Editing Propositional formula (section)

== Propositional connectives ==

Arbitrary propositional formulas are built from propositional variables and other propositional formulas using [[logical connective|propositional connective]]s. Examples of connectives include:
* The unary negation connective. If <math>\alpha</math> is a formula, then <math>\lnot \alpha</math> is a formula.
* The classical binary connectives <math>\land, \lor, \to, \leftrightarrow</math>. Thus, for example, if <math>\alpha</math> and <math>\beta</math> are formulas, so is <math>(\alpha \to \beta)</math>.
* Other binary connectives, such as NAND, NOR, and XOR
* The ternary connective IF ... THEN ... ELSE ...
* Constant 0-ary connectives ⊤ and ⊥ (alternately, constants { T, F }, { 1, 0 } etc. )
* The "theory-extension" connective EQUALS (alternately, IDENTITY, or the sign " = " as distinguished from the "logical connective" <math>\leftrightarrow</math>)

=== Connectives of rhetoric, philosophy and mathematics ===
The following are the connectives common to rhetoric, philosophy and mathematics together with their [[truth table]]s. The symbols used will vary from author to author and between fields of endeavor. In general the abbreviations "T" and "F" stand for the evaluations TRUTH and FALSITY applied to the variables in the propositional formula (e.g. the assertion: "That cow is blue" will have the truth-value "T" for Truth or "F" for Falsity, as the case may be.).

The connectives go by a number of different word-usages, e.g. "a IMPLIES b" is also said "IF a THEN b". Some of these are shown in the table.

