language L of propositional logic) is important, that is specified as follows: the symbol '1' represents the truth value 'true' and the symbol '0'represents the truth value 'false'.

(1) Conjunction is true if and only if (iff) both its components are true, otherwise it is

false.

1val vtt 1val a 1val

0val vtt 0val alebo 0val

(2) Disjunction is true iff at least one component of it is true, otherwise it is false.

1val vtt 1val alebo 1val

0val vtt 0val a 0val

(3) Implication is true iff the first component () is false or the second component ()

is true, otherwise (the first component is true and the second component is false) is false.

1val vtt 0val alebo 1val

0val vtt 1val a 0val

(4) Equivalence is true iff both components have the same truth value, otherwise it is false.

1val vtt val val

0val vtt val val

(5) Negation is true iff the component is false, otherwise it is false.

1val vtt 0val

0val vtt 1val

On this occasion it is worth mentioning psychologists B. Inhelderovú and J. Piaget [4], who argued that no children aged 10 years cope with the principles of the semantic approach to propositional logic. They are able to correctly reproduce the table of truth values for disjunction, conjunction and negation. Problems with the semantic interpretation of the implications persist to adulthood; the correct interpretation requires basic logical education. For these reasons, the semantic approach to the development of mental logic model is suitable for those ( "lay in the logic") who are able to reproduce correctly the truth table and Boolean operators and do not have any other special knowledge of logic. Now, just enough interpretation of the implications of a text is highly dependent, in some extreme cases natural language is interpreted as a conjunction, which can cause in other cases some problems with the correct interpretation of the semantic implications. Thus, the only possible solution to this problem is that small children should "drill" and fix: implication p q is true if and only if p is false or q is true and also that the implication p q

is false if and only if p is true and q is false (these two facts can be expressed by analysis of Aristotle, truth implies that only the truth and false implies anything, so statement 1 0 must be

false). Truth value formula can be easily determined using the tabular method (see Table 1), where procedures will allow the truth values sub format as the next steps we use to design a truth-value "subformats" containing the previous sub form as its part.

Formula is called a tautology (or law, declared ) for any valid interpretation of the 1val

1def val ‘

In tab. 1 formula 2 is a tautology, it is true for all possible interpretations i, i = 1, 2, 3, 4.

Otherwise, if for every interpretation true 0val , formula is called a contradiction. If there

is at least one interpretation such that, 1val formula is satisfiable (i.e. that tautology is a

special case of satisfiability, the enrolment ‘ ). In tab. 1 formula 1 is feasible for two

interpretations 1 and 4. We can say that all the formulas that are not contradictions are feasible

and tautology is also satisfiable, which are all possible to interpret truth. Table 7.1. Tabulation method for calculating the truth values of formulas

1 p q p q a 2 p q p q

p q p q p q 1 p q p q 2 p q p q

1 0 0 0 0 1 1

2 0 1 1 0 0 1

3 1 0 1 0 0 1

4 1 1 1 1 1 1

Tautology of propositional logic has special status laws of logic; these formulas are always true for any truth values of variables. Some tautologies are frequently used not only in the propositional logic, but also in everyday reasoning and are usually referred to the proper name. Usually it is a tautology form of equivalences that allows one to replace other formulas without losing their properties of tautology.

Any non-empty set of formulas, 1 2 n, ,..., , is called the theory of propositional logic. If

the theory is in an interpretation of , for which all formulas are true, 1ival , for

1 2i , ,...,n , then this interpretation is called model theory. The theory is called consistent.

Table 7.2. The design model for the three formulas of propositional logic

1 p q p q

1 2 3 4 5

p q p q p q 3 4

0 0 0 0 1

0 1 1 0 0

1 0 1 0 0

1 1 1 1 1

Example 7.1.

1 2 3p q p q , p q q p , p q p q

To find out if this theory has a model. Using the tabular method we determine the truth values of formulas for all possible interpretations, see table 2.

From these tables we see that there are two interpretations of variables, 1 0 0p ,q and

4 1 1p ,q , for which all the formulas is true, i.e. interpretation 1 and 4 are the model

theory of . We can also say that the theory is consistent, which follows directly the fact that it has a model.

Formula is called tautological consequence of the theory of (the mark ‘ ) if and only if

every model theory is the model formula (i.e. formula is true). Let be a tautological

consequence of theory, then for each model – interpretation applies : 1ival val ,

for 1 2i , ,...,n .

