Formal Logic/Print version

Preliminaries

Sets

Presentations of logic vary in how much set theory they use. Some are heavily laden with set theory. Though most are not, it is nearly impossible to avoid it completely. It will not be a very important focal point for this book, but we will use a little set theory vocabulary here and there. This section introduces the vocabulary and notation used.

Sets and elements

Mathematicians use 'set' as an undefined primitive term. Some authors resort to quasi-synonyms such as 'collection'.

A set has elements. 'Element' is also undefined in set theory. We say that an element is a member of a set, also an undefined expression. The following are all used synonymously:

x is a member of y

x is contained in y

x is included in y

y contains x

y includes x

Notation

A set can be specified by enclosing its members within curly braces.

$\{1,2,3\}\,\!$

is the set containing 1, 2, and 3 as members. The curly brace notation can be extended to specify a set using a rule for membership.

$\{x:x=1{\text{ or }}x=2{\text{ or }}x=3\}\,\!$ (The set of all x such that x = 1 or x = 2 or x = 3)

is again the set containing 1, 2, and 3 as members.

$\{x:x{\text{ is a positive integer}}\}\,\!$, and

$\{1,2,3,\ldots \}\,\!$

both specify the set containing 1, 2, 3, and onwards.

A modified epsilon is used to denote set membership. Thus

$x\in y\,\!$

indicates that "x is a member of y". We can also say that "x is not a member of y" in this way:

$x\notin y\,\!$

Characteristics of sets

A set is uniquely identified by its members. The expressions

$\{x:x\ \mathrm {is\ an\ even\ prime} \}\,\!$

$\{x:x\ \mathrm {is\ a\ positive\ square\ root\ of} \ 4\}\,\!$

$\{2\}\,\!$

all specify the same set even though the concept of an even prime is different from the concept of a positive square root. Repetition of members is inconsequential in specifying a set. The expressions

$\{1,\ 2,\ 3\}\,\!$

$\{1,\ 1,\ 1,\ 1,\ 2,\ 3\}\,\!$

$\{x:\mathrm {x\ is\ an\ even\ prime\ or} \ x\ \mathrm {is\ a\ positive\ square\ root\ of} \ 4\ \mathrm {or} \ x=1\ \mathrm {or} \ x=2\ \mathrm {or} \ x=3\}\,\!$

all specify the same set.

Sets are unordered. The expressions

$\{1,\ 2,\ 3\}\,\!$

$\{3,\ 2,\ 1\}\,\!$

$\{2,\ 1,\ 3\}\,\!$

all specify the same set.

Sets can have other sets as members. There is, for example, the set

$\{\{1,\ 2\},\{2,\ 3\},\ \{1,\ \mathrm {George\ Washington} \}\}\,\!$

Some special sets

As stated above, sets are defined by their members. Some sets, however, are given names to ease referencing them.

The set with no members is the empty set. The expressions

$\{\}\,\!$

$\varnothing \,\!$

$\{x:x\neq x\}\,\!$

all specify the empty set. Empty sets can also express oxymora ("four-sided triangles" or "birds with radial symmetry") and factual non-existence ("the King of Czechoslovakia in 1994").

A set with exactly one member is called a singleton. A set with exactly two members is called a pair. Thus {1} is a singleton and {1, 2} is a pair.

ω is the set of natural numbers, {0, 1, 2, ...}.

Subsets, power sets, set operations

Subsets

A set s is a subset of set a if every member of s is a member of a. We use the horseshoe notation to indicate subsets. The expression

$\{1,\ 2\}\subseteq \{1,\ 2,\ 3\}\,\!$

says that {1, 2} is a subset of {1, 2, 3}. The empty set is a subset of every set. Every set is a subset of itself. A proper subset of a is a subset of a that is not identical to a. The expression

$\{1,\ 2\}\subset \{1,\ 2,\ 3\}\,\!$

says that {1, 2} is a proper subset of {1, 2, 3}.

Power sets

A power set of a set is the set of all its subsets. A script 'P' is used for the power set.

${\mathcal {P}}\{1,\ 2,\ 3\}=\{\varnothing ,\ \{1\},\ \{2\},\ \{3\},\ \{1,\ 2\},\ \{1,\ 3\},\ \{2,\ 3\},\ \{1,\ 2,\ 3\}\}\,\!$

Union

The union of two sets a and b, written a ∪ b, is the set that contains all the members of a and all the members of b (and nothing else). That is,

$a\cup b=\{x:x\in a\ \mathrm {or} \ x\in b\}\,\!$

As an example,

$\{1,\ 2,\ 3\}\ \cup \ \{2,\ 3,\ 4\}=\{1,\ 2,\ 3,\ 4\}\,\!$

Intersection

The intersection of two sets a and b, written a ∩ b, is the set that contains everything that is a member of both a and b (and nothing else). That is,

$a\cap b=\{x:x\in a\ \mathrm {and} \ x\in b\}\,\!$

As an example,

$\{1,\ 2,\ 3\}\ \cap \ \{2,\ 3,\ 4\}=\{2,\ 3\}\,\!$

Relative complement

The relative complement of a in b, written b \ a (or b − a) is the set containing all the members of b that are not members of a. That is,

$b\setminus a=\{x:x\in b\ \mathrm {and} \ x\notin a\}\,\!$

As an example,

$\{2,\ 3,\ 4\}\setminus \{1,\ 3\}=\{2,\ 4\}\,\!$

Ordered sets, relations, and functions

The intuitive notions of ordered set, relation, and function will be used from time to time. For our purposes, the intuitive mathematical notion is the most important. However, these intuitive notions can be defined in terms of sets.

Ordered sets

First, we look at ordered sets. We said that sets are unordered:

$\{a,\ b\}=\{b,\ a\}\,\!$

But we can define ordered sets, starting with ordered pairs. The angle bracket notation is used for this:

$\langle a,\ b\rangle \ \neq \ \langle b,\ a\rangle \,\!$

Indeed,

$\langle x,\ y\rangle \ =\ \langle u,\ v\rangle \ {\mbox{if and only if}}\ x=u\ {\mbox{and}}\ y=v\,\!$

Any set theoretic definition giving ⟨a, b⟩ this last property will work. The standard definition of the ordered pair ⟨a, b⟩ runs:

$\langle a,\ b\rangle \ =\ \{\{a\},\ \{a,\ b\}\}\,\!$

This means that we can use the latter notation when doing operations on an ordered pair.

There are also bigger ordered sets. The ordered triple ⟨a, b, c⟩ is the ordered pair ⟨⟨a, b⟩, c⟩. The ordered quadruple ⟨a, b, c, d⟩ is the ordered pair ⟨⟨a, b, c⟩, d⟩. This, in turn, is the ordered triple ⟨⟨⟨a, b⟩, c⟩, d⟩. In general, an ordered n-tuple ⟨a₁, a₂, ..., a_n⟩ where n greater than 1 is the ordered pair ⟨⟨a₁, a₂, ..., a_n-1⟩, a_n⟩.

It can be useful to define an ordered 1-tuple as well: ⟨a⟩ = a.

These definitions are somewhat arbitrary, but it is nonetheless convenient for an n-tuple, n ⟩ 2, to be an n-1 tuple and indeed an ordered pair. The important property that makes them serve as ordered sets is:

$\langle x_{1},\ x_{2},\ ...,\ x_{n}\rangle \ =\ \langle y_{1},\ y_{2},\ ...,\ y_{n}\rangle \ \mathrm {if\ and\ only\ if} \ x_{1}=y_{1},\ x_{2}=y_{2},\ ...,\ x_{n}=y_{n}\,\!$

Relations

We now turn to relations. Intuitively, the following are relations:

x < y

x is a square root of y

x is a brother of y

x is between y and z

The first three are binary or 2-place relations; the fourth is a ternary or 3-place relation. In general, we talk about n-ary relations or n-place relations.

First consider binary relations. A binary relation is a set of ordered pairs. The less than relation would have among its members ⟨1, 2⟩, ⟨1, 3⟩, ⟨16, 127⟩, etc. Indeed, the less than relation defined on the natural numbers ω is:

$\{\langle x,\ y\rangle :x\in \omega ,\ y\in \omega ,\ \mathrm {and} \ x<y\}\,\!$

Intuitively, ⟨x, y⟩ is a member of the less than relation if x < y. In set theory, we do not worry about whether a relation matches an intuitive concept such as less than. Rather, any set of ordered pairs is a binary relation.

We can also define a 3-place relation as a set of 3-tuples, a 4-place relation as a set of 4-tuples, etc. We only define n-place relations for n ≥ 2. An n-place relation is said to have an arity of n. The following example is a 3-place relation.

$\{\langle 1,\ 2,\ 3\rangle ,\ \langle 8,\ 2,\ 1\rangle ,\ \langle 653,\ 0,\ 927\rangle \}\,\!$

Because all n-tuples where n > 1 are also ordered pairs, all n-place relations are also binary relations.

Functions

Finally, we turn to functions. Intuitively, a function is an assignment of values to arguments such that each argument is assigned at most one value. Thus the + 2 function assigns a numerical argument x the value x + 2. Calling this function f, we say f(x) = x + 2. The following define specific functions.

$f_{1}(x)=x\times x\,\!$

$f_{2}(x)=\ \mathrm {the\ smallest\ prime\ number\ larger\ than} \ x\,\!$

$f_{3}(x)=6/x\ \!$

$f_{4}(x)=\ \mathrm {the\ father\ of\ x} \,\!$

Note that f₃ is undefined when x = 0. According to biblical tradition, f₄ is undefined when x = Adam or x = Eve. The following do not define functions.

$f_{5}(x)=\pm {\sqrt {x}}\,\!$

$f_{6}(x)=\ \mathrm {a\ son\ of\ x} \,\!$

Neither of these assigns unique values to arguments. For every positive x, there are two square roots, one positive and one negative, so f₅ is not a function. For many x, x will have multiple sons, so f₆ is not a function. If f₆ is assigned the value the son of x then a unique value is implied by the rules of language, therefore f₆ will be a function.

A function f is a binary relation where, if ⟨x, y⟩ and ⟨x, z⟩ are both members of f, then y = z.

We can define many place functions. Intuitively, the following are definitions of specific many place functions.

$f_{7}(x,\ y)=x+y\,\!$

$f_{8}(x,\ y,\ z)=(x+y)\times z\,\!$

Thus ⟨4, 7, 11⟩ is a member of the 2-place function f₇. ⟨3, 4, 5, 35⟩ is a member of the 3 place function f₈

The fact that all n-tuples, n ≥ 2, are ordered pairs (and hence that all n-ary relations are binary relations) becomes convenient here. For n ≥ 1, an n-place function is an n+1 place relation that is a 1-place function. Thus, for a 2-place function f,

$\langle x,\ y,\ z_{1}\rangle \ \in f\ \mathrm {and} \ \langle x,\ y,\ z_{2}\rangle \ \in f\ \mathrm {if\ and\ only\ if} \ z_{1}=z_{2}\,\!$

Sentential Logic

Sentential Logic

Goals

Sentential logic

Sentential logic attempts to capture certain logical features of natural languages. In particular, it covers truth-functional connections for sentences. Its formal language specifically recognizes the sentential connections

It is not the case that _____

_____ and _____

Either _____ or _____

_____ or _____ (or both)

if _____, then _____

_____ if and only if _____

The blanks are to be filled with statements that can be true or false. For example, "it is raining today" or "it will snow tomorrow". Whether the final sentence is true or false is entirely determined on whether the filled statements are true or false. For example, if it is raining today, but it will not snow tomorrow, then it is true to say that "Either it is raining today or it will snow tomorrow". On the other hand, it is false to say "it is raining today and it will snow tomorrow", since it won't snow tomorrow.