{|class="wikitable"
|- style="font-size:9pt" align="center" valign="bottom"
| width="48" Height="12" | 
| width="43.5" | 
| width="45" | 
| width="42" | 
| width="60" | 
| width="57.75" | 
|style="background-color:#E5E0EC" width="114.75" | b only if a
| width="187.5" | 
| width="93" | 
| width="87.75" | 
| width="48" | 
| width="63" | 
|- style="font-size:9pt" align="center"
| Height="12" | 
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|style="background-color:#E5E0EC" | b IS SUFFICIENT FOR a
|style="background-color:#F2F2F2" | b PRECISELY WHEN a
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|- style="font-size:9pt" align="center"
| Height="12" | 
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|style="background-color:#E5E0EC" | {{not a typo|a}} IS NECESSARY FOR b
|style="background-color:#F2F2F2" | b IF AND ONLY IF a;  b IFF a
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| 
|- style="font-size:9pt" align="center"
| Height="12" | 
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|style="background-color:#FDE9D9" | inclusive OR
|style="background-color:#E5E0EC" | IF b THEN a
|style="background-color:#F2F2F2" | b IS NECESSARY AND SUFFICIENT FOR a
| 
| 
| 
| 
|- style="font-size:9pt" align="center"
| Height="12" | 
| 
|style="background-color:#EAF1DD" | negation
|style="background-color:#EAF1DD" | negation
|style="background-color:#DBE5F1" | conjunction
|style="background-color:#FDE9D9" | disjunction
|style="background-color:#E5E0EC" | implication
|style="background-color:#F2F2F2" | biconditional
| 
| 
| 
| 
|- style="font-size:9pt" align="center"
! Height="12" colspan="2" | variables
|style="background-color:#EAF1DD" | NOT b
|style="background-color:#EAF1DD" | NOT a
|style="background-color:#DBE5F1" | b AND a
|style="background-color:#FDE9D9" | b OR a
|style="background-color:#E5E0EC" | b IMPLIES a
|style="background-color:#F2F2F2" | b IS [[Logical equivalence|logically equivalent]] TO a ***
| f IS A tautology
| NEITHER a NOR b
| b stroke a
| exclusive OR
|- style="font-size:9pt" align="center"
|style="font-weight:bold" Height="12" | b
|style="font-weight:bold" | a
|style="background-color:#EAF1DD;font-weight:bold" | &not;(b)
|style="background-color:#EAF1DD;font-weight:bold" | &not;(a)
|style="background-color:#DBE5F1;font-weight:bold" | (b &and; a)
|style="background-color:#FDE9D9;font-weight:bold" | (b &or; a)
|style="background-color:#E5E0EC;font-weight:bold" | (b → a)
|style="background-color:#F2F2F2;font-weight:bold" | (b ↔ a)
| (f = formula)
| (a NOR b)
|style="font-weight:bold" | (b|a)
|style="font-weight:bold" | various
|-  align="center"
| Height="12" | F
| F
|style="background-color:#EAF1DD" | T
|style="background-color:#EAF1DD" | T
|style="background-color:#DBE5F1" | F
|style="background-color:#FDE9D9" | F
|style="background-color:#E5E0EC;font-size:9pt" | T
|style="background-color:#F2F2F2;font-size:9pt" | T
|style="font-size:9pt" | T
|style="font-size:9pt" | T
| T
|style="font-size:9pt" | F
|-  align="center"
| Height="12" | F
|style="font-size:9pt" | T
|style="background-color:#EAF1DD" | T
|style="background-color:#EAF1DD;font-size:9pt" | F
|style="background-color:#DBE5F1" | F
|style="background-color:#FDE9D9;font-size:9pt" | T
|style="background-color:#E5E0EC;font-size:9pt" | T
|style="background-color:#F2F2F2;font-size:9pt" | F
|style="font-size:9pt" | T
|style="font-size:9pt" | F
| T
|style="font-size:9pt" | T
|-  align="center"
|style="font-size:9pt" Height="12" | T
| F
|style="background-color:#EAF1DD" | F
|style="background-color:#EAF1DD" | T
|style="background-color:#DBE5F1" | F
|style="background-color:#FDE9D9;font-size:9pt" | T
|style="background-color:#E5E0EC;font-size:9pt" | F
|style="background-color:#F2F2F2;font-size:9pt" | F
|style="font-size:9pt" | T
|style="font-size:9pt" | F
| T
|style="font-size:9pt" | T
|-  align="center"
|style="font-size:9pt" Height="12" | T
|style="font-size:9pt" | T
|style="background-color:#EAF1DD" | F
|style="background-color:#EAF1DD;font-size:9pt" | F
|style="background-color:#DBE5F1;font-size:9pt" | T
|style="background-color:#FDE9D9;font-size:9pt" | T
|style="background-color:#E5E0EC;font-size:9pt" | T
|style="background-color:#F2F2F2;font-size:9pt" | T
|style="font-size:9pt" | T
|style="font-size:9pt" | F
|style="font-size:9pt" | F
|style="font-size:9pt" | F
|}

=== Engineering connectives ===
[[File:Propositional formula connectives 1.png|313px|thumb|right| Engineering symbols have varied over the years, but these are commonplace. Sometimes they appear simply as boxes with symbols in them. "a" and "b" are called "the inputs" and "c" is called "the output".]]

In general, the engineering connectives are just the same as the mathematics connectives excepting they tend to evaluate with "1" = "T" and "0" = "F". This is done for the purposes of analysis/minimization and synthesis of formulas by use of the notion of ''minterms'' and [[Karnaugh map]]s (see below). Engineers also use the words '''logical product''' from [[Boole]]'s notion (a*a = a) and '''logical sum''' from [[William Stanley Jevons|Jevons]]' notion (a+a = a).<ref>While the notion of logical product is not so peculiar (e.g. 0*0=0, 0*1=0, 1*0=0, 1*1=1), the notion of (1+1=1 ''is'' peculiar; in fact (a "+" b) = (a + (b - a*b)) where "+" is the "logical sum" but + and - are the true arithmetic counterparts. Occasionally all four notions do appear in a formula: A AND B = 1/2*( A plus B minus ( A XOR B ) ] (cf p. 146 in John Wakerly 1978, ''Error Detecting Codes, Self-Checking Circuits and Applications'', North-Holland, New York, {{isbn|0-444-00259-6}} pbk.)</ref>