Example 7.2. Let theory be defined as in the previous example, the two models for variables and interpretations

1 0 0p ,q a 4 1 1p ,q . Consider the formula in form p q , then this formula is not a

tautological consequence of theory, because only the model 2 formula 2

1val is true for model

1 is no longer true, 1

0val .

2 p q q p

1 2 3 4 5

p q p q q p 3 4

0 0 1 1 1

0 1 1 0 0

1 0 0 1 0

1 1 1 1 1

3 p q p q

1 2 3 4 5 6 7

p q p q 3 4 p q 5 6

0 0 1 1 1 1 1

0 1 1 0 0 1 1

1 0 0 1 0 0 1

1 1 0 0 0 1 1

In the previous chapters were formulated two independent terms:

(1) In the syntactic approach the concept of logical consequence of the formula set of

assumptions , which formally written are ‘ .

(2) The semantic approach the notion of tautological consequence of the formula theory , written down formally ‘ .

Although these terms are defined in an independent manner (syntactic and semantic),there are close links between them. By postal sentences of propositional logic[5,7,10,13,16] valid claim:

For any formula relationship � is valid if and only if, ‘ ,

‘ �

Opposite implication:

� ‘

is trivial (easy to check that the axioms are tautologies and use modus ponens rule of tautological assumptions which provides a tautological conclusion). Combining these two implications, we obtain the equivalence between logical and tautological entailment.

‘ �

7.5 Theory of proof - a natural deduction (syntactic approach) Classical way to solve a logical proof in the propositional calculus is Hilbert's axiomatic method [2,5,7,10,16], which postulates a system of 10 axiom and one inference rule. Although this syntactic approach has simple principles, the use of proof of new laws of propositional logic (which are not included in the set of axioms) in many cases is non-trivial matter, which usually requires a lot of "subtle" tricks and procedures that we also managed to do with relatively simple laws of propositional logic, for example p p� . An alternative approach to the structure theory of

evidence is Gentzen’s natural deduction from year 1935 [15], which is based on one trivial axiom (for example p p� or p p ) and about a dozen rules of derivation). The rules of inference in

the natural deduction are made up of the scheme (see Table 3.) Premise 1 ……………… …… Premise n

Conclusion (7.2)

which contains n premises 1 n,..., and a conclusion . This scheme is formalized by means

of reasoning "session" � logical proof (or resulting) {premise 1,… , premise n} conclusion or

1 n,..., � or � . (7.3)

Relatively simple means, we can prove that the relation ("logical proof” - consequence) is equivalent to the formula

1 n... � 1 2 n... �

In both of these formulas use the symbol (?) with empty set of assumptions, , which means

that formula 1 n... (or 1 2 n... ) is the law of propositional logic,

which structure do not need any premises. The left formula, conjunction assumption implies the conclusion. Formalize our considerations with the following sequence of definitions:

(1) Formula is called an immediate result of a set of formulas 1 2 n, ,..., if and only

if there is one application of the rules of logical proof of the formula

(2) Formula is called a logical result of a set of formulas (to indicate if and only if or

is the direct result or is a direct result of expanded some of its result.)

(3) The final sequence of formulas 1 2 n, ,..., is called a proof of the formula from the set

if and only if n every formula i and from this sequence is either an immediate

result of some formulas or formulas 1 2 1i, ,...,

Table 7.3. The rules of inference of natural deduction

# Scheme reasoning

Equivalent law of propositional logic

Name

1 p

p q p p q

Addition

2 p q

p

,

p q

q

p q p , p q q

Simplification

3 pq

p q p q p q

Introduction of conjunction

4

pp q

q

p p q q Modus ponens

5

qp q

p

q p q p

Modus tollens

6 q

p q q p q

Introduction to implication

7

p qq r

p r

p q q r p r

Hypothetical syllogism

8 p qp

q

,

p qq

p

p q p q

p q q p

Elimination of disjunction

9 p q

p q

,

p q

p q

p q p q Disjunctive syllogism

10

p q

q p

p q q p

Inversion implications

11 p qp q

p

p q p q p

Reduction ad absurdum

12 p

p

,

p

p p p

Double negation

Example 7.3 Let there be two simple statements (1) When it will rain, then I’ll go to the movies (2) When it will rain, then I go to cafe What conclusion follows from them? The first steps to perform the formalization of these statements, will introduce three atomic propositional variables p = 'will rain', q = 'go to the cinema', r = 'I go to coffee shop'

Then the set of premises has the form p q, p r ,, we are interested in what nontrivial

result follows from these assumptions ?p q, p r � The set of premises expand the

auxiliary assumption (we say it is activated).