"Whether a statement is true or false" is called the truth value in logician slang. Thus "Either it is raining today or it is not raining today" has a truth value of true and "it is raining today and it is not raining today" has truth value of false.

Note that the above listed sentential connections do not include all possible truth value combinations. For example, there is no connection that is true when both sub-statements are true, both sub-statements are false or the first sub-statement is true while the other is false, and that is false else. However, you can combine the above connections together to build any truth combination of any number of sub-statements.

Issues

Already we have tacitly taken a position in ongoing controversy. Some questions already raised by the seemingly innocuous beginning above are listed.

• Should we admit into our logic only sentences that are true or false? Multi-valued logics admit a greater range of sentences.

• Are the connections listed above truly truth functional? Should we admit connections that are not truth functional sentences into our logic?

• What should logic take as its truth-bearers (objects that are true or false)? The two leading contenders today are sentences and propositions.

• Sentences. These consist of a string of words and perhaps punctuation. The sentence 'The cat is on the mat' consists of six elements: 'the', 'cat', 'is', 'on', another 'the', and 'mat'.

• Propositions. These are the meanings of sentences. They are what is expressed by a sentence or what someone says when he utters a sentence. The proposition that the cat is on the mat consists of three elements: a cat, a mat, and the on-ness relation.

Elsewhere in Wikibooks and Wikipedia, you will see the name 'Propositional Logic' (or rather 'Propositional Calculus', see below) and the treatment of propositions much more often than you will see the name 'Sentential Logic' and the treatment of sentences. Our choice here represents the contributor's view as to which position is more popular among current logicians and what you are most likely to see in standard textbooks on the subject. Considerations as to whether the popular view is actually correct are not taken up here.

Some authors will use talk about statements instead of sentences. Most (but not all) such authors you are likely to encounter take statements to be a subset of sentences, namely those sentences that are either true or false. This use of 'statement' does not represent a third position in the controversy, but rather places such authors in the sentences camp. (However, other—particularly older—uses of 'statement' may well place its authors in a third camp.)

Sometimes you will see 'calculus' rather than 'logic' such as in 'Sentential Calculus' or 'Propositional Calculus' as opposed to 'Sentential Logic' or 'Propositional Logic'. While the choice between 'sentential' and 'propositional' is substantive and philosophical, the choice between 'logic' and 'calculus' is merely stylistic.

The Sentential Language

This page informally describes our sentential language which we name ${\mathcal {L_{S}}}\,\!$. A more formal description will be given in Formal Syntax and Formal Semantics

Language components

Sentence letters

Sentences in ${\mathcal {L_{S}}}\,\!$ are represented as sentence letters, which are single letters such as $\mathrm {P} ,\ \mathrm {Q} ,\ \mathrm {R} ,$ and so on. Some texts restrict these to lower case letters, and others restrict them to capital letters. We will use capital letters.

Intuitively, we can think of sentence letters as English sentences that are either true or false. Thus, $\mathrm {P} \,\!$ may translate as 'The Earth is a planet' (which is true), or 'The moon is made of green cheese' (which is false). But $\mathrm {P} \,\!$ may not translate as 'Great ideas sleep furiously' because such a sentence is neither true nor false. Translations between English and ${\mathcal {L_{S}}}\,\!$ work best if they are restricted to timelessly true or false present tense sentences in the indicative mood. You will see in the translation section below that we do not always follow that advice, wherein we present sentences whose truth or falsity is not timeless.

Sentential connectives

Sentential connectives are special symbols in Sentential Logic that represent truth functional relations. They are used to build larger sentences from smaller sentences. The truth or falsity of the larger sentence can then be computed from the truth or falsity of the smaller ones.

${\mbox{Conjunction:}}\ \land \,\!$

• Translates to English as 'and'.
• $\mathrm {P} \land \mathrm {Q} \,\!$ is called a conjunction and $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are its conjuncts.
• $\mathrm {P} \land \mathrm {Q} \,\!$ is true if both $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are true—and is false otherwise.
• Some authors use an & (ampersand), • (heavy dot) or juxtaposition. In the last case, an author would write

$\mathrm {PQ} \,\!$

instead of our

$\mathrm {P} \land \mathrm {Q} \ .\,\!$

${\mbox{Disjunction:}}\ \lor \,\!$

• Translates to English as 'or'.
• $\mathrm {P} \lor \mathrm {Q} \,\!$ is called a disjunction and $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are its disjuncts.
• $\mathrm {P} \lor \mathrm {Q} \,\!$ is true if at least one of $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are true—is false otherwise.
• Some authors may use a vertical stroke: |. However, this comes from computer languages rather than logicians' usage. Logicians normally reserve the vertical stroke for nand (alternative denial). When used as nand, it is called the Sheffer stroke.

${\mbox{Negation:}}\ \lnot \,\!$

• Translates to English as 'it is not the case that' but is normally read 'not'.
• $\lnot \mathrm {P} \,\!$ is called a negation.
• $\lnot \mathrm {P} \,\!$ is true if $\mathrm {P} \,\!$ is false—and is false otherwise.
• Some authors use ~ (tilde) or −. Some authors use an overline, for example writing

${\bar {\mathrm {P} }}\ \ {\mbox{and}}\ \ ({\overline {(\mathrm {P} \land \mathrm {Q} )}}\lor \mathrm {R} )\,\!$

instead of

$\lnot \mathrm {P} \ \ {\mbox{and}}\ \ (\lnot (\mathrm {P} \land \mathrm {Q} )\lor \mathrm {R} )\ .\,\!$

${\mbox{Conditional:}}\ \rightarrow \,\!$

• Translates to English as 'if...then' but is often read 'arrow'.
• $\mathrm {P} \rightarrow \mathrm {Q} \,\!$ is called a conditional. Its antecedent is $\mathrm {P} \,\!$ and its consequent is $\mathrm {Q} \,\!$.
• $\mathrm {P} \rightarrow \mathrm {Q} \,\!$ is false if $\mathrm {P} \,\!$ is true and $\mathrm {Q} \,\!$ is false—and true otherwise.
• By that definition, $\mathrm {P} \rightarrow \mathrm {Q} \,\!$ is equivalent to $(\lnot \mathrm {P} )\lor \mathrm {Q} \,\!$
• Some authors use ⊃ (hook).

${\mbox{Biconditional:}}\ \leftrightarrow \,\!$

• Translates to English as 'if and only if'
• $\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$ is called a biconditional.
• $\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$ is true if $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ both are true or both are false—and false otherwise.
• By that definition, $\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$ is equivalent to the more verbose $(\mathrm {P} \land \mathrm {Q} )\lor ((\lnot \mathrm {P} )\land (\lnot \mathrm {Q} ))\,\!$. It is also equivalent to $(\mathrm {P} \rightarrow \mathrm {Q} )\land (\mathrm {Q} \rightarrow \mathrm {P} )\,\!$, the conjunction of two conditionals where in the second conditional the antecedent and consequent are reversed from the first.
• Some authors use ≡.

Grouping

Parentheses $(\,\!$ and $)\,\!$ are used for grouping. Thus

$((\mathrm {P} \land \mathrm {Q} )\rightarrow \mathrm {R} )\,\!$

$(\mathrm {P} \land (\mathrm {Q} \rightarrow \mathrm {R} ))\,\!$

are two different and distinct sentences. Each negation, conjunction, disjunction, conditional, and biconditionals gets a single pair or parentheses.

Notes

(1) An atomic sentence is a sentence consisting of just a single sentence letter. A molecular sentence is a sentence with at least one sentential connective. The main connective of a molecular formula is the connective that governs the entire sentence. Atomic sentences, of course, do not have a main connective.

(2) The ⊃ and ≡ signs for conditional and biconditional are historically older, perhaps a bit more traditional, and definitely occur more commonly in WikiBooks and Wikipedia than our arrow and double arrow. They originate with Alfred North Whitehead and Bertrand Russell in Principia Mathematica. Our arrow and double arrow appear to originate with Alfred Tarski, and may be a bit more popular today than the Whitehead and Russell's ⊃ and ≡.

(3) Sometimes you will see people reading our arrow as implies. This is fairly common in WikiBooks and Wikipedia. However, most logicians prefer to reserve 'implies' for metalinguistic use. They will say:

If P then Q

or even

P arrow Q

They approve of:

'P' implies 'Q'

but will frown on:

P implies Q

Translation

Consider the following English sentences:

If it is raining and Jones is out walking, then Jones has an umbrella.

If it is Tuesday or it is Wednesday, then Jones is out walking.

To render these in ${\mathcal {L_{S}}}\,\!$, we first specify an appropriate English translation for some sentence letters.

$\mathrm {P} \ :\,\!$ It is raining.

$\mathrm {Q} \ :\,\!$ Jones is out walking.

$\mathrm {R} \ :\,\!$ Jones has an umbrella.

$\mathrm {S} \ :\,\!$ It is Tuesday.

$\mathrm {T} \ :\,\!$ It is Wednesday.

We can now partially translate our examples as:

${\mbox{If}}\ \mathrm {P} \ {\mbox{and}}\ \mathrm {Q} ,\ {\mbox{then}}\ \mathrm {R} \,\!$

${\mbox{If}}\ \mathrm {S} \ {\mbox{or}}\ \mathrm {T} ,\ {\mbox{then}}\ \mathrm {Q} \,\!$

Then finish the translation by adding the sentential connectives and parentheses:

$((\mathrm {P} \land \mathrm {Q} )\rightarrow \mathrm {R} )\,\!$

$((\mathrm {S} \lor \mathrm {T} )\rightarrow \mathrm {Q} )\,\!$

Quoting convention

For English expressions, we follow the logical tradition of using single quotes. This allows us to use ' 'It is raining' ' as a quotation of 'It is raining'.

For expressions in ${\mathcal {L_{S}}}\,\!$, it is easier to treat them as self-quoting so that the quotation marks are implicit. Thus we say that the above example translates $\mathrm {S} \rightarrow \mathrm {P} \,\!$ (note the lack of quotes) as 'If it is Tuesday, then It is raining'.

Formal Syntax

In The Sentential Language, we informally described our sentential language. Here we give its formal syntax or grammar. We will call our language ${\mathcal {L_{S}}}\,\!$.