{|class="wikitable" style="text-align:center"
|- style="font-size:9pt" 
|  Height="12" | 
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! rowspan="2"  | logical product
! rowspan="2"  | logical sum
|  | 
|  | 
!  | half-adder<br/>(no carry)
|- style="font-size:9pt" 
| Height="12" | 
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! exclusive OR
|- style="font-size:9pt" 
! Height="12" | row number
! colspan="2" | variables
|style="background-color:#EAF1DD" | NOT
|style="background-color:#EAF1DD" | NOT
|style="background-color:#DBE5F1" | AND
|style="background-color:#FDE9D9" | OR
| NAND
| NOR
| XOR
|-  
|style="font-size:9pt;white-space:nowrap;" Height="12" | b*2<sup>1</sup>+a*2<sup>0</sup>
|style="font-size:9pt;font-weight:bold" | b
|style="font-size:9pt;font-weight:bold" | a
|style="background-color:#EAF1DD;font-size:9pt;font-weight:bold;white-space:nowrap;" | ~(b)
|style="background-color:#EAF1DD;font-size:9pt;font-weight:bold;white-space:nowrap;" | ~(a)
|style="background-color:#DBE5F1;font-size:9pt;font-weight:bold;white-space:nowrap;" | (b & a)
|style="background-color:#FDE9D9;font-size:9pt;font-weight:bold;white-space:nowrap;" | (b &or; a)
|style="font-size:9pt;font-weight:bold;white-space:nowrap;" | ~(b & a)
|style="font-size:9pt;font-weight:bold;white-space:nowrap;" | ~(b &or; a)
|style="font-size:14pt" | ⊕
|-  
|style="font-size:9pt" Height="12" | 0
| 0
| 0
|style="background-color:#EAF1DD" | 1
|style="background-color:#EAF1DD" | 1
|style="background-color:#DBE5F1" | 0
|style="background-color:#FDE9D9" | 0
|style="font-size:9pt" | 1
|style="font-size:9pt" | 1
|style="font-size:9pt" | 0
|-  
|style="font-size:9pt" Height="12" | 1
| 0
|style="font-size:9pt" | 1
|style="background-color:#EAF1DD" | 1
|style="background-color:#EAF1DD;font-size:9pt" | 0
|style="background-color:#DBE5F1" | 0
|style="background-color:#FDE9D9;font-size:9pt" | 1
|style="font-size:9pt" | 1
|style="font-size:9pt" | 0
|style="font-size:9pt" | 1
|-  
|style="font-size:9pt" Height="12" | 2
|style="font-size:9pt" | 1
| 0
|style="background-color:#EAF1DD" | 0
|style="background-color:#EAF1DD" | 1
|style="background-color:#DBE5F1" | 0
|style="background-color:#FDE9D9;font-size:9pt" | 1
|style="font-size:9pt" | 1
|style="font-size:9pt" | 0
|style="font-size:9pt" | 1
|-  
|style="font-size:9pt" Height="12" | 3
|style="font-size:9pt" | 1
|style="font-size:9pt" | 1
|style="background-color:#EAF1DD" | 0
|style="background-color:#EAF1DD" | 0
|style="background-color:#DBE5F1;font-size:9pt" | 1
|style="background-color:#FDE9D9;font-size:9pt" | 1
|style="font-size:9pt" | 0
|style="font-size:9pt" | 0
|style="font-size:9pt" | 0
|}

===  CASE connective: IF ... THEN ... ELSE ... ===
The IF ... THEN ... ELSE ... connective appears as the simplest form of CASE operator of [[recursion theory]] and [[computation theory]] and is the connective responsible for conditional goto's (jumps, branches). From this one connective all other connectives can be constructed (see more below). Although " IF c THEN b ELSE a " sounds like an implication it is, in its most reduced form, a switch that makes a decision and offers as outcome only one of two alternatives "a" or "b" (hence the name [[switch statement]] in the [[C (programming language)|C]] programming language).<ref>A careful look at its Karnaugh map shows that IF...THEN...ELSE can also be expressed, in a rather round-about way, in terms of two exclusive-ORs: ( (b AND (c XOR a)) OR (a AND (c XOR b)) ) = d.</ref>