1. p q (1. premise 1)

2. p r (2. premise 2)

3. p (activation of the auxiliary premise )

4. q (using the rule 4 modus ponens on the premises 1 and 3)

5. r (using the rule 4 modus ponens on the premises 2 and 3)

6. q r (introduction of conjunction of implication 4 and 5)

7. p q r (deactivation of the auxiliary premise 3 with a result of 6)

The conclusion which follows from the premises = 'if it rains I go to the cinema and cafe' Finally, this chapter summarizes the main conclusions of the syntactic approach for logical proof of formulas of propositional logic:

(1) Original Hilbert formal system of logical evidence for cognitive science is worthless, it is very difficult to put it to use as a mental model of reasoning. (2) Gentzen's natural deduction system (which is formally equivalent to Hilbert system) bears all the characteristics of simplicity and intuitiveness, makes it suitable as a mental model for the logic of those who have completed basic training in logic. 7.6 Model of propositional logic - semantic table (semantic approach) In the previous chapter there were formulated basic principles of syntactic approach to propositional logic. This chapter will show the implementation of the semantic approach using diagrammatic techniques called semantic table [7, 10, 12]. In this method, we solve the session

1 n,..., ‘ so that the formula 1 n... decomposes gradually to sub forms

and this recurrent process ends when subforms contain only atomic variables or their negation (called literals), see Fig. 1.

A disjunkcia ( ) B konjunkcia ( ) C implik( )ácia

D ekvivalencia( )

Figure 1.7. There are three primary partitions for creating semantic form. If the subform has form or

, use the De Morgan relations to their equivalent transcript on the transcript

respectively on for already existing diagrammatic decompositions. Diagram D

represents the decomposition of equivalence, which was used in the definition formula

def , the implications were distributed by diagram C and thus two closed

branches formed were removed as irrelevant.

Using the method of semantic task form will resolve a given propositional formula model, i.e.

there is at least one interpretation such that 1val . This task is generally solvable with a

tabular methods (see Table 7.2.), but the method of semantic panes is a very effective alternative

to the tabular method, where we investigate 2n lines (if the formula have n atomic propositional

variables). The method of semantic panes is based on the gradual transformation of the

diagrammatic formula which is equivalent to DNF [7] form DNF (disjunctive normal form,

which contains the disjunction conjunction literals).

Example 7.4. Rewrite the formula p q r r p to equivalent DNF form.

1 2 30 0 1 111 11 0

DNF

, , , , , ,

p q r p p r q q r q p r

p q r p p p r p q q r p

q p r p

p q r q p r q p r

We see that the adjusted DNF formula has clauses that are false (conjunct contains a variable and its negation), which are marked ´ ´. For the remaining clauses, which are marked ´ ´ there is

always interpretation , for which they are true (see Fig. 2 and Tab. 4).

Table 7.4. Table of truth values of formulas p q r r p

# P Q r p q r p (r p) r (r p)

1 0 0 0 1 1 0 0 0

2 0 0 1 1 1 0 1 1=(0,0,1)

3 0 1 0 0 1 0 0 0

4 0 1 1 0 1 0 1 0

5 1 0 0 0 0 1 1 0

6 1 0 1 0 1 0 1 0

7 1 1 0 1 0 1 1 3=(1,1,0)

8 1 1 1 1 1 0 1 2=(1,1,1)

In example 4, the formula p q r r p was rewritten in the form of DNF (disjunctive

normal form, which contains disjunction conjunctive clauses). The clauses of the DNF form formulas are

clearly identifiable with branches of semantic table. Note that in the event that a branch at an early stage

comprises a pair of atomic variable and its negation may be closed prematurely because further extension

of this bond can no longer change the truth value. The importance of semantic pane is based on the

following sentence:

Formula is a tautological consequence of formulas 1 2 n, ,..., , 1 2 n, ,..., , if and only if

the semantic tableau for an formula constructed 1 2 n, ,..., , contains only closed branches.