Vocabulary

• Sentence letters: Capital letters 'A' – 'Z', each with (1) a superscript '0' and (2) a natural number subscript. (The natural numbers are the set of positive integers and zero.) Thus the sentence letters are:

$\mathrm {A_{0}^{0}} ,\ \mathrm {A_{1}^{0}} ,\ ...,\ \mathrm {B_{0}^{0}} ,\ \mathrm {B_{1}^{0}} ,\ ...,\ ...,\ \mathrm {Z_{0}^{0}} ,\ \mathrm {Z_{1}^{0}} ,\ ...\,\!$

• Sentential connectives:

$\land ,\ \lor ,\ \lnot ,\ \rightarrow ,\ \leftrightarrow \,\!$

• Grouping symbols:

$(,\ )\,\!$

The superscripts on sentence letters are not important until we get to the predicate logic, so we won't really worry about those here. The subscripts on sentence letters are to ensure an infinite supply of sentence letters. On the next page, we will abbreviate away most superscripts and subscripts.

Expressions

Any string of characters from the ${\mathcal {L_{S}}}\,\!$ vocabulary is an expression of ${\mathcal {L_{S}}}\,\!$. Some expressions are grammatically correct. Some are as incorrect in ${\mathcal {L_{S}}}\,\!$ as 'Over talks David Mary the' is in English. Still other expressions are as hopelessly ill-formed in ${\mathcal {L_{S}}}\,\!$ as 'jmr.ovn asgj as;lnre' is in English.

We call a grammatically correct expression of ${\mathcal {L_{S}}}\,\!$ a well-formed formula. When we get to Predicate Logic, we will find that only some well formed formulas are sentences. For now though, we consider every well formed formula to be a sentence.

Construction rules

An expression of ${\mathcal {L_{S}}}\,\!$ is called a well-formed formula of ${\mathcal {L_{S}}}\,\!$ if it is constructed according to the following rules.

${\mbox{i.}}$ The expression consists of a single sentence letter

${\mbox{ii.}}$ The expression is constructed from other well-formed formulae $\varphi \,\!$ and $\psi \,\!$ in one of the following ways:

${\mbox{ii-a.}}\ \ \lnot \varphi \,\!$

${\mbox{ii-b.}}\ \ (\varphi \land \psi )\,\!$

${\mbox{ii-c.}}\ \ (\varphi \lor \psi )\,\!$

${\mbox{ii-d.}}\ \ (\varphi \rightarrow \psi )\,\!$

${\mbox{ii-e.}}\ \ (\varphi \leftrightarrow \psi )\,\!$

In general, we will use 'formula' as shorthand for 'well-formed formula'. Since all formulae in ${\mathcal {L_{S}}}\,\!$ are sentences, we will use 'formula' and 'sentence' interchangeably.

Quoting convention

We will take expressions of ${\mathcal {L_{S}}}\,\!$ to be self-quoting and so regard

$(\mathrm {P_{0}^{0}} \rightarrow \mathrm {Q_{0}^{0}} )\,\!$

to include implicit quotation marks. However, something like

$(1)\quad (\varphi \rightarrow \psi )\,\!$

requires special consideration. It is not itself an expression of ${\mathcal {L_{S}}}\,\!$ since $\varphi \,\!$ and $\psi \,\!$ are not in the vocabulary of ${\mathcal {L_{S}}}\,\!$. Rather they are used as variables in English which range over expressions of ${\mathcal {L_{S}}}\,\!$. Such a variable is called a metavariable, and an expression using a mix of vocabulary from ${\mathcal {L_{S}}}\,\!$ and metavariables is called a metalogical expression. Suppose we let $\varphi \,\!$ be $\mathrm {P_{0}^{0}} \,\!$ and $\psi \,\!$ be $(\mathrm {Q_{0}^{0}} \lor \mathrm {R_{0}^{0}} )\ .\,\!$ Then (1) becomes

$(\,\!$'$\mathrm {P_{0}^{0}} \,\!$' $\rightarrow (\,\!$'$\mathrm {Q_{0}^{0}} \,\!$' $\lor \,\!$'$\mathrm {R_{0}^{0}} \,\!$'$))\,\!$

which is not what we want. Instead we take (1) to mean (using explicit quotes):

the expression consisting of '$(\,\!$' followed by $\varphi \,\!$ followed by '$\rightarrow \,\!$' followed by $\psi \,\!$ followed by '$)\,\!$' .

Explicit quotes following this convention are called Quine quotes or corner quotes. Our corner quotes will be implicit.

Additional terminology

We introduce (or, in some cases, repeat) some useful syntactic terminology.

• We distinguish between an expression (or a formula) and an occurrence of an expression (or formula). The formula

$((\mathrm {P_{0}^{0}} \land \mathrm {P_{0}^{0}} )\land \lnot \mathrm {P_{0}^{0}} )\,\!$

is the same formula no matter how many times it is written. However, it contains three occurrences of the sentence letter $\mathrm {P_{0}^{0}} \,\!$ and two occurrences of the sentential connective $\land \,\!$.

• $\psi \,\!$ is a subformula of $\varphi \,\!$ if and only if $\varphi \,\!$ and $\psi \,\!$ are both formulae and $\varphi \,\!$ contains an occurrence of $\psi \,\!$. $\psi \,\!$ is a proper subformula of $\varphi \,\!$ if and only if (i) $\psi \,\!$ is a subformula of $\varphi \,\!$ and (ii) $\psi \,\!$ is not the same formula as $\varphi \,\!$.

• An atomic formula or atomic sentence is one consisting solely of a sentence letter. Or put the other way around, it is a formula with no sentential connectives. A molecular formula or molecular sentence is one which contains at least one occurrence of a sentential connective.

• The main connective of a molecular formula is the last occurrence of a connective added when the formula was constructed according to the rules above.

• A negation is a formula of the form $\lnot \varphi \,\!$ where $\varphi \,\!$ is a formula.

• A conjunction is a formula of the form $(\varphi \land \psi )\,\!$ where $\varphi \,\!$ and $\psi \,\!$ are both formulae. In this case, $\varphi \,\!$ and $\psi \,\!$ are both conjuncts.

• A disjunction is a formula of the form $(\varphi \lor \psi )\,\!$ where $\varphi \,\!$ and $\psi \,\!$ are both formulae. In this case, $\varphi \,\!$ and $\psi \,\!$ are both disjuncts.

• A conditional is a formula of the form $(\varphi \rightarrow \psi )\,\!$ where $\varphi \,\!$ and $\psi \,\!$ are both formulae. In this case, $\varphi \,\!$ is the antecedent, and $\psi \,\!$ is the consequent. The converse of $(\varphi \rightarrow \psi )\,\!$ is $(\psi \rightarrow \varphi )\,\!$. The contrapositive of $(\varphi \rightarrow \psi )\,\!$ is $(\lnot \psi \rightarrow \lnot \varphi )\,\!$.

• A biconditional is a formula of the form $(\varphi \leftrightarrow \psi )\,\!$ where $\varphi \,\!$ and $\psi \,\!$ are both formulae.

Examples

$(1)\quad (\lnot (\mathrm {P_{0}^{0}} \lor \mathrm {Q_{0}^{0}} )\rightarrow (\mathrm {R_{0}^{0}} \land \lnot \mathrm {Q_{0}^{0}} ))\,\!$

By rule (i), all sentence letters, including

$\mathrm {P_{0}^{0}} ,\ \mathrm {Q_{0}^{0}} ,\ {\mbox{and}}\ \mathrm {R_{0}^{0}} \ ,\,\!$

are formulae. By rule (ii-a), then, the negation

$\lnot \mathrm {Q_{0}^{0}} \,\!$

is also a formula. Then by rules (ii-c) and (ii-b), we get the disjunction and conjunction

$(\mathrm {P_{0}^{0}} \lor \mathrm {Q_{0}^{0}} ),\ {\mbox{and}}\ (\mathrm {R_{0}^{0}} \land \lnot \mathrm {Q_{0}^{0}} )\,\!$

as formulae. Applying rule (ii-a) again, we get the negation

$\lnot (\mathrm {P_{0}^{0}} \lor \mathrm {Q_{0}^{0}} )\,\!$

as a formula. Finally, rule (ii-c) generates the conditional of (1), so it too is a formula.

$(2)\quad ((\mathrm {P_{0}^{0}} \lnot \land \mathrm {Q_{0}^{0}} )\lor \mathrm {S_{0}^{0}} )\,\!$

This appears to be generated by rule (ii-c) from

$(\mathrm {P_{0}^{0}} \lnot \land \mathrm {Q_{0}^{0}} ),\ {\mbox{and}}\ \mathrm {S_{0}^{0}} \ .\,\!$

The second of these is a formula by rule (i). But what about the first? It would have to be generated by rule (ii-b) from

$\mathrm {P_{0}^{0}} \lnot \ {\mbox{and}}\ \mathrm {Q_{0}^{0}} \ .\,\!$

But

$\mathrm {P_{0}^{0}} \lnot \,\!$

cannot be generated by rule (ii-a). So (2) is not a formula.

Informal Conventions

In The Sentential Language, we gave an informal description of a sentential language, namely ${\mathcal {L_{S}}}\,\!$. We have also given a Formal Syntax for ${\mathcal {L_{S}}}\,\!$. Our official grammar generates a large number of parentheses. This makes formal definitions and other specifications easier to write, but it makes the language rather cumbersome to use. In addition, all the subscripts and superscripts quickly get to be unnecessarily tedious. The end result is an ugly and difficult to read language.

We will continue to use official grammar for specifying formalities. However, we will informally use a less cumbersome variant for other purposes. The transformation rules below convert official formulae of ${\mathcal {L_{S}}}\,\!$ into our informal variant.

Transformation rules

We create informal variants of official ${\mathcal {L_{S}}}\,\!$ formulae as follows. The examples are cumulative.

• The official grammar required sentence letters to have the superscript '0'. Superscripts aren't necessary or even useful until we get to the predicate logic, so we will always omit them in our informal variant. We will write, for example, $\mathrm {P_{0}} \,\!$ instead of $\mathrm {P_{0}^{0}} \,\!$.

• We will omit the subscript if it is '0'. Thus we will write $\mathrm {P} \,\!$ instead of $\mathrm {P_{0}^{0}} \,\!$. However, we cannot omit all subscripts; we still need to write, for example, $\mathrm {P_{1}} \,\!$.

• We will omit outermost parentheses. For example, we will write

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$

instead of

$(\mathrm {P_{0}^{0}} \rightarrow \mathrm {Q_{0}^{0}} )\ .\,\!$

• We will let a series of the same binary connective associate on the right. For example, we can transform the official

$(\mathrm {P_{0}^{0}} \land (\mathrm {Q_{0}^{0}} \land \mathrm {R_{0}^{0}} ))\,\!$

into the informal

$\mathrm {P} \land \mathrm {Q} \land \mathrm {R} \ .\,\!$

However, the best we can do with

$((\mathrm {P_{0}^{0}} \land \mathrm {Q_{0}^{0}} )\land \mathrm {R^{0}} )\,\!$

is

$(\mathrm {P} \land \mathrm {Q} )\land \mathrm {R} \ .\,\!$

• We will use precedence rankings to omit internal parentheses when possible. For example, we will regard $\rightarrow \,\!$ as having lower precedence (wider scope) than $\lor \,\!$. This allows us to write

$\mathrm {P} \rightarrow \mathrm {Q} \lor \mathrm {R} \,\!$

instead of

$(\mathrm {P_{0}^{0}} \rightarrow (\mathrm {Q_{0}^{0}} \lor \mathrm {R_{0}^{0}} ))\ .\,\!$

However, we cannot remove the internal parentheses from

$((\mathrm {P_{0}^{0}} \rightarrow \mathrm {Q_{0}^{0}} )\lor \mathrm {R_{0}^{0}} )\ .\,\!$

Our informal variant of this latter formula is

$(\mathrm {P} \rightarrow \mathrm {Q} )\lor \mathrm {R} \ .\,\!$

Full precedence rankings are given below.