The following three propositions are equivalent (as indicated by the logical equivalence sign ≡ ):

# ( IF 'counter is zero' THEN 'go to instruction ''b'' ' ELSE 'go to instruction ''a'' ') ≡
# ( (c → b) & (~c → a) ) ≡  ( ( IF 'counter is zero' THEN 'go to instruction ''b'' ' ) AND ( IF 'It is NOT the case that counter is zero' THEN 'go to instruction ''a'' ) "  ≡
# ( (c & b) &or; (~c & a) ) ≡  " ( 'Counter is zero' AND 'go to instruction ''b'' ) OR ( 'It is NOT the case that 'counter is zero' AND 'go to instruction ''a'' ) "

Thus IF ... THEN ... ELSE—unlike implication—does not evaluate to an ambiguous "TRUTH" when the first proposition is false i.e. c = F in (c → b). For example, most people would reject the following compound proposition as a nonsensical ''non sequitur'' because the second sentence is ''not connected in meaning'' to the first.<ref>Robbin p. 3.</ref>
: Example: The proposition " IF 'Winston Churchill was Chinese' THEN 'The sun rises in the east' " evaluates as a TRUTH given that 'Winston Churchill was Chinese' is a FALSEHOOD and 'The sun rises in the east' evaluates as a TRUTH.

In recognition of this problem, the sign → of formal implication in the propositional calculus is called [[material conditional|material implication]] to distinguish it from the everyday, intuitive implication.{{efn|Rosenbloom discusses this problem of implication at some length. Most philosophers and mathematicians just accept the material definition as given above. But some do not, including the [[Intuitionism|intuitionists]]; they consider it a form of the law of excluded middle misapplied.{{sfn|Rosenbloom|1950|pp=30 and 54ff}}}}

The use of the IF ... THEN ... ELSE construction avoids controversy because it offers a completely deterministic choice between two stated alternatives; it offers two "objects" (the two alternatives b and a), and it ''selects'' between them exhaustively and unambiguously.<ref>Indeed, exhaustive selection between alternatives -- '''mutual exclusion''' -- is required by the definition that Kleene gives the CASE operator (Kleene 1952229)</ref> In the truth table below, d1 is the formula: ( (IF c THEN b) AND (IF NOT-c THEN a) ). Its fully reduced form d2 is the formula: ( (c AND b) OR (NOT-c AND a). The two formulas are equivalent as shown by the columns "=d1" and "=d2". Electrical engineers call the fully reduced formula the AND-OR-SELECT operator. The CASE (or SWITCH) operator is an extension of the same idea to ''n'' possible, but mutually exclusive outcomes. Electrical engineers call the CASE operator a [[multiplexer]].

