  The year 1948 witnessed the historic moment of the birth of classic
information theory (CIT). Guided by CIT, modern communication techniques have
approached the theoretic limitations, such as, entropy function $H(U)$, channel
capacity $C=\max_{p(x)}I(X;Y)$ and rate-distortion function
$R(D)=\min_{p(\hat{x}|x):\mathbb{E}d(x,\hat{x})\leq D} I(X;\hat{X})$. Semantic
communication paves a new direction for future communication techniques whereas
the guided theory is missed. In this paper, we try to establish a systematic
framework of semantic information theory (SIT). We investigate the behavior of
semantic communication and find that synonym is the basic feature so we define
the synonymous mapping between semantic information and syntactic information.
Stemming from this core concept, synonymous mapping $f$, we introduce the
measures of semantic information, such as semantic entropy $H_s(\tilde{U})$,
up/down semantic mutual information $I^s(\tilde{X};\tilde{Y})$
$(I_s(\tilde{X};\tilde{Y}))$, semantic capacity
$C_s=\max_{f_{xy}}\max_{p(x)}I^s(\tilde{X};\tilde{Y})$, and semantic
rate-distortion function
$R_s(D)=\min_{\{f_x,f_{\hat{x}}\}}\min_{p(\hat{x}|x):\mathbb{E}d_s(\tilde{x},\hat{\tilde{x}})\leq
D}I_s(\tilde{X};\hat{\tilde{X}})$. Furthermore, we prove three coding theorems
of SIT by using random coding and (jointly) typical decoding/encoding, that is,
the semantic source coding theorem, semantic channel coding theorem, and
semantic rate-distortion coding theorem. We find that the limits of SIT are
extended by using synonymous mapping, that is, $H_s(\tilde{U})\leq H(U)$,
$C_s\geq C$ and $R_s(D)\leq R(D)$. All these works composite the basis of
semantic information theory. In addition, we discuss the semantic information
measures in the continuous case. For the band-limited Gaussian channel, we
obtain a new channel capacity formula,
$C_s=B\log\left[S^4\left(1+\frac{P}{N_0B}\right)\right]$.