Precedence and scope

Precedence rankings indicate the order that we evaluate the sentential connectives. $\lor \,\!$ has a higher precedence than $\rightarrow \,\!$. Thus, in calculating the truth value of

$(1)\quad \mathrm {P} \rightarrow \mathrm {Q} \lor \mathrm {R} \ ,\,\!$

we start by evaluating the truth value of

$(2)\quad \mathrm {Q} \lor \mathrm {R} \,\!$

first. Scope is the length of expression that is governed by the connective. The occurrence of $\rightarrow \,\!$ in (1) has a wider scope than the occurrence of $\lor \,\!$. Thus the occurrence of $\rightarrow \,\!$ in (1) governs the whole sentence while the occurrence of $\lor \,\!$ in (1) governs only the occurrence of (2) in (1).

The full ranking from highest precedence (narrowest scope) to lowest precedence (widest scope) is:

	$\lnot \,\!$		highest precedence (narrowest scope)
	$\land \,\!$
	$\lor \,\!$
	$\rightarrow \,\!$
	$\leftrightarrow \,\!$		lowest precedence (widest scope)

Examples

Let's look at some examples. First,

$((\mathrm {P_{0}^{0}} \rightarrow \mathrm {Q_{0}^{0}} )\leftrightarrow ((\mathrm {P_{0}^{0}} \land \mathrm {Q_{0}^{0}} )\lor ((\lnot \mathrm {P_{0}^{0}} \land \mathrm {Q_{0}^{0}} )\lor (\lnot \mathrm {P_{0}^{0}} \land \lnot \mathrm {Q_{0}^{0}} ))))\,\!$

can be written informally as

$\mathrm {P} \rightarrow \mathrm {Q} \leftrightarrow \mathrm {P} \land \mathrm {Q} \lor \lnot \mathrm {P} \land \mathrm {Q} \lor \lnot \mathrm {P} \land \lnot \mathrm {Q} \ .\,\!$

Second,

$((\mathrm {P_{0}^{0}} \leftrightarrow \mathrm {Q_{0}^{0}} )\leftrightarrow ((\mathrm {P_{0}^{0}} \land \mathrm {Q_{0}^{0}} )\lor (\lnot \mathrm {P_{0}^{0}} \land \lnot \mathrm {Q_{0}^{0}} )))\,\!$

can be written informally as

$(\mathrm {P} \leftrightarrow \mathrm {Q} )\leftrightarrow \mathrm {P} \land \mathrm {Q} \lor \lnot \mathrm {P} \land \lnot \mathrm {Q} \ .\,\!$

Some unnecessary parentheses may prove helpful. In the two examples above, the informal variants may be easier to read as

$(\mathrm {P} \rightarrow \mathrm {Q} )\leftrightarrow (\mathrm {P} \land \mathrm {Q} )\lor (\lnot \mathrm {P} \land \mathrm {Q} )\lor (\lnot \mathrm {P} \land \lnot \mathrm {Q} )\,\!$

and

$(\mathrm {P} \leftrightarrow \mathrm {Q} )\leftrightarrow (\mathrm {P} \land \mathrm {Q} )\lor (\lnot \mathrm {P} \land \lnot \mathrm {Q} )\ .\,\!$

Note that the informal formula

$\lnot \mathrm {P} \rightarrow \mathrm {Q} \,\!$

is restored to its official form as

$(\lnot \mathrm {P_{0}^{0}} \rightarrow \mathrm {Q_{0}^{0}} )\ .\,\!$

By contrast, the informal formula

$\lnot (\mathrm {P} \rightarrow \mathrm {Q} )\,\!$

is restored to its official form as

$\lnot (\mathrm {P_{0}^{0}} \rightarrow \mathrm {Q_{0}^{0}} )\ .\,\!$

Formal Semantics

English syntax for 'Dogs bark' specifies that it consists of a plural noun followed by an intransitive verb. English semantics for 'Dogs bark' specify its meaning, namely that dogs bark.

In The Sentential Language, we gave an informal description of ${\mathcal {L_{S}}}\,\!$. We also gave a Formal Syntax. However, at this point our language is just a toy, a collection of symbols we can string together like beads on a necklace. We do have rules for how those symbols are to be ordered. But at this point those might as well be aesthetic rules. The difference between well-formed formulae and ill-formed expressions is not yet any more significant than the difference between pretty and ugly necklaces. In order for our language to have any meaning, to be usable in saying things, we need a formal semantics.

Any given formal language can be paired with any of a number of competing semantic rule sets. The semantics we define here is the usual one for modern logic. However, alternative semantic rule-sets have been proposed. Alternative semantic rule-sets of ${\mathcal {L_{S}}}\,\!$ have included (but are certainly not limited to) intuitionistic logics, relevance logics, non-monotonic logics, and multi-valued logics.

Formal semantics

The formal semantics for a formal language such as ${\mathcal {L_{S}}}\,\!$ goes in two parts.

• Rules for specifying an interpretation. An interpretation assigns semantic values to the non-logical symbols of a formal syntax. The semantics for a formal language will specify what range of values can be assigned to which class of non-logical symbols. ${\mathcal {L_{S}}}\,\!$ has only one class of non-logical symbols, so the rule here is particularly simple. An interpretation for a sentential language is a valuation, namely an assignment of truth values to sentence letters. In predicate logic, we will encounter interpretations that include other elements in addition to a valuation.

• Rules for assigning semantic values to larger expressions of the language. For sentential logic, these rules assign a truth value to larger formulae based on truth values assigned to smaller formulae. For more complex syntaxes (such as for predicate logic), values are assigned in a more complex fashion.

An extended valuation assigns truth values to the molecular formulae of ${\mathcal {L_{S}}}\,\!$ (or similar sentential language) based on a valuation. A valuation for sentence letters is extended by a set of rules to cover all formulae.

Valuations

We can give a (partial) valuation ${\mathfrak {v}}\,\!$ as:

$\mathrm {P_{0}} \ :\ {\mbox{True}}\,\!$

$\mathrm {P_{1}} \ :\ {\mbox{False}}\,\!$

$\mathrm {P_{2}} \ :\ {\mbox{False}}\,\!$

$\mathrm {P_{3}} \ :\ {\mbox{False}}\,\!$

(Remember that we are abbreviating our sentence letters by omitting superscripts.)

Usually, we are only interested in the truth values of a few sentence letters. The truth values assigned to other sentence letters can be random.

Given this valuation, we say:

${\mathfrak {v}}[\mathrm {P_{0}} ]\ =\ {\mbox{True}}\,\!$

${\mathfrak {v}}[\mathrm {P_{1}} ]\ =\ {\mbox{False}}\,\!$

${\mathfrak {v}}[\mathrm {P_{2}} ]\ =\ {\mbox{False}}\,\!$

${\mathfrak {v}}[\mathrm {P_{3}} ]\ =\ {\mbox{False}}\,\!$

Indeed, we can define a valuation as a function taking sentence letters as its arguments and truth values as its values (hence the name 'truth value'). Note that ${\mathcal {L_{S}}}\,\!$ does not have a fixed interpretation or valuation for sentence letters. Rather, we specify interpretations for temporary use.

Extended valuations

An extended interpretation generates the truth values of longer sentences given an interpretation. For sentential logic, an interpretation is a valuation, so an extended interpretation is an extended valuation. We define an extension ${\overline {\mathfrak {v}}}\,\!$ of valuation ${\mathfrak {v}}\,\!$ as follows.

For all sentence letters $\varphi$ and $\psi$ from ${\mathcal {L_{S}}}:$

${\mbox{i.}}\quad {\overline {\mathfrak {v}}}[\varphi ]={\mathfrak {v}}[\varphi ].\,\!$

${\mbox{ii.}}\quad {\overline {\mathfrak {v}}}[\lnot \varphi ]\ =\ {\begin{cases}{\mbox{True}}&{\mbox{if}}\ {\overline {\mathfrak {v}}}[\varphi ]={\mbox{False}};\\{\mbox{False}}&{\mbox{otherwise (i.e., if}}\ {\overline {\mathfrak {v}}}[\varphi ]={\mbox{True).}}\end{cases}}\,\!$

${\mbox{iii.}}\quad {\overline {\mathfrak {v}}}[(\varphi \land \psi )]\ =\ {\begin{cases}{\mbox{True}}&{\mbox{if}}\ {\overline {\mathfrak {v}}}[\varphi ]={\overline {\mathfrak {v}}}[\psi ]={\mbox{True}};\\{\mbox{False}}&{\mbox{otherwise}}.\end{cases}}\,\!$

${\mbox{iv.}}\quad {\overline {\mathfrak {v}}}[(\varphi \lor \psi )]\ =\ {\begin{cases}{\mbox{True}}&{\mbox{if}}\ {\overline {\mathfrak {v}}}[\varphi ]={\mbox{True}}\ {\mbox{or}}\ {\overline {\mathfrak {v}}}[\psi ]={\mbox{True}}\ {\mbox{(or both)}};\\{\mbox{False}}&{\mbox{otherwise}}.\end{cases}}\,\!$

${\mbox{v.}}\quad {\overline {\mathfrak {v}}}[(\varphi \rightarrow \psi )]\ =\ {\begin{cases}{\mbox{True}}&{\mbox{if}}\ {\overline {\mathfrak {v}}}[\varphi ]={\mbox{False}}\ \ {\mbox{or}}\ \ {\overline {\mathfrak {v}}}[\psi ]={\mbox{True}}\ \ {\mbox{(or both)}};\\{\mbox{False}}&{\mbox{otherwise.}}\end{cases}}\,\!$

${\mbox{vi.}}\quad {\overline {\mathfrak {v}}}[(\varphi \leftrightarrow \psi )]\ =\ {\begin{cases}{\mbox{True}}&{\mbox{if}}\ {\overline {\mathfrak {v}}}[\varphi ]={\overline {\mathfrak {v}}}[\psi ];\\{\mbox{False}}&{\mbox{otherwise}}.\end{cases}}\,\!$

Example

We will determine the truth value of this example sentence given two valuations.