{|
|- style="font-size:9pt" align="center"
| width="27.75" Height="12" | 
| width="20.25" | 
| width="18.75" | 
| width="18.75" | 
| width="6.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="16.5" | 
| width="12.75" | 
| width="12.75" | 
|style="background-color:#FDE9D9" width="17.25" | d1
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="18" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="24.75" | 
| width="5.25" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
|style="background-color:#FDE9D9" width="15.75" | d2
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="12.75" | 
| width="27" | 
|- style="font-size:9pt;font-weight:bold" align="center"
!style="background-color:#F2F2F2" Height="12" | row
! c
! b
! a
|style="background-color:#A5A5A5" | 
| (
| (
| c
| →
| b
| )
|style="background-color:#FDE9D9" | &
| (
|style="background-color:#EAF1DD" | ~
| (
| c
| )
| →
| a
| )
| )
|style="background-color:#FDE9D9" |  =d1
|style="background-color:#A5A5A5" | 
| (
| (
| c
|style="background-color:#DBEEF3" | &
| b
| )
|style="background-color:#FDE9D9" | &or;
| (
|style="background-color:#EAF1DD" | ~
| (
| c
| )
|style="background-color:#DBE5F1" | &
| a
| )
| )
|style="background-color:#FDE9D9" |  =d2
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2" Height="12" | 0
| 0
| 0
|style="background-color:#C5D9F1" | 0
|style="background-color:#A5A5A5" | 
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| 
| 0
| 1
| 0
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|style="background-color:#B8CCE4" | 0
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| 1
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| 0
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|style="background-color:#B8CCE4" | 0
|style="background-color:#B8CCE4" | 0
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| 
|style="background-color:#B8CCE4" | 0
|style="background-color:#A5A5A5" | 
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| 0
| 0
| 0
| 
|style="background-color:#B8CCE4" | 0
| 
| 1
| 
| 0
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|style="background-color:#B8CCE4" | 0
|style="background-color:#B8CCE4" | 0
| 
| 
|style="background-color:#B8CCE4" | 0
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2" Height="12" | 1
| 0
| 0
|style="background-color:#C5D9F1" | 1
|style="background-color:#A5A5A5" | 
| 
| 
| 0
| 1
| 0
| 
|style="background-color:#B8CCE4" | 1
| 
| 1
| 
| 0
| 
|style="background-color:#B8CCE4" | 1
|style="background-color:#B8CCE4" | 1
| 
| 
|style="background-color:#B8CCE4" | 1
|style="background-color:#A5A5A5" | 
| 
| 
| 0
| 0
| 0
| 
|style="background-color:#B8CCE4" | 1
| 
| 1
| 
| 0
| 
|style="background-color:#B8CCE4" | 1
|style="background-color:#B8CCE4" | 1
| 
| 
|style="background-color:#B8CCE4" | 1
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2" Height="12" | 2
| 0
| 1
|style="background-color:#C5D9F1" | 0
|style="background-color:#A5A5A5" | 
| 
| 
| 0
| 1
| 1
| 
|style="background-color:#B8CCE4" | 0
| 
| 1
| 
| 0
| 
|style="background-color:#B8CCE4" | 0
|style="background-color:#B8CCE4" | 0
| 
| 
|style="background-color:#B8CCE4" | 0
|style="background-color:#A5A5A5" | 
| 
| 
| 0
| 0
| 1
| 
|style="background-color:#B8CCE4" | 0
| 
| 1
| 
| 0
| 
|style="background-color:#B8CCE4" | 0
|style="background-color:#B8CCE4" | 0
| 
| 
|style="background-color:#B8CCE4" | 0
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2" Height="12" | 3
| 0
| 1
|style="background-color:#C5D9F1" | 1
|style="background-color:#A5A5A5" | 
| 
| 
| 0
| 1
| 1
| 
|style="background-color:#B8CCE4" | 1
| 
| 1
| 
| 0
| 
|style="background-color:#B8CCE4" | 1
|style="background-color:#B8CCE4" | 1
| 
| 
|style="background-color:#B8CCE4" | 1
|style="background-color:#A5A5A5" | 
| 
| 
| 0
| 0
| 1
| 
|style="background-color:#B8CCE4" | 1
| 
| 1
| 
| 0
| 
|style="background-color:#B8CCE4" | 1
|style="background-color:#B8CCE4" | 1
| 
| 
|style="background-color:#B8CCE4" | 1
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2" Height="12" | 4
| 1
|style="background-color:#DBEEF3" | 0
| 0
|style="background-color:#A5A5A5" | 
| 
| 
| 1
|style="background-color:#DBEEF3" | 0
|style="background-color:#DBEEF3" | 0
| 
|style="background-color:#DBEEF3" | 0
| 
| 0
| 
| 1
| 
| 1
| 0
| 
| 
|style="background-color:#DBEEF3" | 0
|style="background-color:#A5A5A5" | 
| 
| 
| 1
|style="background-color:#DBEEF3" | 0
|style="background-color:#DBEEF3" | 0
| 
|style="background-color:#DBEEF3" | 0
| 
| 0
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| 1
| 
| 0
| 0
| 
| 
|style="background-color:#DBEEF3" | 0
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2" Height="12" | 5
| 1
|style="background-color:#DBEEF3" | 0
| 1
|style="background-color:#A5A5A5" | 
| 
| 
| 1
|style="background-color:#DBEEF3" | 0
|style="background-color:#DBEEF3" | 0
| 
|style="background-color:#DBEEF3" | 0
| 
| 0
| 
| 1
| 
| 1
| 1
| 
| 
|style="background-color:#DBEEF3" | 0
|style="background-color:#A5A5A5" | 
| 
| 
| 1
|style="background-color:#DBEEF3" | 0
|style="background-color:#DBEEF3" | 0
| 
|style="background-color:#DBEEF3" | 0
| 
| 0
| 
| 1
| 
| 0
| 1
| 
| 
|style="background-color:#DBEEF3" | 0
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2" Height="12" | 6
| 1
|style="background-color:#DBEEF3" | 1
| 0
|style="background-color:#A5A5A5" | 
| 
| 
| 1
|style="background-color:#DBEEF3" | 1
|style="background-color:#DBEEF3" | 1
| 
|style="background-color:#DBEEF3" | 1
| 
| 0
| 
| 1
| 
| 1
| 0
| 
| 
|style="background-color:#DBEEF3" | 1
|style="background-color:#A5A5A5" | 
| 
| 
| 1
|style="background-color:#DBEEF3" | 1
|style="background-color:#DBEEF3" | 1
| 
|style="background-color:#DBEEF3" | 1
| 
| 0
| 
| 1
| 
| 0
| 0
| 
| 
|style="background-color:#DBEEF3" | 1
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2" Height="12" | 7
| 1
|style="background-color:#DBEEF3" | 1
| 1
|style="background-color:#A5A5A5" | 
| 
| 
| 1
|style="background-color:#DBEEF3" | 1
|style="background-color:#DBEEF3" | 1
| 
|style="background-color:#DBEEF3" | 1
| 
| 0
| 
| 1
| 
| 1
| 1
| 
| 
|style="background-color:#DBEEF3" | 1
|style="background-color:#A5A5A5" | 
| 
| 
| 1
|style="background-color:#DBEEF3" | 1
|style="background-color:#DBEEF3" | 1
| 
|style="background-color:#DBEEF3" | 1
| 
| 0
| 
| 1
| 
| 0
| 1
| 
| 
|style="background-color:#DBEEF3" | 1
|}

