Logical Systems

Question 1

Consistency

Answer 1

No theorem of that system contradicts another. Or a set of sentences is logically consistent if and only if it its possible for all the members of that set to be true.

Question 2

Semantics

Answer 2

Interpretations of formal and natural languages usually trying to capture the pre-theoretic notion of entailment

Question 3

Entailment

Answer 3

also, implication, material conditional are logical consequences. It is the relationship between statements that hold true when one statement logically follows from one or more statements. The concluding sentence must be true if every sentence in the set is true

Question 4

Inductive Strength

Answer 4

an argument has inductive strength to the extent that the conclusion is probable given the premises

Question 5

Logically Indeterminate

Answer 5

if and only if it is neither logically true nor logically false

Question 6

Equivalence

Answer 6

iff it is not possible for one of the sentences to be true while the other sentence is false. Equivalent sets of sentences can be all false or all true.

Question 7

Truth Functionally True

Answer 7

iff P is true on every truth-value assignment

Question 8

Truth Functionally False

Answer 8

iff p is false on every truth value assignment

Question 9

Truth Functionally Indeterminate

Answer 9

iff p is neither truth functionally true nor truth functionally false

Question 10

Truth Functional Equivalence

Answer 10

P and Q are equivalent iff there is no truth value assignment on which P and Q have different truth-values

Question 11

Truth Functional Consistency

Answer 11

A set of sentences is consistent iff there is at least one truth value assignment on which all the members of the set are true

Question 12

Quantificationally Truth

Answer 12

A sentence P of Pl is Quant. true iff P is true on every interpretation. A sentence P of PL is Quant. True iff (-P) has a closed truth-tree

Question 13

Quantificationally False

Answer 13

A sentence P of PL is Quant. false iff P is false on every interpretation. A sentence P of PL is quant. false iff (P) has a closed truth tree

Question 14

Quantificationally Indeterminate

Answer 14

A sentence P of PL is quant. indeterminate iff P is neither quant. true nor quant. false.

Question 15

Quantificationally Equivalent

Answer 15

Sentences P and Q of PL are quant. equivalent iff there is no interpretation on which P and Q have different truth-values. Sentences P and Q of PL are quant. equivalent iff both set (P, -Q) and (Q, -P) have closed truth trees

Question 16

Quantificationally Valid

Answer 16

An argument of PL is quant. valid iff there is no interpretation on which every premise is true and the conclusion is false. An argument of PL from premises to conclusion is quant. valid iff (-C) has a closed truth tree.

Question 17

Quantificational Decomposing Strategy

Answer 17

1. Decompose the existentially quantified sentence first
2. Decompose sentences whose decomposition does not require branching (negation of quantified sentences, Conj. Decomp, Neg. Disjunct. Decomp, DD Neg.
3. Give Priority to decomposing sentences whose results in the closing of one or more branches
4. Stop when the tree yields an answer
5. Lastly, decompose the more complex sentences

Question 18

Negated Universal Decomposition (-AD)

Answer 18

nonbranching rule. (Ex)-P

“It is not the case that each thing is such and such” is equivalent to “something is not such and such”

Question 19

Negated Existential Decomposition (-ED)

Answer 19

nonbranching rule. -(Ex)P = (Ax)-P

“It is not the case that something is such and such” is equivalent to “each thing is such that it is not such and such”

Question 20

Universal Decomposition (AD)

Answer 20

Select a substitution instance in which the constant is already occurring on the open branch in question.

Question 21

Literal

Answer 21

a sentence that is either an atomic sentence or negation of an atomic sentence

Question 22

Compound Sentences

Answer 22

Sentences generated from other sentences by means of sentential connectives are compound sentences

Question 23

Existential Decomposition (ED)

Answer 23

(Ex)P
P(a/x)
*a is any individual constant of PL that is foreign to the branch

Question 24

Universal Elimination

Answer 24

Also, instantiation. The inference from a universal claim to a specific instance

Question 25

Existential Introduction

Answer 25

Also, Existential Generalization

Question 26

Universal Introduction

Answer 26

Also, Universal Generalization. introduces a universal quantified claim from a substitution instance. For proper generalization, an arbitrary constant must be introduced