$(1)\quad \mathrm {P} \land \mathrm {Q} \rightarrow \lnot (\mathrm {Q} \lor \mathrm {R} )\,\!$

First, consider the following valuation:

$\mathrm {P} \ :\ {\mbox{True}}\,\!$

$\mathrm {Q} \ :\ {\mbox{True}}\,\!$

$\mathrm {R} \ :\ {\mbox{False}}\,\!$

(2)  By clause (i):

${\overline {\mathfrak {v}}}[\mathrm {P} ]\ =\ {\mbox{True}}\,\!$

${\overline {\mathfrak {v}}}[\mathrm {Q} ]\ =\ {\mbox{True}}\,\!$

${\overline {\mathfrak {v}}}[\mathrm {R} ]\ =\ {\mbox{False}}\,\!$

(3)  By (1) and clause (iii),

${\overline {\mathfrak {v}}}[\mathrm {P} \land \mathrm {Q} ]\ =\ {\mbox{True}}.\,\!$

(4)  By (1) and clause (iv),

${\overline {\mathfrak {v}}}[\mathrm {Q} \lor \mathrm {R} ]\ =\ {\mbox{True}}.\,\!$

(5)  By (4) and clause (v),

${\overline {\mathfrak {v}}}[\lnot (\mathrm {Q} \lor \mathrm {R} )]\ =\ {\mbox{False}}.\,\!$

(6)  By (3), (5) and clause (v),

${\overline {\mathfrak {v}}}[\mathrm {P} \land \mathrm {Q} \rightarrow \lnot (\mathrm {Q} \lor \mathrm {R} )]\ =\ {\mbox{False}}.\,\!$

Thus (1) is false in our interpretation.

Next, try the valuation:

$\mathrm {P} \ :\ {\mbox{True}}\,\!$

$\mathrm {Q} \ :\ {\mbox{False}}\,\!$

$\mathrm {R} \ :\ {\mbox{True}}\,\!$

(7) By clause (i):

${\overline {\mathfrak {v}}}[\mathrm {P} ]\ =\ {\mbox{True}}\,\!$

${\overline {\mathfrak {v}}}[\mathrm {Q} ]\ =\ {\mbox{False}}\,\!$

${\overline {\mathfrak {v}}}[\mathrm {R} ]\ =\ {\mbox{True}}\,\!$

(8) By (7) and clause (iii),

${\overline {\mathfrak {v}}}[\mathrm {P} \land \mathrm {Q} ]\ =\ {\mbox{False}}.\,\!$

(9) By (7) and clause (iv),

${\overline {\mathfrak {v}}}[\mathrm {Q} \lor \mathrm {R} ]\ =\ {\mbox{True}}.\,\!$

(10) By (9) and clause (v),

${\overline {\mathfrak {v}}}[\lnot (\mathrm {Q} \lor \mathrm {R} )]\ =\ {\mbox{False}}.\,\!$

(11) By (8), (10) and clause (v),

${\overline {\mathfrak {v}}}[\mathrm {P} \land \mathrm {Q} \rightarrow \lnot (\mathrm {Q} \lor \mathrm {R} )]\ =\ {\mbox{True}}.\,\!$

Thus (1) is true in this second interpretation. Note that we did a bit more work this time than necessary. By clause (v), (8) is sufficient for the truth of (1).

Truth Tables

In the Formal Syntax, we earlier gave a formal semantics for sentential logic. A truth table is a device for using this form syntax in calculating the truth value of a larger formula given an interpretation (an assignment of truth values to sentence letters). Truth tables may also help clarify the material from the Formal Syntax.

Basic tables

Negation

We begin with the truth table for negation. It corresponds to clause (ii) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\lnot \mathrm {P} \,\!$ T F F T

T and F represent True and False respectively. Each row represents an interpretation. The first column shows what truth value the interpretation assigns to the sentence letter $\mathrm {P} \,\!$. In the first row, the interpretation assigns $\mathrm {P} \,\!$ the value True. In the second row, the interpretation assigns $\mathrm {P} \,\!$ the value False.

The second column shows the value $\lnot \mathrm {P} \,\!$ receives under a given row's interpretation. Under the interpretation of the first row, $\lnot \mathrm {P} \,\!$ has the value False. Under the interpretation of the second row, $\lnot \mathrm {P} \,\!$ has the value True.

We can put this more formally. The first row of the truth table above shows that when ${\mathfrak {v}}[\mathrm {P} ]\,\!$ = True, ${\overline {\mathfrak {v}}}[\lnot \mathrm {P} ]\,\!$ = False. The second row shows that when ${\mathfrak {v}}[\mathrm {P} ]\,\!$ = False, ${\overline {\mathfrak {v}}}[\lnot \mathrm {P} ]\,\!$ = True. We can also put things more simply: a negation has the opposite truth value than that which is negated.

Conjunction

The truth table for conjunction corresponds to clause (iii) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \land \mathrm {Q} \,\!$ T T T T F F F T F F F F

Here we have two sentence letters and so four possible interpretations, each represented by a single row. The first two columns show what the four interpretations assign to $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$. The interpretation represented by the first row assigns both sentence letters the value True, and so on. The last column shows the value assigned to $\mathrm {P} \land \mathrm {Q} \,\!$. You can see that the conjunction is true when both conjuncts are true—and the conjunction is false otherwise, namely when at least one conjunct is false.

Disjunction

The truth table for disjunction corresponds to clause (iv) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \lor \mathrm {Q} \,\!$ T T T T F T F T T F F F

Here we see that a disjunction is true when at least one of the disjuncts is true—and the disjunction is false otherwise, namely when both disjuncts are false.

Conditional

The truth table for conditional corresponds to clause (v) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \rightarrow \mathrm {Q} \,\!$ T T T T F F F T T F F T

A conditional is true when either its antecedent is false or its consequent is true (or both). It is false otherwise, namely when the antecedent is true and the consequent is false.

Biconditional

The truth table for biconditional corresponds to clause (vi) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$ T T T T F F F T F F F T

A biconditional is true when both parts have the same truth value. It is false when the two parts have opposite truth values.

Example

We will use the same example sentence from Formal Semantics:

$\mathrm {P} \land \mathrm {Q} \rightarrow \lnot (\mathrm {Q} \lor \mathrm {R} )\ .\,\!$

We construct its truth table as follows:

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {R} \,\!$ $\mathrm {P} \land \mathrm {Q} \,\!$ $\mathrm {Q} \lor \ \mathrm {R} \,\!$ $\lnot (\mathrm {Q} \lor \mathrm {R} )\,\!$ $(\mathrm {P} \land \mathrm {Q} )\rightarrow \lnot (\mathrm {Q} \lor \mathrm {R} )\,\!$ T T T T T F F T T F T T F F T F T F T F T T F F F F T T F T T F T F T F T F F T F T F F T F T F T F F F F F T T

With three sentence letters, we need eight valuations (and so lines of the truth table) to cover all cases. The table builds the example sentence in parts. The $\mathrm {P} \land \mathrm {Q} \,\!$ column was based on the $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ columns. The $\mathrm {Q} \lor \mathrm {R} \,\!$ column was based on the $\mathrm {Q} \,\!$ and $\mathrm {R} \,\!$ columns. This in turn was the basis for its negation in the next column. Finally, the last column was based on the $\mathrm {P} \land \mathrm {Q} \,\!$ and $\lnot (\mathrm {Q} \lor \mathrm {R} )\,\!$ columns.

We see from this truth table that the example sentence is false when both $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are true, and it is true otherwise.

This table can be written in a more compressed format as follows.

$\mathrm {P} \,\!$      $\mathrm {Q} \,\!$      $\mathrm {R} \,\!$      $\mathrm {P} \,\!$ (1)    $\land \,\!$      $\mathrm {Q} \,\!$ (4)    $\rightarrow \,\!$ (3)    $\lnot \,\!$      $(\mathrm {Q} \,\!$ (2)    $\lor \,\!$      $\mathrm {R} )\,\!$ T T T   T   F F   T   T T F   T   F F   T   T F T   F   T F   T   T F F   F   T T   F   F T T   F   T F   T   F T F   F   T F   T   F F T   F   T F   T   F F F   F   T T   F

The numbers above the connectives are not part of the truth table but rather show what order the columns were filled in.

Satisfaction and validity of formulae

Satisfaction

In sentential logic, an interpretation under which a formula is true is said to satisfy that formula. In predicate logic, the notion of satisfaction is a bit more complex. A formula is satisfiable if and only if it is true under at least one interpretation (that is, if and only if at least one interpretation satisfies the formula). The example truth table of Truth Tables showed that the following sentence is satisfiable.

$\mathrm {P} \land \mathrm {Q} \rightarrow \lnot (\mathrm {Q} \lor \mathrm {R} )$

For a simpler example, the formula $\mathrm {P} \,\!$ is satisfiable because it is true under any interpretation that assigns $\mathrm {P} \,\!$ the value True.

We use the notation ${\mathfrak {v}}\vDash \varphi$ to say that the interpretation ${\mathfrak {v}}\,\!$ satisfies $\varphi \,\!$. If ${\mathfrak {v}}$ does not satisfy $\varphi ,$ then we write ${\mathfrak {v}}\not \vDash \varphi .$

The concept of satisfaction is also extended to sets of formulae. A set of formulae is satisfiable if and only if there is an interpretation under which every formula of the set is true (that is, the interpretation satisfies every formula of the set).

A formula is unsatisfiable if and only if there is no interpretation under which it is true. A trivial example is

$\mathrm {P} \land \lnot \mathrm {P} \,\!$

You can easily confirm by doing a truth table that the formula is false no matter what truth value an interpretation assigns to $\mathrm {P} \,\!$. We say that an unsatisfiable formula is logically false. One can say that an unsatisfiable formula of sentential logic (but not one of predicate logic) is tautologically false.

Validity

A formula is valid if and only if it is satisfied under every interpretation. For example,

$\mathrm {P} \lor \lnot \mathrm {P} \,\!$

is valid. You can easily confirm by a truth table that it is true no matter what the interpretation assigns to $\mathrm {P} \,\!$. We say that a valid sentence is logically true. We call a valid formula of sentential logic—but not one of predicate logic—a tautology.

We use the notation $\vDash \varphi$ to say that $\varphi$ is valid and $\not \vDash \varphi$ to indicate $\varphi$ is not valid.

Equivalence

Two formulae are equivalent if and only if they are true under exactly the same interpretations. You can easily confirm by truth table that any interpretation that satisfies $\mathrm {P} \land \mathrm {Q} \,\!$ also satisfies $\mathrm {Q} \land \mathrm {P} \,\!$. In addition, any interpretation that satisfies $\mathrm {Q} \land \mathrm {P} \,\!$ also satisfies $\mathrm {P} \land \mathrm {Q} \,\!$. Thus they are equivalent.

We can use the following convenient notation to say that $\varphi \,\!$ and $\psi \,\!$ are equivalent.

$\varphi \,\!$ $\psi \,\!$

which is true if and only if

$\vDash \ (\varphi \leftrightarrow \psi )\,\!$

Validity of arguments

An argument is a set of formulae designated as premises together with a single sentence designated as the conclusion. Intuitively, we want the premises jointly to constitute a reason to believe the conclusion. For our purposes an argument is any set of premises together with any conclusion. That can be a bit artificial for some particularly silly arguments, but the logical properties of an argument do not depend on whether it is silly or whether anyone actually does or might consider the premises to be a reason to believe the conclusion. We consider arguments as if one does or might consider the premises to be a reason for the conclusion independently of whether anyone actually does or might do so. Even an empty set of premises together with a conclusion counts as an argument.