=== IDENTITY and evaluation ===

The first table of this section stars *** the entry logical equivalence to note the fact that "[[Logical equivalence]]" is not the same thing as "identity". For example, most would agree that the assertion "That cow is blue" is identical to the assertion "That cow is blue". On the other hand, ''logical'' equivalence sometimes appears in speech as in this example: " 'The sun is shining' means 'I'm biking' " Translated into a propositional formula the words become: "IF 'the sun is shining' THEN 'I'm biking', AND IF 'I'm biking' THEN 'the sun is shining'":<ref>The use of quote marks around the expressions is not accidental. Tarski comments on the use of quotes in his "18. Identity of things and identity of their designations; use of quotation marks" p. 58ff.</ref>
: "IF 's' THEN 'b' AND IF 'b' THEN 's' " is written as ((s → b) & (b → s)) or in an abbreviated form as (s ↔ b). As the rightmost symbol string is a '''definition''' for a new symbol in terms of the symbols on the left, the use of the IDENTITY sign = is appropriate:
:: ((s → b) & (b → s)) = (s ↔ b)

Different authors use different signs for logical equivalence: ↔ (e.g. Suppes, Goodstein, Hamilton), ≡ (e.g. Robbin), ⇔ (e.g. Bender and Williamson). Typically identity is written as the equals sign =. One exception to this rule is found in ''Principia Mathematica''. For more about the philosophy of the notion of IDENTITY see [[Identity of indiscernibles|Leibniz's law]].