This theorem allows us to verify whether the formula is a tautological consequence of the theory with

the model 1 2 n, ,..., . Significantly interesting problem is solving session 1 ?n,..., ‘ , i.e.

finally, we search such that the theory is tautological consequence of. constructs a solution using the

following procedure:

Construct the semantic tableau for a formula of the theory 1 2 n, ,..., , and assume that there is at

least one open branch pane (i.e. the theory is consistent). Then the solution , ‘ , constructs a

conjunction as disjunction of literals to peaks of the open branches.

p q

p q

q p q p

q p q p

r r p ( ) r r p ( )

r

r

p

r

r

p

1 =

(0,0

,1)

2 =

(1,1

,1)

3 =(1

,1,0

)

r p r p

Figure 7.2 Construction of the semantic formula p q r r p table. Branches table

that is marked × is not considered as they represent false clauses. Branches marked , represent the true interpretation of clauses (see Tab. 4). Example 7.5 Using semantic table session we will verify the tautological because

p q,q r p r ‘ . Using Semantic we study table conjunction

p q q r p r , if we can show that the semantic tableau has all branches closed,

then the session p q,q r p r ‘ , see Fig. 3.

Example 7.6 With semantic search pane will conclude from the theory p q, p r , i.e. we

will deal with the session ?p q, p r � . The results are shown in Fig. 4, after the theory

p q, p r has four different interpretations - models for which the premises are true

theory

1 0 ?, ?p ,q r

2 0 ?, 1p ,q r

3 0 1, ?p ,q r

4 ? 1, 1p ,q r

where the symbol '?' means that the truth value of the variable is not specified (that it can be arbitrary). Each model is assigned to a solution that is true

1 p

2 p r

3 p q

4 q r

This means that we have four solutions of ?p q, p r � in the form

ip q, p r � for i = 1, 2, 3, 4. These solutions can be "composed" by the disjunction

of the new solution, disjunction for all i we get a solution that is true for all models of the theory

p q, p r

1

1

p p r p q q r

p r q q r

p q r p q r

Then it holds p q, p r p q r ‘ . Note that this solution is easily constructed and by natural

deduction. When a set of assumptions is extended by an additional assumption p, then

p q, p r, p q r � .

Figure 7.3. Semantic tableau for p q q r p r (see Example 5), each branch is

closed, then the session is tautological because p q,q r p r ‘ is valid.

Example 7.7 Using semantic panes we will find a solution, resulting from p q,q r , i.e.

we will deal with the session ?p q,q r � . The results are shown in Fig. 5

The semantic pane that opened three branches produces interpretations (models).

1 0 0, ?p ,q r

2 0 ?, 1p ,q r

3 ? 1, 1p ,q r

Each model can be attributed to a solution that is true

1 p q

2 p r

3 q r

If these three independent solutions are connected by disjunction

1 1

p q p r q r p q r r p q p r p r

We demonstrated that this is true p q,q r p r � which is nothing more than a

hypothetical syllogism.

Figure 7.4. Semantic tableau for theory p q, p r

Figure 7.5. Semantic tableau for theory p q, p r

Finally, we can conclude that the semantic table provides a simple and effective means to control the relationship "tautological entailment”, ‘ , without having to know a relatively complex

syntactic theory of evidence “logical entailment ". Also, semantic tableau is a suitable technique

for solving session tautological entailment 1 ?n,..., ‘ , using the "particulate" solutions for each

open branch and we get a solution that is a tautological consequence of the theory 1 n,..., .

7.7 Conclusion Development of mental models (mental modelling) in cognitive science is among elementary cognitive activities in modelling activities. The concept of mental modelling provides cognitive science leading features of experimental science. Using the theoretical argumentation system (mostly taken from artificial intelligence) interprets experimental measurements of cognitive science (cognitive psychology) in a manner which closely resembles the relationship between experiment and theory in science. The mental model for the cognitive activity acts as its theory, the criterion validity and effectiveness of the theory - the mental model is the agreement of predictions with experimental observations, the results of observations of Cognitive Psychology. It is required that mental models are able to affect not only the right solutions but also to interpret the error in the construction of solutions. These models have a certain complexity, which is variable, when applied to various problems (e.g. different categorical syllogisms) [6], and we expect that the complexity of the model correlates with the occurrence of errors. In this article we pay attention to the construction of mental models of simple propositional logic. Their design uses two different approaches: (1) Syntactic approach, where the mental model is identical to the natural deduction system of Gentzen, which contains about a dozen elementary rules of reasoning and which are identical to the known laws of logic (modus ponens and modus tollens, hypothetical syllogism, inversion rule and the implications.). This mental model is applied to users who are trained in formal logic, at least on a level they have the basic education in logical introduction to propositional logic. (2) The semantic approach, where the mental model is based on semantic tablets. This approach requires the user only the ability to correctly (semantically) interpret elementary logical connectives, according to Inhelderovej and Piaget like children aged around 10 years. We therefore conclude that the semantic approach to the construction of a mental model of logic is applied to users who have not completed training in propositional logic, the only cognitive ability to correctly answer is (semantically) interpreted from elementary logical connectives. Of course, this approach is still has problem with the correct interpretation of the implications that can be a source of error (experimentally observed) in reasoning.