The following example show the same argument using several notations.

Notation 1

$\mathrm {P} \,\!$

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$

Therefore $\mathrm {Q} \,\!$

Notation 2

$\mathrm {P} \,\!$

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$

∴ $\mathrm {Q} \,\!$

Notation 3

$\mathrm {P} \,\!$

${\underline {\mathrm {P} \rightarrow \mathrm {Q} }}\,\!$

$\mathrm {Q} \,\!$

Notation 4

$\mathrm {P} ,\ \mathrm {P} \rightarrow \mathrm {Q} \,\!$   ∴   $\mathrm {Q} \,\!$

An argument is valid if and only if every interpretation that satisfies all the premises also satisfies the conclusion. A conclusion of a valid argument is a logical consequence of its premises. We can express the validity (or invalidity) of the argument with $\Gamma \,\!$ as its set of premises and $\psi \,\!$ as its conclusion using the following notation.

(1)    $\Gamma \vDash \psi \,\!$

(2)    $\Gamma \ \not \vDash \ \psi \,\!$

For example, we have

$\{\mathrm {P} ,\ \mathrm {P} \rightarrow \mathrm {Q} \}\ \vDash \ \mathrm {Q} \,\!$

Validity for arguments, or logical consequence, is the central notion driving the intuitions on which we build a logic. We want to know whether our arguments are good arguments, that is, whether they represent good reasoning. We want to know whether the premises of an argument constitute good reason to believe the conclusion. Validity is one essential feature of a good argument. It is not the only essential feature. A valid argument with at least one false premise is useless. Validity is the truth-preserving feature. It does not tell us that the conclusion is true, only that the logical features of the argument are such that, if the premises are true, then the conclusion is. A valid argument with true premises is sound.

There are other less formal features that a good argument needs. Just because the premises are true does not mean that they are believed, that we have any reason to believe them, or that we could collect evidence for them. It should also be noted that validity only applies to certain types of arguments, particularly deductive arguments. Deductive arguments are intended to be valid. The archetypical example for a deductive argument is a mathematical proof. Inductive arguments, of which scientific arguments provide the archetypical example, are not intended to be valid. The truth of the premises are not intended to guarantee that the conclusion is true. Rather, the truth of the premises are intended to make the truth of the conclusion highly probably or likely. In science, we do not intend to offer mathematical proofs. Rather, we gather evidence.

Formulae and arguments

For every valid formula, there is a corresponding valid argument having the valid formula as its conclusion and the empty set as its set of premises. Thus

$\vDash \psi \,\!$

if and only if

$\varnothing \vDash \psi \,\!$

For every valid argument with finitely many premises, there is a corresponding valid formula. Consider a valid argument with $\psi \,\!$ as the conclusion and having as its premises $\varphi _{1},\varphi _{2},...,\varphi _{n}\,\!$. Then

$\varphi _{1},\ \varphi _{2},\ ...,\ \varphi _{n}\vDash \psi \,\!$

There is then the corresponding valid formula

$\varphi _{1}\land \varphi _{2}\land ...\land \varphi _{n}\rightarrow \psi \,\!$

There corresponds to the valid argument

$\mathrm {P} ,\ \mathrm {P} \rightarrow \mathrm {Q} \,\!$   ∴   $\mathrm {Q} \,\!$

the following valid formula:

$\mathrm {P} \land (\mathrm {P} \rightarrow \mathrm {Q} )\rightarrow \mathrm {Q} \,\!$

Implication

You may see some text reading our arrow $\rightarrow \,\!$ as 'implies' and using 'implications' as an alternative for 'conditional'. This is generally decried as a use-mention error. In ordinary English, the following are considered grammatically correct:

(3)    'That there is smoke implies that there is fire'.

(4)    'There is smoke' implies 'there is fire'.

In (3), we have one fact or proposition or whatever (the current favorite among philosophers appears to be proposition) implying another of the same species. In (4), we have one sentence implying another.

But the following is considered incorrect:

There is smoke implies there is fire.

Here, in contrast to (3), there are no quotation marks. Nothing is the subject doing the implying and nothing is the object implied. Rather, we are composing a larger sentence out of smaller ones as if 'implies' were a grammatical conjunction such as 'only if'.

Thus logicians tend to avoid using 'implication' to mean conditional. Rather, they use 'implies' to mean has as a logical consequence and 'implication' to mean valid argument. In doing this, they are following the model of (4) rather than (3). In particular, they read (1) and (2) as '$\Gamma \,\!$ implies (or does not imply) $\psi \,\!$.

Expressibility

Formula truth tables

A formula with n sentence letters requires $2^{n}\,\!$ lines in its truth table. And, for a truth table of m lines, there are $2^{m}\,\!$ possible formulas. Thus, for a sentence of n letters, $m=2^{n}$ and the number of possible formulas is $2^{2^{n}}$.

For example, there are four possible formulas of one sentence letter (requiring a two-line truth table) and 16 possible formulas of two sentence letters (requiring a four-line truth table). We illustrate this with the following tables. The numbered columns represent the different possibilities for the column of a main connective.

$\mathrm {P} \,\!$ (i) (ii) (iii) (iv) T T T F F F T F T F

Column (iii) is the negation formula presented earlier.

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (xv) (xvi) T T T T T T T T T T F F F F F F F F T F T T T T F F F F T T T T F F F F F T T T F F T T F F T T F F T T F F F F T F T F T F T F T F T F T F T F

Column (ii) represents the formula for disjunction, column (v) represents conditional, column (vii) represents biconditional, and column (viii) represents conjunction.

Expressing arbitrary formulas

The question arises whether we have enough connectives to represent all the formulas of any number of sentence letters. Remember that each row represents one valuation. We can express that valuation by conjoining sentence letters assigned True under that valuation and negations of sentence letters assigned false under that valuation. The four valuations of the second table above can be expressed as

$\mathrm {P} \land \mathrm {Q} \,\!$

$\mathrm {P} \land \lnot \mathrm {Q} \,\!$

$\lnot \mathrm {P} \land \mathrm {Q} \,\!$

$\lnot \mathrm {P} \land \lnot \mathrm {Q} \,\!$

Now we can express an arbitrary formula by disjoining the valuations under which the formula has the value true. For example, we can express column (x) with:

(1)      $(\mathrm {P} \land \lnot \mathrm {Q} )\lor (\lnot \mathrm {P} \land \mathrm {Q} )\,\!$

You can confirm by completing the truth table that this produces the desired result. The formula is true when either (a) $\mathrm {P} \,\!$ is true and $\mathrm {Q} \,\!$ is false or (b) $\mathrm {P} \,\!$ is false and $\mathrm {Q} \,\!$ is true. There is an easier way to express this same formula: $\mathrm {P} \leftrightarrow \lnot \mathrm {Q} \,\!$. Coming up with a simple way to express an arbitrary formula may require insight, but at least we have an automatic mechanism for finding some way to express it.

Now consider a second example. We want to express a formula of $\mathrm {P} \,\!$, $\mathrm {Q} \,\!$, and $\mathrm {R} \,\!$, and we want this to be true under (and only under) the following three valuations.

			(i)		(ii)		(iii)
	$\mathrm {P} \,\!$		True		False		False
	$\mathrm {Q} \,\!$		True		True		False
	$\mathrm {R} \,\!$		False		False		True

We can express the three conditions which yield true as

$\mathrm {P} \land \mathrm {Q} \land \lnot \mathrm {R} \,\!$

$\lnot \mathrm {P} \land \mathrm {Q} \land \lnot \mathrm {R} \,\!$

$\lnot \mathrm {P} \land \lnot \mathrm {Q} \land \mathrm {R} \,\!$

Now we need to say that either the first condition holds or that the second condition holds or that the third condition holds:

(2)      $(\mathrm {P} \land \mathrm {Q} \land \lnot \mathrm {R} )\lor (\lnot \mathrm {P} \land \mathrm {Q} \land \lnot \mathrm {R} )\lor (\lnot \mathrm {P} \land \lnot \mathrm {Q} \land \mathrm {R} )\,\!$

You can verify by a truth table that it yields the desired result, that the formula is true in just the interpretation above.

This technique for expressing arbitrary formulas does not work for formulas evaluating to False in every interpretation. We need at least one interpretation yielding True in order to get the formula started. However, we can use any tautologically false formula to express such formulas. $\mathrm {P} \land \lnot \mathrm {P} \,\!$ will suffice.

Normal forms

A normal form provides a standardized rule of expression where any formula is equivalent to one which conforms to the rule. It will be useful in the following to define a literal as a sentence letter or its negation (e.g. $\mathrm {P}$, and $\mathrm {Q}$ as well as $\neg \mathrm {P}$, and $\neg \mathrm {Q}$).

Disjunctive normal form

We say a formula is in disjunctive normal form if it is a disjunction of conjunctions of literals. An example is $(\mathrm {P} \land \mathrm {Q} )\lor (\lnot \mathrm {R} \land \mathrm {S} )$. For the purpose of this definition we admit so called degenerate disjunctions and conjunctions of only one disjunct or conjunct. Thus we count $\mathrm {P} \,\!$ as being in disjunctive normal form because it is a degenerate (one-place) disjunction of a degenerate (one-place) conjunction. The degeneracy can be removed by converting it to the equivalent formula $(\mathrm {P} \land \mathrm {P} )\lor (\mathrm {P} \land \mathrm {P} )\,\!$. We also admit many-place disjunctions and conjunctions for the purposes of this definition, such as $(\lnot \mathrm {P} \land \mathrm {Q} \land \lnot \mathrm {R} )\lor (\mathrm {S} \land \mathrm {R} )$. A method for finding the disjunctive normal form of a arbitrary formula is shown above.

Conjunctive normal form

There is another common normal form in sentential logic, namely conjunctive normal form. A formula is in conjunctive normal form if it is a conjunction of disjunctions of literals. An example is $(\mathrm {P} \lor \lnot \mathrm {Q} )\land (\mathrm {R} \lor \mathrm {S} )$. Again, we can express arbitrary formulas in conjunctive normal form. First, take the valuations for which the formula evaluates to False. For each such valuation, form a disjunction of sentence letters the valuation assigns False together with the negations of sentence letters the valuation assign true. For the valuation

$\mathrm {P} \,\!$   :    False

$\mathrm {Q} \,\!$   :    True

$\mathrm {R} \,\!$   :    False

we form the disjunction

$\lnot \mathrm {P} \lor \mathrm {Q} \lor \lnot \mathrm {R} \,\!$

The conjunctive normal form expression of an arbitrary formula is the conjunction of all such disjunctions matching the interpretations for which the formula evaluates to false. The conjunctive normal form equivalent of (1) above is

$(\lnot \mathrm {P} \lor \mathrm {Q} )\land (\mathrm {P} \lor \lnot \mathrm {Q} )\,\!$

The conjunctive normal form equivalent of (2) above is

$(\lnot \mathrm {P} \lor \lnot \mathrm {Q} \lor \lnot \mathrm {R} )\land (\mathrm {P} \lor \lnot \mathrm {Q} \lor \lnot \mathrm {R} )\land (\mathrm {P} \lor \lnot \mathrm {Q} \lor \mathrm {R} )\land (\mathrm {P} \lor \mathrm {Q} \lor \lnot \mathrm {R} )\land (\mathrm {P} \lor \mathrm {Q} \lor \mathrm {R} )$

Interdefinability of connectives

Negation and conjunction are sufficient to express the other three connectives and indeed any arbitrary formula.