As noted above, Tarski considers IDENTITY to lie outside the propositional calculus, but he asserts that without the notion, "logic" is insufficient for mathematics and the deductive sciences. In fact the sign comes into the propositional calculus when a formula is to be evaluated.<ref>Hamilton p. 37. Bender and Williamson p. 29 state "In what follows, we'll replace "equals" with the symbol " ⇔ " (equivalence) which is usually used in logic. We use the more familiar " = " for assigning meaning and values."</ref>

In some systems there are no truth tables, but rather just formal axioms (e.g. strings of symbols from a set { ~, →, (, ), variables p<sub>1</sub>, p<sub>2</sub>, p<sub>3</sub>, ... } and formula-formation rules (rules about how to make more symbol strings from previous strings by use of e.g. substitution and [[modus ponens]]). the result of such a calculus will be another formula (i.e. a well-formed symbol string). Eventually, however, if one wants to use the calculus to study notions of validity and truth, one must add axioms that define the behavior of the symbols called "the truth values" {T, F} ( or {1, 0}, etc.) relative to the other symbols.

For example, Hamilton uses two symbols = and ≠ when he defines the notion of a '''valuation v''' of any [[well-formed formula]]s (wffs) ''A'' and ''B'' in his "formal statement calculus" L. A valuation '''v''' is a ''[[Function (mathematics)|function]]'' from the wffs of his system L to the range (output) { T, F }, given that each variable  p<sub>1</sub>, p<sub>2</sub>, p<sub>3</sub> in a wff is assigned an arbitrary truth value { T, F }.
{{NumBlk|*|  '''v'''(''A'') ≠ '''v'''(~''A'')|{{EquationRef|i}}}}
{{NumBlk|*|  '''v'''(''A'' → ''B'') {{=}} F if and only if '''v'''(''A'') {{=}} T and '''v'''(''B'') {{=}} F|{{EquationRef|ii}}}}

The two definitions ({{EquationNote|i}}) and ({{EquationNote|ii}}) define the equivalent of the truth tables for the ~ (NOT) and → (IMPLICATION) connectives of his system. The first one derives F ≠ T and T ≠ F, in other words " '''v'''(''A'') does not '''mean''' '''v'''(~''A'')". Definition ({{EquationNote|ii}}) specifies the third row in the truth table, and the other three rows then come from an application of definition ({{EquationNote|i}}). In particular ({{EquationNote|ii}}) '''assigns''' the value F (or a meaning of "F") to the entire expression. The definitions also serve as formation rules that allow substitution of a value previously derived into a formula:
{|
|- style="font-size:9pt" align="center"
| width="8.25" Height="12" | 
| width="25.5" | 
|style="background-color:#E5E0EC" width="50" | v(A→B)
| width="29.25" | 
| width="6.75" | 
|- style="font-size:9pt" align="center"
| Height="12" | (
| v(A)
|style="background-color:#E5E0EC" |  →
| v(B)
| )
|- style="font-size:9pt" align="center"
| Height="12" | 
| F
|style="background-color:#E5E0EC" | T
| F
| 
|- style="font-size:9pt" align="center"
| Height="12" | 
| F
|style="background-color:#E5E0EC" | T
| T
| 
|- style="font-size:9pt" align="center"
| Height="12" | 
| T
|style="background-color:#CCC0DA" | F
| F
| 
|- style="font-size:9pt" align="center"
| Height="12" | 
| T
|style="background-color:#E5E0EC" | T
| T
| 
|}

Some [[formal system]]s specify these valuation axioms at the outset in the form of certain formulas such as the [[law of contradiction]] or laws of identity and nullity. The choice of which ones to use, together with laws such as commutation and distribution, is up to the system's designer as long as the set of axioms is '''complete''' (i.e. sufficient to form and to evaluate any well-formed formula created in the system).