Questions

Question 1.7 What is specified mental model in cognitive science?

It is important to have a model that is able not only find the right solution but also to interpret the error in

the constructed solution. Mental model theory of reasoning use arguments together with general

knowledge to construct mental models. It consists of two premises and a conclusion.

There are two approaches to create a mental model in cognitive science:

Syntactic approach and semantic approach.

Question 7.2 What is specified syntactic approach to creating a mental model of reasoning?

It is based on Gentzen's natural deduction which contains about dozen rules of inference based on the

elementary laws of logic. The set of premises 1 2 n, ,..., using the rules of natural deduction

derive a conclusion . This method is more suitable for people who have some kind of basic training in

logic.

Question 7.3 What is specified as a semantic approach to creating a mental model of reasoning?

The mental model is identified with semantic model, a set of premises and tautological result -

written � . It is based on graphical method called semantic table. To construct the model necessary is

to know Boolean operators.

Question 4.7 What is specified logic language (syntax of propositional logic)?

Syntax - method of the structure of language theory. This language consists propositional logic formulas,

which are defined by a set of variables {p,q,...p,q,...} and set of Boolean operators , , , , .

Syntax is usually associated with the rules (or grammar) governing the composition of texts in a formal

language that constitute the well-formed formulas of a formal system.

Question 5.7 What are specified truth values of formulas of propositional logic language (semantics of

propositional logic)?

The symbol '1' represents the truth value 'true' and the symbol '0'represents the truth value 'false'.

(1) Conjunction is true if and only if (iff) both its components are true, otherwise it is false.

(2) Disjunction is true iff at least one component of it is true, otherwise it is false.

(3) Implication is true iff the first component () is false or the second component () is

true, otherwise (their first component is true and the second component is false) is false.

(4) Equivalence is true iff both components have the same truth value,

otherwise it is false.

(5) Negation is true iff the component is false, otherwise it is false.

Question 6.7 What is specified by the theory?

Question 7.7 What is defined by the tautological notion result theory?

Question 7.8 Using the natural conclusion of inference, construct the following two assumptions:

(3) When it rains, then I will go to the movies

(4) When it rains, then I will go to café

1. p q (1. premise 1)

2. p r (2. premise 2)

3. p (activation of the auxiliary premise )

4. q (using the rule 4 modus ponens on the premises 1 and 3)

5. r (using the rule 4 modus ponens on the premises 2 and 3)

6. q r (introduction of conjunction of implication 4 and 5)

7. p q r (deactivation of the auxiliary premise 3 with a result of 6)

Conclusion: If it rains I will go to the movies and to the café.

Question 7.9 Using the natural conclusion of inference, construct the following two assumptions:

(5) When it rains, then I will go to the movies

(6) When it snows, then I will go to the movies

P= ’it will rain’, q= ’go to the movies’, r= ’it will snow’

1.

2.

3.

4.

Conclusion: If I go to the movies then it is raining or snowing.

Question 7.10 How is defined semantic table?

In this method, solving the session 1 n,..., ‘ so that the formula 1 n...

decomposes gradually to sub forms and this recurrent process ends when subforms contain only atomic

variables or their negation (called literals). It adds some useful computational features:

• The expansion structure is represented using a tree, rather than a sequence of formulas.

• The expansion may be halted upon finding a satisfiable disjunction.

Question 7.11 Copy the formula p q p r r into an equivalent DNF form.

p q r p≡q p

0 0 0 1 1 0 0

0 0 1 1 1 1 1 1=(0,0,1)

0 1 0 0 1 0 0

0 1 1 0 1 1 0

1 0 0 0 0 1 0

1 0 1 0 1 1 0

1 1 0 1 0 1 1 3=(1,1,0)

1 1 1 1 1 1 1 2=(1,1,1)

Question 7.12 Using semantic table authenticate session tautological consequence

p q,q r p r ‘ .

Question 7.13 Using semantic construct table conclusion of the theory p q, p r , i.e. we will

deal with the session ?p q, p r � .

The theory p q, p r has four different interpretations - models for which the premises

are true theory

1 0 ?, ?p ,q r

2 0 ?, 1p ,q r

3 0 1, ?p ,q r

4 ? 1, 1p ,q r

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