$\mathrm {P} \lor \mathrm {Q} \,\!$       $\lnot (\lnot \mathrm {P} \land \lnot \mathrm {Q} )\,\!$

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$       $\lnot (\mathrm {P} \land \lnot \mathrm {Q} )\,\!$

$\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$       $(\mathrm {P} \rightarrow \mathrm {Q} )\land (\mathrm {Q} \rightarrow \mathrm {P} )\,\!$       $\lnot (\mathrm {P} \land \lnot \mathrm {Q} )\land \lnot (\mathrm {Q} \land \lnot \mathrm {P} )\,\!$

Negation and disjunction are sufficient to express the other three connectives and indeed any arbitrary formula.

$\mathrm {P} \land \mathrm {Q} \,\!$       $\lnot (\lnot \mathrm {P} \lor \lnot \mathrm {Q} )\,\!$

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$       $\lnot \mathrm {P} \lor \mathrm {Q} \,\!$

$\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$       $(\mathrm {P} \land \mathrm {Q} )\lor (\lnot \mathrm {P} \land \lnot \mathrm {Q} )\,\!$       $\lnot (\lnot \mathrm {P} \lor \lnot \mathrm {Q} )\lor \lnot (\mathrm {P} \lor \mathrm {Q} )\,\!$

Negation and conditional are sufficient to express the other three connectives and indeed any arbitrary formula.

$\mathrm {P} \lor \mathrm {Q} \,\!$       $\lnot \mathrm {P} \rightarrow \mathrm {Q} \,\!$

$\mathrm {P} \land \mathrm {Q} \,\!$       $\lnot (\mathrm {P} \rightarrow \lnot \mathrm {Q} )\,\!$

$\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$       $(\mathrm {P} \rightarrow \mathrm {Q} )\land (\mathrm {Q} \rightarrow \mathrm {P} )\,\!$       $\lnot ((\mathrm {P} \rightarrow \mathrm {Q} )\rightarrow \lnot (\mathrm {Q} \rightarrow \mathrm {P} ))\,\!$

Negation and biconditional are not sufficient to express the other three connectives.

Joint and alternative denials

We have seen that three pairs of connectives are each jointly sufficient to express any arbitrary formula. The question arises, is it possible to express any arbitrary formula with just one connective? The answer is yes, but not with any of our connectives. There are two possible binary connectives each of which, if added to ${\mathcal {L_{S}}}\,\!$, would be sufficient.

Alternative denial

Alternative denial, sometimes called NAND. The usual symbol for this is called the Sheffer stroke, written as $\mid$ (some authors use ↑). Temporarily add the symbol $\mid \,\!$ to ${\mathcal {L_{S}}}\,\!$ and let ${\mathfrak {v}}[(\varphi \mid \psi )]\,\!$ be True when at least one of $\varphi \,\!$ or $\psi \,\!$ is false. It has the truth table :

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \mid \mathrm {Q} \,\!$ T T F T F T F T T F F T

We now have the following equivalences.

$\lnot \mathrm {P} \,\!$       $\mathrm {P} \mid \mathrm {P} \,\!$

$\mathrm {P} \land \mathrm {Q} \,\!$       $(\mathrm {P} \mid \mathrm {Q} )\mid (\mathrm {P} \mid \mathrm {Q} )\,\!$

$\mathrm {P} \lor \mathrm {Q} \,\!$       $(\mathrm {P} \mid \mathrm {P} )\mid (\mathrm {Q} \mid \mathrm {Q} )\,\!$

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$       $\mathrm {P} \mid (\mathrm {Q} \mid \mathrm {Q} )\,\!$

$\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$       $(\mathrm {P} \mid \mathrm {Q} )\mid ((\mathrm {P} \mid \mathrm {P} )\mid (\mathrm {Q} \mid \mathrm {Q} ))\,\!$

Joint denial

Joint denial, sometimes called NOR. Temporarily add the symbol $\downarrow \,\!$ to ${\mathcal {L_{S}}}\,\!$ and let ${\mathfrak {v}}[(\varphi \downarrow \psi )]\,\!$ be True when both $\varphi \,\!$ and $\psi \,\!$ are false. It has the truth table :

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \downarrow \mathrm {Q} \,\!$ T T F T F F F T F F F T

We now have the following equivalences.

$\lnot \mathrm {P} \,\!$       $\mathrm {P} \downarrow \mathrm {P} \,\!$

$\mathrm {P} \land \mathrm {Q} \,\!$       $(\mathrm {P} \downarrow \mathrm {P} )\downarrow (\mathrm {Q} \downarrow \mathrm {Q} )\,\!$

$\mathrm {P} \lor \mathrm {Q} \,\!$       $(\mathrm {P} \downarrow \mathrm {Q} )\downarrow (\mathrm {P} \downarrow \mathrm {Q} )\,\!$

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$       $((\mathrm {P} \downarrow \mathrm {Q} )\downarrow \mathrm {Q} )\downarrow ((\mathrm {P} \downarrow \mathrm {Q} )\downarrow \mathrm {Q} )\,\!$

$\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$       $((\mathrm {P} \downarrow \mathrm {P} )\downarrow \mathrm {Q} )\downarrow (\mathrm {P} \downarrow (\mathrm {Q} \downarrow \mathrm {Q} ))\,\!$

Properties of Sentential Connectives

Here we list some of the more famous, historically important, or otherwise useful equivalences and tautologies. They can be added to the ones listed in Interdefinability of connectives. We can go on at quite some length here, but will try to keep the list somewhat restrained. Remember that for every equivalence of $\varphi \,\!$ and $\psi \,\!$, there is a related tautology $\varphi \leftrightarrow \psi \,\!$.

Bivalence

Every formula has exactly one of two truth values.

$\vDash \ \mathrm {P} \lor \lnot \mathrm {P} \,\!$      Law of Excluded Middle

$\vDash \ \lnot (\mathrm {P} \land \lnot \mathrm {P} )\,\!$      Law of Non-Contradiction

Analogues to arithmetic laws

Some familiar laws from arithmetic have analogues in sentential logic.

Reflexivity

Conditional and biconditional (but not conjunction and disjunction) are reflexive.

$\vDash \ \mathrm {P} \rightarrow \mathrm {P} \,\!$

$\vDash \ \mathrm {P} \leftrightarrow \mathrm {P} \,\!$

Commutativity

Conjunction, disjunction, and biconditional (but not conditional) are commutative.

$\mathrm {P} \land \mathrm {Q} \,\!$       $\mathrm {Q} \land \mathrm {P} \,\!$

$\mathrm {P} \lor \mathrm {Q} \,\!$       $\mathrm {Q} \lor \mathrm {P} \,\!$

$\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$       $\mathrm {Q} \leftrightarrow \mathrm {P} \,\!$

Associativity

Conjunction, disjunction, and biconditional (but not conditional) are associative.

$(\mathrm {P} \land \mathrm {Q} )\land \mathrm {R} \,\!$       $\mathrm {P} \land (\mathrm {Q} \land \mathrm {R} )\,\!$

$(\mathrm {P} \lor \mathrm {Q} )\lor \mathrm {R} \,\!$       $\mathrm {P} \lor (\mathrm {Q} \lor \mathrm {R} )\,\!$

$(\mathrm {P} \leftrightarrow \mathrm {Q} )\leftrightarrow \mathrm {R} \,\!$       $\mathrm {P} \leftrightarrow (\mathrm {Q} \leftrightarrow \mathrm {R} )\,\!$

Distribution

We list ten distribution laws. Of these, probably the most important are that conjunction and disjunction distribute over each other and that conditional distributes over itself.

$\mathrm {P} \land (\mathrm {Q} \land \mathrm {R} )\,\!$       $(\mathrm {P} \land \mathrm {Q} )\land (\mathrm {P} \land \mathrm {R} )\,\!$

$\mathrm {P} \land (\mathrm {Q} \lor \mathrm {R} )\,\!$       $(\mathrm {P} \land \mathrm {Q} )\lor (\mathrm {P} \land \mathrm {R} )\,\!$

$\mathrm {P} \lor (\mathrm {Q} \land \mathrm {R} )\,\!$       $(\mathrm {P} \lor \mathrm {Q} )\land (\mathrm {P} \lor \mathrm {R} )\,\!$

$\mathrm {P} \lor (\mathrm {Q} \lor \mathrm {R} )\,\!$       $(\mathrm {P} \lor \mathrm {Q} )\lor (\mathrm {P} \lor \mathrm {R} )\,\!$

$\mathrm {P} \lor (\mathrm {Q} \rightarrow \mathrm {R} )\,\!$       $(\mathrm {P} \lor \mathrm {Q} )\rightarrow (\mathrm {P} \lor \mathrm {R} )\,\!$

$\mathrm {P} \lor (\mathrm {Q} \leftrightarrow \mathrm {R} )\,\!$       $(\mathrm {P} \lor \mathrm {Q} )\leftrightarrow (\mathrm {P} \lor \mathrm {R} )\,\!$

$\mathrm {P} \rightarrow (\mathrm {Q} \land \mathrm {R} )\,\!$       $(\mathrm {P} \rightarrow \mathrm {Q} )\land (\mathrm {P} \rightarrow \mathrm {R} )\,\!$

$\mathrm {P} \rightarrow (\mathrm {Q} \lor \mathrm {R} )\,\!$       $(\mathrm {P} \rightarrow \mathrm {Q} )\lor (\mathrm {P} \rightarrow \mathrm {R} )\,\!$

$\mathrm {P} \rightarrow (\mathrm {Q} \rightarrow \mathrm {R} )\,\!$       $(\mathrm {P} \rightarrow \mathrm {Q} )\rightarrow (\mathrm {P} \rightarrow \mathrm {R} )\,\!$

$\mathrm {P} \rightarrow (\mathrm {Q} \leftrightarrow \mathrm {R} )\,\!$       $(\mathrm {P} \rightarrow \mathrm {Q} )\leftrightarrow (\mathrm {P} \rightarrow \mathrm {R} )\,\!$

Transitivity

Conjunction, conditional, and biconditional (but not disjunction) are transitive.

$\vDash \ (\mathrm {P} \land \mathrm {Q} )\land (\mathrm {Q} \land \mathrm {R} )\rightarrow \mathrm {P} \land \mathrm {R} \,\!$

$\vDash \ (\mathrm {P} \rightarrow \mathrm {Q} )\land (\mathrm {Q} \rightarrow \mathrm {R} )\rightarrow (\mathrm {P} \rightarrow \mathrm {R} )\,\!$

$\vDash \ (\mathrm {P} \leftrightarrow \mathrm {Q} )\land (\mathrm {Q} \leftrightarrow \mathrm {R} )\rightarrow (\mathrm {P} \leftrightarrow \mathrm {R} )\,\!$

Other tautologies and equivalences

Conditionals

These tautologies and equivalences are mostly about conditionals.

$\vDash \ \mathrm {P} \land \mathrm {Q} \rightarrow \mathrm {P} \qquad \qquad \vDash \ \mathrm {P} \land \mathrm {Q} \rightarrow \mathrm {Q} \,\!$

$\vDash \ \mathrm {P} \rightarrow \mathrm {P} \lor \mathrm {Q} \qquad \qquad \vDash \ \mathrm {Q} \rightarrow \mathrm {P} \lor \mathrm {Q} \,\!$

$\vDash \ (\mathrm {P} \rightarrow \mathrm {Q} )\lor (\mathrm {Q} \rightarrow \mathrm {R} )\,\!$

$\vDash \ \lnot \mathrm {P} \rightarrow (\mathrm {P} \rightarrow \mathrm {Q} )\,\!$      Conditional addition

$\vDash \ \mathrm {Q} \rightarrow (\mathrm {P} \rightarrow \mathrm {Q} )\,\!$      Conditional addition

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$       $\lnot \mathrm {Q} \rightarrow \lnot \mathrm {P} \,\!$      Contraposition

$\mathrm {P} \land \mathrm {Q} \rightarrow \mathrm {R} \,\!$       $\mathrm {P} \rightarrow (\mathrm {Q} \rightarrow \mathrm {R} )\,\!$      Exportation

Biconditionals

These tautologies and equivalences are mostly about biconditionals.

$\vDash \ \mathrm {P} \land \mathrm {Q} \rightarrow (\mathrm {P} \leftrightarrow \mathrm {Q} )\,\!$      Biconditional addition

$\vDash \ \lnot \mathrm {P} \land \lnot \mathrm {Q} \rightarrow (\mathrm {P} \leftrightarrow \mathrm {Q} )\,\!$      Biconditional addition

$\vDash (\mathrm {P} \leftrightarrow \mathrm {Q} )\lor (\mathrm {P} \leftrightarrow \lnot \mathrm {Q} )\,\!$

$\lnot (\mathrm {P} \leftrightarrow \mathrm {Q} )\,\!$       $\lnot \mathrm {P} \leftrightarrow \mathrm {Q} \,\!$       $\mathrm {P} \leftrightarrow \lnot \mathrm {Q} \,\!$

Miscellaneous

We repeat DeMorgan's Laws from the Interdefinability of connectives section of Expressibility and add two additional forms. We also list some additional tautologies and equivalences.

$\vDash \ \mathrm {P} \leftrightarrow \mathrm {P} \land \mathrm {P} \,\!$      Idempotence for conjunction

$\vDash \ \mathrm {P} \leftrightarrow \mathrm {P} \lor \mathrm {P} \,\!$      Idempotence for disjunction

$\vDash \ \mathrm {P} \rightarrow \mathrm {P} \lor \mathrm {Q} \,\!$      Disjunctive addition

$\vDash \ \mathrm {Q} \rightarrow \mathrm {P} \lor \mathrm {Q} \,\!$      Disjunctive addition

$\vDash \ \mathrm {P} \land \lnot \mathrm {P} \rightarrow \mathrm {Q} \,\!$

$\mathrm {P} \land \mathrm {Q} \,\!$       $\lnot (\lnot \mathrm {P} \lor \lnot \mathrm {Q} )\,\!$      Demorgan's Laws

$\mathrm {P} \lor \mathrm {Q} \,\!$       $\lnot (\lnot \mathrm {P} \land \lnot \mathrm {Q} )\,\!$      Demorgan's Laws

$\lnot (\mathrm {P} \land \mathrm {Q} )\,\!$       $\lnot \mathrm {P} \lor \lnot \mathrm {Q} \,\!$      Demorgan's Laws

$\lnot (\mathrm {P} \lor \mathrm {Q} )\,\!$       $\lnot \mathrm {P} \land \lnot \mathrm {Q} \,\!$      Demorgan's Laws

$\mathrm {P} \,\!$       $\lnot \lnot \mathrm {P} \,\!$      Double Negation

Deduction and reduction principles

The following two principles will be used in constructing our derivation system on a later page. They can easily be proven, but—since they are neither tautologies nor equivalences—it takes more than a mere truth table to do so. We will not attempt the proof here.

Deduction principle

Let $\varphi \,\!$ and $\psi \,\!$ both be formulae, and let $\Gamma \,\!$ be a set of formulae.

$\mathrm {If} \ \ \Gamma \cup \{\varphi \}\vDash \psi \mathrm {,\ then} \ \ \Gamma \vDash (\varphi \rightarrow \psi )\,\!$

Reduction principle

Let $\varphi \,\!$ and $\psi \,\!$ both be formulae, and let $\Gamma \,\!$ be a set of formulae.

$\mathrm {If} \ \ \Gamma \cup \{\varphi \}\vDash \psi \ \ \mathrm {and} \ \ \Gamma \cup \{\varphi \}\vDash \lnot \psi \mathrm {,\ then} \ \ \Gamma \vDash \lnot \varphi ,\!$

$\mathrm {If} \ \ \Gamma \cup \{\lnot \varphi \}\vDash \psi \ \ \mathrm {and} \ \ \Gamma \cup \{\lnot \varphi \}\vDash \lnot \psi \mathrm {,\ then} \ \ \Gamma \vDash \varphi \,\!$

Substitution and Interchange

This page will use the notions of occurrence and subformula introduced at the Additional terminology section of Formal Syntax. These notions have been little used if at all since then, so you might want to review them.

Substitution

Tautological forms

We have introduced a number of tautologies, one example being

(1)    $\mathrm {Q} \rightarrow (\mathrm {P} \rightarrow \mathrm {Q} )\,\!$

Use the metavariables $\psi$ and $\varphi$ to replace $\mathrm {Q}$ and $\mathrm {R}$ in (1). This produces the form

(2)    $\psi \rightarrow (\varphi \rightarrow \psi )\,\!$

As it turns out, any formula matching this form is a tautology. Thus, for example, let $\psi =\mathrm {R} \land \mathrm {S}$ and $\varphi =\mathrm {P} \lor \mathrm {Q}$. Then,

(3)    $\mathrm {R} \land \mathrm {S} \rightarrow (\mathrm {P} \lor \mathrm {Q} \rightarrow \mathrm {R} \land \mathrm {S} )\,\!$

is a tautology. This process can be generalized to all tautologies: for any tautology, find its explicit form by replacing each sentence letters with distinct metavariables (written as Greek letters, as shown in (2)). We can call this a tautological form, which is a metalogical expression rather than a formula. Any instance of this tautological form is a tautology.

Substitution instances

The preceding illustrated how we can generate new tautologies from old ones via tautological forms. Here, we will show how to generate tautologies without resorting to tautological forms. To do this, we define a substitution instance of a formula. Any substitution instance of a tautology is also a tautology.

First, we define the simple substitution instance of a formula for a sentence letter. Let $\varphi \,\!$ and $\psi \,\!$ be formulae and $\pi \,\!$ be a sentence letter. The simple substitution instance $\varphi [\pi /\psi ]\,\!$ is the result of replacing every occurrence of $\pi \,\!$ in $\varphi \,\!$ with an occurrence of $\psi \,\!$. A substitution instance of formulae for a sentence letters is the result of a chain of simple substitution instances. In particular, a chain of zero simple substitutions instances starting from $\varphi \,\!$ is a substitution instance and indeed is just $\varphi \,\!$ itself. Thus, any formula is a substitution instance of itself.

It turns out that if $\varphi \,\!$ is a tautology, then so is any simple substitution instance $\varphi [\pi /\psi ]\,\!$. If we start with a tautology and generate a chain of simple substitution instances, then every formula in the chain is also a tautology. Thus any (not necessarily simple) substitution instance of a tautology is also a tautology.

Substitution examples

Consider (1) again. We substitute $\mathrm {R} \land \mathrm {S} \,\!$ for every occurrence of $\mathrm {Q} \,\!$ in (1). This gives us the following simple substitution instance of (1):

(4)    $\mathrm {R} \land \mathrm {S} \rightarrow (\mathrm {P} \rightarrow \mathrm {R} \land \mathrm {S} )\,\!$

In this, we substitute $\mathrm {P} \lor \mathrm {Q} \,\!$ for $\mathrm {P} \,\!$. That gives us (3) as a simple substitution instance of (4). Since (3) is the result of a chain of two simple substitution instances, it is a (non-simple) substitution instance of (1) Since (1) is a tautology, so is (3). We can express the chain of substitutions as

$\mathrm {Q} \rightarrow (\mathrm {P} \rightarrow \mathrm {Q} )[\mathrm {Q} /\mathrm {R} \land \mathrm {S} ][\mathrm {P} /\mathrm {P} \lor \mathrm {Q} ]\,\!$

Take another example, also starting from (1). We want to obtain

(5)    $\mathrm {P} \rightarrow (\mathrm {Q} \rightarrow \mathrm {P} )\,\!$

Our first attempt might be to substitute $\mathrm {P} \,\!$ for $\mathrm {Q} \,\!$,

(6)    $\mathrm {P} \rightarrow (\mathrm {P} \rightarrow \mathrm {P} )\,\!$

This is indeed a tautology, but it is not the one we wanted. Instead, we substitute $\mathrm {R} \,\!$ for $\mathrm {P} \,\!$ in (1), obtaining

$\mathrm {Q} \rightarrow (\mathrm {R} \rightarrow \mathrm {Q} )\,\!$

Now substitute $\mathrm {P} \,\!$ for $\mathrm {Q} \,\!$ obtaining

$\mathrm {P} \rightarrow (\mathrm {R} \rightarrow \mathrm {P} )\,\!$

Finally, substituting $\mathrm {Q} \,\!$ for $\mathrm {R} \,\!$ gets us the result we wanted, namely (5). Since (1) is a tautology, so is (5). We can express the chain of substitutions as

$\mathrm {Q} \rightarrow (\mathrm {P} \rightarrow \mathrm {Q} )[\mathrm {P} /\mathrm {R} ][\mathrm {Q} /\mathrm {P} ][\mathrm {R} /\mathrm {Q} ]\!$

Simultaneous substitutions

We can compress a chain of simple substitutions into a single complex substitution. Let $\varphi \,\!$, $\psi _{1}\,\!$, $\psi _{2}\,\!$, ... be formulae; let $\pi _{1}\,\!$, $\pi _{2}\,\!$, ... be sentence letters. We define a simultaneous substitution instance of formulas for sentence letters $\varphi [\pi _{1}/\psi _{1},\ \pi _{2}/\psi _{2},\ ...]\,\!$ be the result of starting with $\varphi \,\!$ and simultaneously replacing $\pi _{1}\,\!$ with $\psi _{1}\,\!$, $\pi _{2}\,\!$ with $\psi _{2}\,\!$, .... We can regenerate our examples.

The previously generated formula (3) is